Encyclopedia of Mathematical Physics Vol.3 I-O Ed. Fran Oise Et Al

I Image Processing: Mathematics G Aubert, Universite´ de Nice Sophia Antipolis, Nice, France P Kornprobst, INRIA, Sophia Antipolis, France ª 2006 Elsevier Ltd. All rights reserved.

Our society is often designated as being an ‘‘information society.’’ It could also be defined as an ‘‘image society.’’ This is not only because image is a powerful and widely used medium of communication, but also because it is an easy, compact, and widespread way to represent the physical world. If we think about it, it is indeed striking to realize just how much images are omnipresent in our lives through numerous applications such as medical and satellite imaging, videosurveillance, cinema, robotics, etc. Many approaches have been developed to process these digital images, and it is difficult to say which one is more natural than the other. Image processing has a long history. Maybe the oldest methods come from 1D signal processing techniques. They rely on filter theory (linear or not), on spectral analysis, or on some basic concepts of probability and statistics. For an overview, we refer the interested reader to the book by Gonzalez and Woods (1992). In this article, some recent mathematical concepts will be revisited and illustrated by the image restoration problem, which is presented below. We first discuss stochastic modeling which is widely based on Markov random field theory and deals directly with digital images. This is followed by a discussion of variational approaches where the general idea is to define some cost functions in a continuous setting. Next we show how the scale space theory is connected with partial differential equations (PDEs). Finally, we present the wavelet theory, which is inherited from signal processing and relies on decomposition techniques.

kinds of ‘‘textures,’’ progressive or sharp contours, and fine objects. This gives an idea of the complexity of finding an approach that allows to cope with the different structures at the same time. It also highlights the discrete nature of images which will be handled differently depending on the chosen mathematical tools. For instance, PDEs based approaches are written in a continuous setting, referring to analogous images, and once the existence and the uniqueness of the solution have been proved, we need to discretize them in order to find a numerical solution. On the contrary, stochastic approaches will directly consider discrete images in the modeling of the cost functions. The Image Restoration Problem

It is well known that during formation, transmission, and recording processes images deteriorate. Classically, this degradation is the result of two phenomena. The first one is deterministic and is related to the image acquisition modality, to possible defects of the imaging system (e.g., blur created by an incorrect lens adjustment or by motion). The second phenomenon is random and corresponds to the noise coming from any signal transmission. It can also come from image quantization. It is important to choose a degradation model as close as possible to reality. The random noise is usually modeled by a probabilistic distribution. In many cases, a Gaussian distribution is assumed. However, some applications require more specific ones, like the gamma distribution for radar images (speckle noise) or the Poisson distribution for tomography. Unfortunately, it is usually impossible to identify the kind of noise involved for a given real image. A commonly used model is the following. Let u :   R2 ! R be an original image describing a real scene, and let f be the observed image of the same scene (i.e., a degradation of u). We assume that f ¼ Au þ 

Introduction As in the real world, a digital image is composed of a wide variety of structures. Figure 1 shows different

½1

where  stands for a white additive Gaussian noise and A is a linear operator representing the blur (usually a convolution). Given f, the problem is

2 Image Processing: Mathematics

probability p(u=f ) (MAP: Maximum A Posteriori). Thanks to the Bayes’ rule, we have pðu=f Þ ¼

(c) (b)

(d)

pðf =uÞpðuÞ pðf Þ

½3

Let us explain the meaning of the different terms in [3]:

 The term p(f =u) expresses the probability, the (e)

(a)

Figure 1 Digital image example. 1 the close-ups show examples of low resolution, low contrasts, graduated shadings, sharp transitions, and fine elements. (a) low resolution, (b) low contrasts, (c) graduated shadings, (d) sharp transitions, and (e) fine elements.

then to reconstruct u knowing [1]. This problem is ill-posed, and we are able to carry out only an approximation of u. In this article, we will focus on the simplified model of pure denoising: f ¼uþ

½2

The Probabilistic Approach The Bayesian Framework

In this section, we show how the problem of pure denoising, that is, recovering u from the equation f = u þ  knowing only some statistical information on  can be solved by using a probabilistic approach. In this context, f, u, and  are considered as random variables. The general idea for recovering u is to maximize some prior probability. Most models involve two parts: a prior model of possible restored images u and a data model expressing consistency with the observed data.

 The prior model is given by a probability space (u , p), where u is the set of all values of u. The model is specified by giving the probability p(u) on all these values.  The data model is a larger probability space (u, f , p), where u, f is the set of all possible values of u and all possible values of the observed image f. This model is completed by giving the conditional probability p(f =u) of any image f given u, resulting in the joint probabilities p(f , u) = p(f =u)p(u). Implicitly, we assume that the spaces (u ) and (u, f ) are finite although huge. The next step is to use a Bayesian approach introduced in image processing by Besag (1974) and Geman and Geman (1984). The probabilities p(u) and p(f =u) are supposed to be known and, given an observed image f, we seek the image u which maximizes the conditional a posteriori

likelihood, that an image u is realized in f. It also quantifies the lack of total precision of the model and the presence of noise.  The term p(u) expresses our incomplete a priori information about the ideal image u (it is the probability of the model, i.e., the propensity that u be realized independently of the observation f ).  The term p(f ) which is the probability to observe f is a constant and does not play any role when maximizing the conditional probability p(u=f ) with respect to u. Let us remark that the problem maxu p(u=f ) is equivalent to minu E(u) = log p(f =u)  log p(u). So Bayesian models lead to a minimization process. Then the main question is how to assign these probabilities? The easiest probability to determine is p(f =u). If the images u and f consist in a set of values u = (ui, j ), i, j = 1, N and f = (fi, j ), i, j = 1, N, we suppose the conditional independence of (fi, j =ui, j ) in any pixel: pðf =uÞ ¼

N Y

pðfi;j =ui;j Þ

i¼1

and if the restoration model is of the form f = u þ  where  is a white Gaussian noise with variance  2 , then ðfi;j  ui;j Þ2 1 pðfi;j =ui;j Þ ¼ pffiffiffiffiffiffiffiffiffi exp  22 2 and pðf =uÞ ¼

1 ð2ÞN=2

exp 

N X ðfi;j  ui;j Þ2 i;j

22

Therefore, at this stage, the MAP reduces to minimize EðuÞ ¼ K kf  uk2  log pðuÞ

½4 2

where k.k stands for the Euclidean norm on RN and K is a constant. So, it remains now to assign a probability law p(u). To do that, the most common way is to use the theory of Markov random fields (MRFs).

Image Processing: Mathematics The Theory of Markov Random Fields

In this approach, an image is described as a finite set S of sites corresponding to the pixels. For each site, we associate a descriptor representing the state of the site, for example, its gray level. In order to take into account local interaction between sites, one needs to endow S with a system of neighborhoods V. Definition 1 For each site s, we define its neighborhood V(s) as: VðsÞ ¼ ftg such that s 2 = VðsÞ and t 2 VðsÞ ) s 2 VðtÞ Then we associate to this neighborhood system the notion of clique: a clique is either a singleton or a set of sites which are all neighbors of each other. Depending on the neighborhood system, the family of cliques will be different and involve more and less sites. We will denote by C the set of all the cliques relative to a neighborhood system V (see Figure 2). Before introducing the general framework of MRFs, let us define some notations. For a site s, Xs will stand for a random variable taking its values in some set E (e.g., E = {0, 1, . . . , 255}) and xs will be a realization of Xs and xs = (xt )t6¼s will denote an image configuration where site s has been removed. Finally, we will denote by X the random variable X = (Xs , Xt , . . . ) with values in  = E jSj . Definition 2 We say that X is an MRF if the local conditional probability at a site s is only a function of V(s), that is, s

s

pðXs ¼ xs =X ¼ x Þ ¼ pðXs ¼ xs =xt ; t 2 VðsÞÞ Therefore, the gray level at a site depends only on gray levels of neighboring pixels. Now we give the following fundamental theorem due to Hammersley– Clifford (Besag 1974) which states the equivalence between MRFs and Gibbs fields. Theorem 1 Let us suppose that S is finite, E is a discrete set and for all x 2  = E jSj , p(X = x) > 0, then X is an MRF relatively to a system of neighborhoods V if and only if there exists a family of potential functions (Vc )c 2 C such that P p(x) = (1=Z) exp( c 2 C VP c (x)). The function V(x) = c 2 C Vc (x) is called the energy potential or the Gibbs measure and Z is a P normalizing constant: Z = exp( x 2  V(x)).

C1

C2

C1

C2

Figure 2 Examples of neighborhood system and cliques.

3

If, for example, the collection of neighborhoods is the set P of 4-neighbors, then P the theorem says that V(x) = c = {s} 2 C1 Vc (xs ) þ c = {(s, t)} 2 C2 Vc (xs , xt ). Application to the Denoising Problem

Now, given this theorem we can reformulate, thanks to [4], the restoration problem (with the change of notation u = x and us = xs ): find u minimizing the global energy EðuÞ ¼ K kf  uk2 þ VðuÞ

½5

The next step is now to precise the Gibbs measure. In restoration, the potential V(u) is often dedicated to impose local regularity constraints, for example, by penalizing differences between neighbors. This can be modeled using cliques of order 2 in the following manner: X VðuÞ ¼  ðus  ut Þ ðs;tÞ 2 C2

where  is a given real function. This term penalizes the difference of intensities between neighbors which may come from an edge or some noise. This discrete cost function is very similar to the gradient penalty terms in the continuous framework (see the next section). The resulting final energy is (sometimes E(u) is written E(u=f )) EðuÞ ¼ K

X X ðfs  us Þ2 þ  ðus  ut Þ s2S

ðs;tÞ 2 C2

where the constant  is a weighting parameter which can be estimated. The difficulty in choosing the strength of the penalty term defined by  is to be able to penalize the noise while keeping the most salient features, that is, edges. Historically, the function  was first chosen as (z) = z2 but this choice is not good since the resulting regularization is too strong introducing a blur in the image and loss of the edges. A better choice is (z) = jzj (Rudin et al. 1992) or a regularized version of this function. Of course, other choices are possible depending on the considered application and the desired degree of smoothness. In this section, it has been shown how to model the restoration problem through MRFs and the Bayesian framework. Numerically, two main types of algorithms can be used to minimize the energy: deterministic algorithms and stochastic algorithms. The former are generally used when the global energy is strictly convex (e.g., algorithms based on

4 Image Processing: Mathematics

gradient descent). The latter are rather used when E(u) is not convex. There are stochastic minimization algorithms mainly based on simulated annealing. Their main interest is that they always converge (almost surely) to a minimizer (this is not the case for deterministic algorithms which give only local minimizers) but they are often strongly time consuming. We refer the reader to Li (1995) for more details about MRFs and Bayesian framework and Kirkpatrick et al. (1983) for more information on stochastic algorithms.

The Variational Approach Minimizing a Cost Function over a Functional Space

One important issue in the previous section was the definition of p(u) which gives some a priori on the solution. In the variational approach, this idea is also present but the way to infer it is in fact to define the more suitable functional space that describes images and their geometrical properties. The choice of a functional space sets a norm which in turn will constrain the solution to a certain smoothness. We illustrate this idea in this section on the denoising problem [2] which can be seen as a decomposition one. This means that given the observation f, we look for u and  such that f = u þ , where  incorporates all oscillations, that is, noise, and also texture. Let us define a functional to be minimized which takes into account the data f and possibly some statistical informations about :  min ðjujE Þ such that ðjjG Þ ¼  ðu;Þ ½6 with f ¼ u þ g This formulation means that we look, among all decompositions f = u þ , for the one which minimizes (jujE ) under the constraint (jjG ) = . Banach spaces E and G, and functions  and will be discussed in the next subsection. Since a minimization problem under constraints can be expressed with an additional term weighted by a

Lagrange multiplier, the formulation [6] can be rewritten as:   ½7 min ðjujE Þ þ  ðjjG Þ; f ¼ u þ  ðu;Þ

A similar writing consists in replacing  by f  u so that [7] rewrites   min ðjujE Þ þ  ðjf  ujG Þ ½8 u

which is the classical formulation in image restoration. From a numerical point of view, the minimization is usually carried out by solving the associated Euler equations but this may be a difficult task. The main concern is the search for E and G and their norm (or seminorm). It is guided by the choice that an image u is composed of various geometric structures (homogeneous regions, edges) while  = f  u represents oscillations (noise and textures). Examples of Functional Spaces

In this section, we revisit some possible choices of functional spaces summarized in Table 1. The first case (a) was inspired by the classical Tikhonov regularization. The functional space H 1 ()(  R2 ) is the space of functions in L2 () such that the distributional gradient Du is in L2 (). Unfortunately, functions in H1 () do not admit discontinuities across curves and this is a major problem with respect to image analysis since images are made of smooth patches separated by sharp variations. Considering the problem reported in (a), Rudin et al. (1992) proposed to work on BV(), the space of bounded variations (BV) Ambrosio et al. (2000) defined by   Z BVðÞ ¼ u 2 L1 ðÞ; jDuj < 1 Z Z with udiv’ dx; jDuj ¼ sup 

’¼

 ð’1 ; ’2 ; . . . ; ’N Þ 2 C10 ðÞN ;



j’jL1 ðÞ  1

½9

Table 1 Examples of functional spaces and their norm (see model [8]) Model (a) (b) (c)

E and jujE R 1=2 H 1 (), jujE = R  jruj2 dx BV (), jujE = R jDuj BV (), jujE =  jDuj

(t)

G and jujG

t2 t t

L2 () with its usual norm L2 () with its usual norm fb 2 L2 (); b = div, jjL1 ()2  1,   Nj@ = 0g

(t) t2 t2 t

Image Processing: Mathematics

It is equivalent to define BV() as the space of L1 () functions whose distributional gradient Du is a bounded measure and [9] is its total variation. The space BV() has some interesting properties: 1. lower semicontinuity of the total variation R Du with respect to the L1 () topology, j j  2. if u 2 BV(), we can define, for H1 almost everywhere x 2 Su , the complement of Lebesgue points (i.e., the jump set of u), a normal nu (x) and two approximate ‘‘right’’ and ‘‘left’’ limits uþ (x) and u (x), and 3. Du can be decomposed as a sum of a regular measure, a jump measure, and a Cantor measure: Du ¼ ru dx þ ðuþ  u Þnu H1=Su þ Cu where ru is the approximate gradient and H1 the one-dimensional Hausdorff measure. This ability to describe functions with discontinuities across a hypersurface Su makes BV() very convenient to describe images with edges. In this context, the image restoration problem is well posed and suitable numerical tools can be proposed (Chambolle and Lions 1997). One criticism of the model (b) in Table 1 pointed out by Meyer (2001) is that if f is a characteristic function and if f is sufficiently small with respect to a suitable norm, then the model (Rudin et al. 1992) gives u = 0 and  = f contrary to what one should expect (u = f and  = 0). In fact, the main reason of this phenomenon is that the L2 -norm for the  component is not the right one since very oscillating functions can have large L2 -norm (e.g., fn (x) = cos(nx)). To better describe such oscillating functions, Meyer (2001) introduced the space of functions which can be expressed as a divergence of L1 -fields. This work was developed in RN and this framework was adapted to bounded 2D domains by Aubert and Aujol (2005) (see (c) in Table 1). An example of image decomposition is shown in Figure 3. In this section, we have shown how the choice of the functional spaces is closely related to the definition of a variational formulation. The

5

functionals are written in a continuous setting and they can usually be minimized by solving the discretized Euler equations iteratively, until convergence. These PDEs and the differential operators are constrained by the energy definition but it is also possible to work directly on the equations, forgetting the formal link with the energy. Such an approach has also been much developed in the computer vision community and it is illustrated in the next section. We refer the reader to Aubert and Kornprobst (2002) for a general review of variational approaches and PDEs as applied to image analysis.

Scale Spaces and PDEs Another approach to perform nonlinear filtering is to define a family of image smoothing operators Tt , depending on a scale parameter t. Given an image f (x), we can define the image u(t, x) = (Tt f )(x) which corresponds to the image f analyzed at scale t. In this section, following Alvarez–Guichard–Lions– Morel (Alvarez et al. 1993), we show that u(t, x) is the solution of a PDE provided some suitable assumptions on Tt . Basic Principles of a Scale Space

This section describes some natural assumptions to be fulfilled by scale spaces. We first assume that the output at scale t can be computed from the output at a scale t  h for very small h. This is natural, since a coarser scale view of the original picture is likely to be deduced from a finer one. Tt is obtained by composition of transition filters, denoted by Ttþh, t . So the first axiom is (A1) Ttþh = Ttþh, t Tt

T0 = Id

Another assumption is that operators act locally, that is, (Ttþh, t f )(x) depends essentially upon the values of f (y) with y in a small neighborhood of x. Taking into account the fact that as the scale increases, no new feature should be created by the scale space, we have the local comparison principle: if an image u is locally brighter than another image v, then this order must be conserved by the analysis. This is expressed by: (A2) For all u and v such that u(y) > v(y) in a neighborhood of x and y 6¼ x, then for h small enough, we have

Original

u

η

Figure 3 Example of image decomposition (see Aubert and Aujol (2005)).

ðTtþh;t uÞðxÞ  ðTtþh;t vÞðxÞ The third assumption states that a very smooth image must evolve in a smooth way with the scale

6 Image Processing: Mathematics

space. Denoting the scalar product of two vectors of RN by < x, y > , this assumption can be written as

([0, T] ). Then u is a viscosity solution of [11] in [0, T]  if and only if

(A3) Let u(y) = 1=2hA(y  x), y  xi þ hp, y  xi þ c be a quadratic form of R 2 , x fixed (A = r2 u(x) 2 S(2) the set of 2 2 symmetric matrices, p = ru(x) a vector of R2 , c = u(x) a constant.). We shall say that a scale space is regular if there exists a function F(t, x, c, p, A), continuous with respect to A, such that

(i) u is a subsolution, that is, 8 2 C2 ([0, T] ), 8(t0 , x0 ) a local strict maximum point of (u  ) (t, x), we have

ðTtþh;t u  uÞðxÞ ! Fðt; x; c; p; AÞ when h ! 0 h

Scale Spaces are Governed by PDEs

In the following theorem, it is stated that the former assumptions are sufficient to prove that scale spaces are in fact governed by PDEs. Theorem 2 Under assumptions A1, A2, A3, there exists a continuous function F : [0, T]  R R2 S(2) !R satisfying F(t, x, c, p, A)  F(t, x, c, p, B) for all p2 R2 , A and B in S(2) with A  B such that t ðuÞ Ttþh;t u  u ! Fðt; x; u; ru; r2 uÞ; ¼ h

@ ðt0 ; x0 Þ þ Hðt0 ; x0 ; uðt0 ; x0 Þ; rðt0 ; x0 Þ; @t r2 ðt0 ; x0 ÞÞ  0 (ii) u is a supersolution, that is, 8 2 C2 ([0, T] ), 8(t0 , x0 ) a local strict minimum point of (u  ) (t, x), we have @ ðt0 ; x0 Þ þ Hðt0 ; x0 ; uðt0 ; x0 Þ; rðt0 ; x0 Þ; @t r2 ðt0 ; x0 ÞÞ  0 In this definition, it is noticeable that derivatives of u are replaced by the derivatives of the test functions . Obviously, it can be verified that this notion of weak solutions coincides with classical solution when u has enough regularity.

Diffusion Operators Coming from the Scale Space

h!0

þ

½10

uniformly for x 2 R2 , uniformly for u. In eqn [10], the left-hand side term can be interpreted as the partial temporal derivative with respect to t so that the notion of PDEs arises. More precisely, if f is continuous and uniformly bounded, then it can be established that u(t, x) = (Tt f )(x) is the viscosity solution(see Definition 3) of @u þ Hðt; x; u; ru; r2 uÞ ¼ 0 ðhere H ¼ FÞ @t ½11 uð0; xÞ ¼ f ðxÞ The map H : [0, T]  R R2 S(2) ! R is called a Hamiltonian and the decreasing property of H with respect to S is called degenerate ellipticity. The theory of viscosity solutions was introduced in the 1980s by Crandall and P L Lions (Crandall and Lions 1981, Crandall et al. 1992). When strong solutions of [11] do not exist, this theory allows to define solutions which are only continuous or even discontinuous. The definition of viscosity solutions is Definition 3 Let H :  R R2 S(2) ! R be continuous and degenerate elliptic and let u 2 C0

A step further is to assume additional properties on the scale spaces and estimate the corresponding operator. Invariance properties include geometric invariance axioms, contrast invariance, or scale invariance. For example, if we assume the axioms A1–A3, gray-level shift invariance: (I1) Tt (0) = 0, Tt (u þ c) = Tt (u) þ c for all u and all constant c. and translation invariance: (I2) Tt ( h .u) = h .(Tt u) for all h in R2 , t  0, where ( h .u)(x) = u(x þ h). Then it can be established that F in [10] is independent of (x, u), that is, u(t, x) = (Tt f )(x) is the unique viscosity solution of @u ¼ Fðru; r2 uÞ @t uð0; xÞ ¼ f ðxÞ With more precise assumptions, one can even recover explicitly the operator F. As an example, if we look for a linear scale space which verifies some isometry assumption: (I3) Tt (R.u)(x) = R.(Tt u)(x) for all orthogonal transformation R on R2 , where (R.u)(x) = u(Rx).

Image Processing: Mathematics

Then it can be proved that the scale space is the unique solution of the heat equation: @u  u ¼ 0 @t uð0; xÞ ¼ f ðxÞ

½12

Figure 4 is an example of [12] applied to a noisy image at different scale, that is, at different time. Note that noise is quickly removed but one has to stop the evolution very early if we would like to preserve some edges. In the nonlinear cases, several operators have also been found based on curvature. For instance, under suitable axioms (Alvarez et al. 1993), including contrast, scale, and affine invariance, the associated scale space is @u  signðÞðtÞ1=3 jruj ¼ 0 @t

ru where  ¼ div jruj

½13

uð0; xÞ ¼ f ðxÞ This equation is called affine morphological scale space (AMSS) and three restored images are shown in Figure 5. Some qualitative differences are shown in Figure 6.

7

Remark Scale space theory has shown the formal link between some operators and PDEs. It has to be noticed that one may propose some PDEs which do not directly come from the scale space framework. Starting from [12] which performs isotropic smoothing and smears edges, many nonlinear diffusion models have been proposed to smooth images while preserving edges (see e.g., Perona and Malik & (1990)). To know more on scale space and PDEs, we refer the reader to Weickert (1998) and Aubert and Kornprobst (2002).

The Wavelet Approach Before the 1980s, the Fourier transform played a major role for analyzing oscillating signals. The interest of such a transform for real application increased after the discovery of the fast Fourier transform. However, the Fourier transform has some limit. The Fourier transform extracts from the signal details of the frequency content but loses all information on the location of particular frequency. Moreover, for computing the Fourier transform F f (), we need to know f(t) for all the real values of t. These difficulties can be overcome by first windowing the signal, and then by taking its Fourier transform: Z F win f ð; tÞ ¼ f ðsÞgðs  tÞeis ds R

Original image

40 iterations

90 iterations

150 iterations

Figure 4 Illustration of heat equation [12].

Original image 40 iterations

90 iterations

150 iterations

Figure 5 Illustration of the AMSS model [13].

where g is a window function. The parameter  plays the role of a frequency localized around the abscissa t of the temporal signal and F win f (, t) give an information about what is happening around s = t, for the frequency . The main drawback of this method is that the window has a fixed length which is a serious disadvantage when we want to treat signals having variations of different orders of magnitude. All these issues highlighted that a mathematical theory of time–frequency representation was necessary. This was achieved with the wavelet representation. In this section, we first recall some elements of this theory (for 1D signal) and then we show how it can be applied for restoring noisy images. The Wavelet Decomposition

Heat

AMSS

Heat

AMSS

Figure 6 Some close-ups of Figures 4 and 5 showing qualitative differences after 40 iterations.

The basic idea is to construct from a function , called mother wavelet, an orthonormal basis { j, k } of L2 (R) deduced from by translation and dilatation. It is required that be regular, oscillating (but not too much), that and F are well localized and that has some null moments. Once this function is

8 Image Processing: Mathematics

chosen, we set j, k (x) = 2j=2 (2j t  k), j, k 2 Z. An elegant and practical way for obtaining such a basis is to construct a multiresolution analysis of L2 (R) (Mallat 1989). Definition 4 A multiresolution analysis of L2 (R) is a sequence Vj , j 2 Z of subspaces of L2 (R), with the following properties: T (i) j Vj = {0}, (ii) S Vj  Vjþ1 , (iii) j Vj = L2 (R), (iv) f (t) 2 Vj if and only if f (2t) 2 Vjþ1 , and (v) There exists a regular function  with compact support such that the family (t  k), k 2 Z, is an orthonormal basis of V0 for the scalar product of L2 (R). Such a function  is called a scaling function. Then it is straightforward to check that the family j, k (t) defined by j, k (t) = 2j=2 (2j t  k) is an orthonormal basis of Vj . A basic example of multiresolution analysis of L2 (R) is to choose V0 as the set of piecewise constant functions on R and take  as the characteristic function of the interval [0, 1): (t) = [0, 1) (t). Let us now look at the link between wavelet basis and multiresolution analysis. We just give main ideas, all details can be found in the work of Mallat (1989). Assume that we have a multiresolution analysis, and let us define W0 as the orthogonal complement of V0 in V1 . We build the mother wavelet by imposing that the family (t  k), k 2 Z, is an orthonormal basis of W0 . For example, if (t) = [0, 1) (t), it can be shown that (t) = [0, 1=2) (t)  [1=2, 1) (t) (called the Haar wavelet). By change of scale, one gets that the family j=2 (2j t  k), k 2 Z, is an orthonormal j, k (t) = 2 basis of Wj , the orthogonal complement of Vj in Vjþ1 , that is, Vj Wj ¼ Vjþ1

½14

Since the Vj ’s are a multiresolution analysis, we have j = þ1 2 VJ = J1 j = 1 Wj and L = j = 1 Wj . It is then clear that j, k (t) is an orthonormal basis of L2 (R), that is, for each function f 2 L2 (R), we get the following decomposition: f ðtÞ ¼

þ1 X X 1

fj;k

j;k ðtÞ

with fj;k ¼hf ;

j;k iL2

k

Let us see now how in practice a multiresolution analysis can be interpreted. Let f be a function in L2 (R). We denote A2j f (resp. D2j f ) the operator which approximates f (resp. gives the details of f ) at

resolution 2j . More precisely, A2j f (resp. D2j f ) is the projection of f on Vj (resp. on Wj ): A2j f ðtÞ ¼

k¼þ1 X

hf ; j;k ij;k ðtÞ

k¼1

A2j f is characterized by the sequence of scalar products Ad2j f = {hf , j, k i}k 2 Z. We call Ad2j f the discrete approximation of f at resolution 2j . In the same way, we have D2j f ðtÞ ¼

k¼þ1 X

hf ;

j;k i j;k ðtÞ

k¼1

D2j f is characterized by the sequence of scalar products Dd2j f = {hf , j, k i}k 2 Z. We call Dd2j f the details of f at resolution 2j . According to [14], approximation and detail are linked by the relation A2jþ1 f ¼ A2j f þ D2j f This means that D2j f represents the details to be added to obtain from one level of approximation to the next level of approximation. Finally, the decomposition of a signal f on a wavelet basis is obtained as an accumulation of details at scale 2j from 0 to þ1: f ¼

j¼þ1 X

D 2j f ¼

j¼1

j¼þ1 X X k¼þ1

hf ;

j;k i j;k

½15

j¼1 k¼1

Instead of considering the sum over all dyadic levels j, one can sum over j  J for a fixed J; in this case, we have f ¼

k¼þ1 X X

hf ;

j;k i

k¼1 jJ

j;k þ

k¼þ1 X

hf ; J;k iJ;k

k¼1

We conclude this section by showing how we can construct a 2D wavelet basis from the 1D case. We can simply use a tensor product. Scaling function and mother wavelet are given, respectively, as follows: ðx; yÞ ¼ ðxÞðyÞ;

¼ð

1

;

2

;

3

Þ

with 1

ðx; yÞ ¼ ðxÞ ðyÞ

2

ðx; yÞ ¼ ðyÞ ðxÞ

3

ðx; yÞ ¼ ðxÞ ðyÞ

As for the 1D case, A2j f denotes the projection of f on Vj , D12j the horizontal details, D22j the vertical

Image Processing: Mathematics

A2–1f

D12–1f

D 22–1f

D 32–1f

Original noisy BV regularization image

9

Wavelet shrinkage

Figure 8 Illustration of two regularization methods. Figure 7 Illustration on the wavelets methodology.

details, and D32j the other details (the indice l in Dl2j is the same as in l ). For a 2D image f, we then have the following decomposition (see Figure 7): X X X k¼þ1 hf ;

f¼

l

þ

j;k i j;k

2  k¼1 jJ k¼þ1 X

&

Let us denote, respectively, by {uj, k, } and {fj, k, } the wavelet coefficients of u and f, then solving [16] amounts to finding the minimizer of the functional X X FðuÞ ¼  juj;k; j þ juj;k;  fj;k; j2 ½17 j;k;

hf ; J;k iJ;k

k¼1

Application to the Denoising Problem

We go back to the denoising problem. Our goal is to solve this problem by using a variational approach and wavelets. We recall that we have an ideal image u that has been corrupted by a white Gaussian noise  resulting in an observation f with f = u þ . As it has been seen in the section ‘‘The variational approach,’’ this question can be tackled by solving the variational problem   min ðjujE Þ þ jf  ujG u

orthonormal basis of L2 (). To avoid further technical complications, we ignore this question.

½16

for suitable choices of E, G, and . Here we propose to choose G = L2 () ( is the domain image) and for E the Besov space B11 (L1 ()) and  = Identity. Besov spaces B q (Lp ()) are used in many domains of mathematics as harmonic analysis or approximation theory. There exist different ways for defining them. Roughly speaking, they consist of functions having derivatives in LP (); the third parameter q allows one to make finer distinctions in smoothness. Here we are only concerned with the Besov space B11 (L1 ()). One important property needed here is that the norm of a function in E = B11 (L1 ()) is equivalent to the l1 -norm of the wavelet coefficients, that is if { j, k } is an orthonormal basis of L2 () and ifP uj, P k, are the wavelet coefficients of u 2 E, then jujE = j k, juj, k, j. Remark When one is concerned with a finite domain, then some changes must be made with respect to the construction given in [15] to obtain an

j;k;

One notes immediately that minimizing problem [17] reduces to finding the minimizer s, given t, of E(s) = js  tj2 þ jsj and that the minimizer of E(s) is given by s = t  (=2) if t > =2, s = 0 if jtj  =2 and s = t þ (=2) if t < (=2). Thus, we shrink the wavelet coefficients fj, k, toward zero by an amount of =2 to obtain the minimizer. This is exactly the wavelet shrinkage algorithm of Donoho and Johnstone (1994). It is remarkable that the wavelet shrinkage algorithm, which has been found by using statistical tools, can also be explained via a variational approach (Chambolle et al. 1998). Figure 8 shows an example of the result on a noisy image. For more details, we refer the reader to Mallat (1998).

Conclusion Image processing is a challenging domain of applied mathematics which has to deal with discrete and continuous representations. In this article, we have covered the core mathematical tools used in the area. The example of gray-scale image restoration allowed us to illustrate and compare the different methodologies. Naturally, as mentioned in the introduction, image processing refers to a wide variety of applications and an intensive research has been carried out on the different topics using the methodologies described here. The reader will find in the references (therein) several illustrations of challenging problems. See also: -Convergence and Homogenization; Convex Analysis and Duality Methods; Elliptic Differential

10

Incompressible Euler Equations: Mathematical Theory

Equations: Linear Theory; Evolution Equations: Linear and Nonlinear; Fluid Mechanics: Numerical Methods; Fractal Dimensions in Dynamics; Free Interfaces and Free Discontinuities: Variational Problems; Geometric Measure Theory; Ginzburg–Landau Equation; Inequalities in Sobolev Spaces; Minimax Principle in the Calculus of Variations; Optimal Transportation; Partial Differential Equations: Some Examples; Stochastic Differential Equations; Variational Techniques for Ginzburg–Landau Energies; Wavelets: Applications; Wavelets: Mathematical Theory.

Further Reading Alvarez L, Guichard F, Lions P, and Morel J (1993) Axioms and fundamental equations of image processing. Archive for Rational Mechanics and Analysis 123(3): 199–257. Ambrosio L, Fusco N, and Pallara D (2000) Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. New York: Clarendon Press. Aubert G and Aujol J (2005) Modeling very oscillating signals – application to image processing. Applied Mathematics and Optimization 51(2): 163–182. Aubert G and Kornprobst P (2002) Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Applied Mathematical Sciences, vol. 147. New York: Springer. Besag J (1974) Spatial interaction and the statistical analysis of lattice systems (with discussion). Journal of Royal Statistical Society 2: 192–236. Chambolle A, DeVore R, Lee N, and Lucier B (1998) Non-linear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Transactions on Image Processing 7(3): 319–334.

Chambolle A and Lions P (1997) Image recovery via total variation minimization and related problems. Nu¨merische Mathematik 76(2): 167–188. Crandall M, Ishii H, and Lions P-L (1992) User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American Society 27: 1–67. Crandall M and Lions P (1981) Condition d’unicite´ pour les solutions ge´ne´ralise´es des e´quations de Hamilton–Jacobi du premier ordre. Comptes Rendus de l’Acade´mie des Sciences 292: 183–186. Donoho D and Johnstone I (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrica 81: 425–455. Geman S and Geman D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6(6): 721–741. Gonzalez RC and Woods RE (1992) Digital Image Processing, 3rd edn. Addison-Wesley. Kirkpatrick S, Gellat C, and Vecchi M (1983) Optimization by simulated annealing. Science 220: 671–680. Li S (1995) Markov Random Field Modeling in Computer Vision. Tokyo: Springer. Mallat S (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(7): 674–693. Mallat S (1998) A Wavelet Tour of Signal Processing. Cambridge: Academic Press. Meyer Y (2001) Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22. Providence, RI: American Mathematical Society. Perona P and Malik J (1990) Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12(7): 629–639. Rudin L, Osher S, and Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D 60: 259–268. Weickert J (1998) Anisotropic Diffusion in Image Processing. Stuttgart: Teubner-Verlag.

Incompressible Euler Equations: Mathematical Theory D Chae, Sungkyunkwan University, Suwon, South Korea ª 2006 Elsevier Ltd. All rights reserved.

Introduction In this article we present comprehensive mathematical results on the incompressible Euler equations. Our presentation is focussed on the two aspects of the equations. The first one is on the theories of classical solutions and the problem of global in time continuation/finite time blow-up of the local classical solutions. The second topic is concerned on the weak solutions, mainly for the two-dimensional (2D) Euler equations for existence and uniqueness questions.

The motion of homogeneous incompressible ideal fluid in a domain   R n is described by the following system of Euler equations: @v þ ðv  rÞv ¼ rp @t

½1

div v ¼ 0

½2

vðx; 0Þ ¼ v0 ðxÞ

½3

where v = (v1 , v2 , . . . , vn ), vj = vj (x, t), j = 1, 2, . . . , n, is the velocity of the fluid flows, p = p(x, t) is the scalar pressure, and v0 (x) is a given initial velocity field satisfying div v0 = 0. Here we use the standard notion of vector calculus, denoting

Incompressible Euler Equations: Mathematical Theory 11



@p @p @p rp ¼ ; ;...; @x1 @x2 @xn n X @vj ðv  rÞvj ¼ vk @xk k¼1 div v ¼



div v ¼ 0;

!ðx; 0Þ ¼ !0 ðxÞ

@xk

Equation [1] represents the balance of momentum for each portion of fluid, while eqn [2] represents the conservation of mass of fluid during its motion, combined with the homogeneity (constant density) assumption on the fluid. Equations [1] and [2] are first obtained by Euler in 1755. Although we could consider, more generally, the inhomogeneous incompressible Euler equations, in mathematical fluid mechanics considerations the incompressible Euler equations usually mean the above system [1]–[2]. For a bounded domain with fixed boundary @, the natural boundary condition is vðx; tÞ  ðxÞ ¼ 0

8ðx; tÞ 2 @  ½0; 1Þ

½4

where (x) is the unit normal vector at the boundary point x 2 @. Several studies are concerned with the Cauchy problem of the system [1]–[3], where we consider the case ( R n ðwhole domain of Rn Þ; or ¼ ½5 R n =Zn ðperiodic domainÞ In this article for simplicity we suppose  = Rn , n = 2, 3 unless otherwise stated. We note that the Euler equation is obtained formally by setting the viscosity = 0, or, equivalently, Reynolds number = 1 in the Navier–Stokes equations. Thus, we may view the Euler equations as the one describing approximately the extremely high Reynolds number turbulent flows. For detailed mathematical studies on the finite Reynolds number Navier–Stokes equations, see Temam (1984) and Lions (1996). For much shorter and more comprehensive review see Constantin (1995). In the study of the Euler equations the notion of vorticity, ! = curl v, plays a very important role. In particular, we can reformulate the system in terms of vorticity fields only as follows. We first suppose we are working in three-dimensional (3D) space, and rewrite [1] as   @v 1  v  curl v ¼ r p þ jvj2 ½6 @t 2 Taking curl of [6], and using elementary vector identities we obtain the following vorticity formulation: @! þ ðv  rÞ! ¼ !  rv @t

½8 ½9

The linear elliptic system [8] for v can be solved explicitly in terms of ! to give the Biot–Savart law Z 1 ðx  yÞ  !ðy; tÞ vðx; tÞ ¼ dy ½10 4 R3 jx  yj3

n X @vk k¼1

curl v ¼ !

½7

Substituting this v into [7] formally, we obtain a integrodifferential system for !. The term in the right-hand side of [7] is called the ‘‘vortex stretching term,’’ and is regarded as the main source of difficulties in the mathematical theory of the 3D Euler equations. In the 2D case we take the vorticity as the scalar, ! = @v2 =@x1  @v1 =@x2 , and the evolution equation of ! becomes @! þ ðv  rÞ! ¼ 0 @t

½11

combined with the 2D Biot–Savart law, Z 1 ðy2 þ x2 ; y1  x1 Þ vðx; tÞ ¼ !ðy; tÞ dy ½12 2 R2 jx  yj2 In many studies of the Euler equations it is convenient to introduce the notion of ‘‘particle trajectory mapping,’’ (, t) defined by @ð; tÞ ¼ vðð; tÞ; tÞ @t ð; 0Þ ¼ ; 2

½13

The mapping (, t) transforms from the location of the initial fluid particles to the location at time t, and the parameter  is called the Lagrangian particle marker. If we denote the Jacobian of the transformation, det (r (, t)) = J(, t), then we can show easily that @J ¼ ðdiv vÞJ @t which implies the fact that the velocity field v satisfies the incompressibility, div v = 0 if and only if the mapping (, t) is volume preserving. At this moment, we note that, although the Euler equations are originally derived by applying the mass conservation and the momentum balance principles, we could also derive them by applying the principle of least action to the action defined by  Z Z  1 t2 @ðx; tÞ2 AðÞ ¼ dx dt 2 t1   @t  Here, (, t) :  !  is a parametrized family of volume-preserving diffeomorphism. This variational approach to the Euler equations implies that we can

12

Incompressible Euler Equations: Mathematical Theory

view solutions of the Euler equations as a geodesic curve in the L2 -metric on the infinite-dimensional manifold of volume-preserving diffeomorphisms (see for more details, e.g., Arnol’d and Khesin (1998)). The 3D Euler equations have many conserved quantities. We list some important ones below. 1. Energy 1 2

EðtÞ ¼

Z

jvðx; tÞj2 dx

½14



and 

1 3

Z

jxj2 ! dx

½20b



respectively. In the 2D ideal incompressible fluids we have extra conserved quantities; namely for any p 2 [1, 1] the integral Z j!ðx; tÞjp dx ½21 

2. Helicity HðtÞ ¼

Z

vðx; tÞ  !ðx; tÞ dx

½15



3. Circulation CðtÞ ¼

I

v  dl

½16

CðtÞ

where C(t) = {(, t)j 2 C} is the curve moving along with the fluid. 4. Impulse Z 1 IðtÞ ¼ x  ! dx ½17 2  5. Moment of impulse Z 1 MðtÞ ¼ x  ðx  !Þ dx 3 

is conserved (as a matter of fact we can R extend this statement by replacing the integral by  f (!(x, t))dx for any continuous function f ). There are many known explicit solutions to the Euler equations (See e.g., Lamb (1932) and Majda and Bertozzi (2002)).

½18

The proof of conservations of the above quantities can be carried out without difficulty by using elementary vector calculus (for details see, e.g., Chorin and Marsden (1993), Majda and Bertozzi (2002), Marchioro and Pulvirenti (1994)). The helicity above, in particular, represents the degree of knotedness of the vortex lines in the fluid, where the vortex lines are the integral curves of the vorticity fields. Arnol’d and Khesin (1998) discuss in detail aspects of helicity and other geometric aspects of the Euler equations. For the 2D Euler equations there is no analog of helicity, while the circulation conservation is replaced by the vorticity flux integral, Z !ðx; tÞ dx ½19 AðtÞ

where A(t) = {(, t)j 2 A} is a planar region moving along the fluid. The impulse and the moment of impulse integrals are replace by Z 1 ðx2 ; x1 Þ! dx ½20a 2 

Local Existence and the Blow-Up Problem The Classical Results

We first introduce some notations of function spaces. The Lebesgue space Lp (), p 2 [1, 1], is the Banach space defined by the norm ( R 1=p p ; p 2 ½1; 1Þ  jf ðxÞj dx kf kLp :¼ ess: supx2 jf ðxÞj; p ¼ 1 Let us set  := (1 , 2 , . . . , n ) 2 (Zþ [ {0})n with jj = 1 þ 2 þ    þ n . Then, D := D1 1 D2 2    Dn n , where Dj = @=@xj , j = 1, 2, . . . , n. For given k 2 Z and p 2 [1, 1) the Sobolev space, W k, p () is the Banach space of functions consisting of functions f 2 Lp () such that Z 1=p p  kf kW k;p :¼ jD f ðxÞj dx 5=2. Now suppose 0 k!(t)kL1 dt < 1. Taking L2 inner product of [7] with !, then after integration by part we obtain

1d k!k2L2 ¼ ðð!  rÞv; !ÞL2 2 dt  k!kL1 krvkL2 k!kL2

where we used the identity krvkL2 = k!kL2 . Applying the Gronwall lemma, we obtain Z T  k!ðtÞkL2  k!0 kL2 exp k!ðsÞkL1 ds 0

 Cð!0 ; T Þ ½22

Further estimate, using the Sobolev inequality, krvkL1  CkvkHm for m > 5=2, gives d kvk2Hm  Ckvk3Hm dt Thanks to Gronwall’s lemma, we have the local-intime uniform estimate kvðtÞkHm

T

¼ k!kL1 k!k2L2

and obtain X I kD ðv  rÞv  ðv  rÞD vkL2 kvkHm

d kvk2Hm  CkrvkL1 kvk2Hm dt

if and only if

t%T

kv0 kHm   2kv0 kHm 1  Ctkv0 kHm

for all t 2 [0, 1=(2Ckv0 kHm )]. This is the key a priori estimate for the construction of the local solutions. The local-in-time solution of the Euler equations in the Sobolev space H m (Rn ) for m > n=2 þ 1, m 2 Z,

½25

for all t 2 [0, T ]. Substituting [24] into [22], and combining this with [25], we have d kvk2Hm dt  C½1 þ k!kL1 ½1 þ logð1 þ kvkHm Þkvk2Hm Applying the Gronwall’s lemma, we obtain kvðtÞkHm  kv0 kHm   Z  exp C1 exp C2 0



T

k!ðÞkL1 d

 Cðv0 ; T Þ for all t 2 [0, T ] and for some constants C1 , C2 . Thus, we proved the ‘‘necessity part’’ of [23], The

14

Incompressible Euler Equations: Mathematical Theory

‘‘sufficiency part’’ is an easy consequence of the Sobolev inequality, Z T k!ðsÞkL1 ds  T sup krvðtÞkL1 0

0tT

 CT sup kvðtÞkHm 0tT

for m > 5=2. Other Related Results

The previous local existence result in H m (Rn ), m > n=2 þ 1, is basically due to T Kato in 1972. He and G Ponce extended this existence result using the fractional Sobolev space, Lsp (Rn ), s > n=2 þ 1, s 2 R in 1986. These results were further extended, using the Besov and the Triebel–Lizorkin spaces, by the present author in 2001. For bounded domain   Rn , R Temam obtained the local-existence result using the space W k, p () in 1975. On the other hand, in the setting of the Ho¨lder space, C1,  (Rn ) L Lichtenstein (1925) and W Wolibner (1933) obtained local existence of solutions of the Euler equations. More recently, J-Y Chemin considered the Zygmund Cs (Rn ), which is identical to the Ho¨lder space C[s], s[s] (Rn ) for noninteger s, where [s] means the largest integer not greater than s, but is different from C[s], 0 (R n ) for integer s. He proved, in 1992, local existence of solutions to the 3D Euler equations in this space in 1992. See Chemin (1998) for details of this proof. The Beale–Kato–Majda criterion for the finitetime blow-up of the classical solutions of the 3D Euler equations has been refined recently by many authors; replacing the L1 -norm of vorticity !(x, t) by the weaker BMO (the space of functions with bounded mean oscillations) norm (H Kozono and Y Taniuchi, 2000), and by the even weaker Besov space or Triebel–Lizorkin space norms by the present author in 2001 (see Triebel (1983) for more details on those spaces). Here we just note that these spaces are refinements of the usual Sobolev spaces. For a bounded domain case, there is a result by A Ferrari in 1993. The blow-up problem is still open even in the case of axisymmetric 3D Euler equations if there is a nonzero swirl (angular velocity). In this case, the blow-up is controlled only by the angular component of the vorticity as shown by the present author (1996). In the region off the axis, in particular, the axisymmetric 3D Euler equation has the same form as the 2D Boussinesq equations. Some researchers also tried to approach to regularity/singularity problem of the 3D Euler equations by investigating the geometric structure

of the vortex stretching term, and obtained a geometric type of blow-up criterion (P Constantin, C Fefferman, and A Majda, 1996). For more detailed review of studies in this direction see Constantin (1995). Since the blow-up problem of the 3D Euler equation itself looks too difficult to solve, it has also been studied on the simplified model problems. In 1985, P Constantin, P D Lax, and A Majda considered the following 1D model problem of the 3D Euler equations: t þ ðHðÞÞx ¼ 0;

ðx; 0Þ ¼ 0 ðxÞ

where H() is the Hilbert transform defined by Hð!Þ ¼

1 PV 

Z

1

!ðyÞ dy 1 x  y

They proved finite-time blow-up of this model problem by explicitly obtaining the solution. There is another, 2D model problem of the 3D Euler equations, the quasigeostrophic equations, t þ ðu  rÞ ¼ 0 u ¼r

?

;  ¼ ðÞ1=2 ðx; 0Þ ¼ 0 ðxÞ

½26

where r? = (@2 , @1 ). Contrary to the above 1D model equation, this 2D model has real physical relevance in the atmospheric science, and (x, t) represents the temperature of the air. The resemblance of this equation to the 3D Euler equation was first observed by P Constantin, A Majda, and E Tabak in 1994, and they derived the finite blowup criterion of the equations. In spite of many interesting partial results, including the work by D Cordoba (1998), the blow-up problem of [26] is still open.

The 2D Euler Equations and the Weak Solutions The Case of W 1, p Weak Solutions

In 2D Euler equations, the problem of global wellposedness of the classical solutions is settled down. This is an immediate consequence of the conservation of k!(t)kL1 as stated in [21] combined with the Beale–Kato–Majda criterion [23]. On the other hand, the notion of weak solutions is not well understood. A weak solution of the Euler equations is a singular (nondifferentiable) solution of the equations. More precisely, by a weak solution of

Incompressible Euler Equations: Mathematical Theory 15

[1]–[2] in   (0, T) we mean a vector field v 2 C([0, T); L2loc ()) satisfying the integral identity: Z TZ @ðx; tÞ dx dt vðx; tÞ   3 @t 0 R Z  vðx; 0Þ  ðx; 0Þ dx 

R3 Z T

R3

Z

T 0

krvkLp  Cp k!kLp

vðx; tÞ vðx; tÞ : rðx; tÞ dx dt ¼ 0

Z R3

vðx; tÞ  r ðx; tÞ dx dt ¼ 0

½27a

½27b

for every vector test function  = (1 , 2 , . . . , n ) 2 C1 0 (  [0, T)) satisfying div  = 0, and for every scalar test function 2 C1 0 (  [0, T)). Here we used the notation (u

v) ij = ui vj , and A : B = Pn A B for n  n matrices A and B. We i, j = 1 ij ij observe that [27a] and [27b] are obtained by multiplying  and to [1] and [2], respectively, and integrating by parts. Thus, even the locally square-integrable vector fields, which are not differentiable in the classical sense, could be solutions of the Euler equations. For the general 3D Euler equations, we do not yet have the global existence theorems for the weak solutions. Actually, it is even suggested that we need more weaker notion of solution (the so-called ‘‘measure-valued solutions’’) to describe generic global solutions for the 3D Euler equations. For the 2D Euler equations, however, we have global existence theorems for !0 2 L1 (R2 ) \ Lp (R2 ) for p 2 [1, 1]. This better situation of 2D Euler equations compared to the 3D case for the weak solutions is mainly due to the conservation law of Lp -norm described in [21]. Here we present briefly the existence proof of the weak solutions for 2D Euler equations in the simplest situation. We will prove the global existence of weak solutions for !0 2 Lp (R2 ), 1 < p < 1. Let " (x) = (1="2 ) (x="), 2 where 2 C1 0 (R ) is a standard mollifier, R satisfying  0, supp  {x 2 R2 jjxj < 1}, and R2 dx = 1. Let v0 be the velocity associated with the initial vorticity !0 , given by the Biot–Savart law [12]. Define Rthe sequence of initial data v"0 (x) = " v0 (x) = R2 " (x  y)v0 (y) dy. For each v"0 we have global-in-time smooth solutions v" (x, t). Moreover, thanks to [21], we have the following estimate of the vorticity that is uniform in ": k!ðtÞ" kLp ¼ k!"0 kLp  k!0 kLp

½28

where we used the property of the mollifier in the second inequality. If we take the (distributional) derivative of the Biot–Savart law [12], we find rv = K ! þ C!, where K(x) is a kernel function

½29

Combining [28] and [29] we have sup krv" ðtÞkLp  Cðv0 Þ;

Z

0

defining a singular integral operator of the convolution type, and C is a constant vector. The wellknown Calderon–Zygmund inequality implies that

8T > 0

½30

0tT

namely the sequence {v" } is uniformly bounded in L1 (0, T; W 1, p (R 2 )). Next, we claim that {v" } satisfies the inequality kv" ðt1 Þ  v" ðt2 ÞkH3 ðR2 Þ  Ckv0 k22 kt1  t2 j

½31

for all t1 , t2 with 0 < t1  t2 < T, where C is an absolute constant. Here the negative-order Sobolev space H m (), m > 0, is defined as the dual of H0m (), and can be identified with the space of m functions C1 0 () completed with metric in H (). 2 2 1 2 Indeed, let  2 C0 (R ). Taking L (R ) inner product of [1] with  we have the estimates Z    @v" ðx; tÞ   ðxÞ dx  2 @t R Z  Z      " " "      ð  rÞp dx þ    ðv  rÞv dx 2 2 ZR  Z R      " " "    ¼ p r dx þ  ðv  rÞv dx R2 "

R2

 kp ðtÞkH2 krkH2 þ kv" ðtÞk2L2 krk1  Cðkp" ðtÞkH2 þ kv"0 k2L2 ÞkkH3

½32

where we used the Sobolev inequality krkL1  CkkH3 and the energy equality in the last step. 2 Since [32] holds for all  2 C1 0 (R ), by taking the 2 2 3 closure of C1 0 (R ) in H (R ) we obtain " dv ðtÞ 2 " ½33 dt 2  Ckp ðtÞkH2 þ kv0 kL2 H We now estimate kp" (t)kH2 . Taking the divergence operation on [1], we have the Poisson equation p" ¼ divðv"  rv" Þ 2 Let 2 C1 0 (R ), then Z  Z      " " "      2 p ðx; tÞ ðxÞ dx ¼  2 divðv  rv Þ dx R R Z    " "  ¼  ðv  rÞv  r dx 2 ZR    " "  ¼  ðv  rÞr  v dx R2 "

 kv ðtÞk2L2 k2 kL1  Ckv0 k2L2 k kH4

½34

16

Incompressible Euler Equations: Mathematical Theory

where we used the energy equality [14] and the Sobolev inequality in the last step. Since [34] holds 2 2 1 for all 2 C1 0 (R ), taking the closure of C0 (R ) in 2 H 4 (R ), we obtain Z    "  p ðx; tÞ ðxÞ dx  Ckv0 k2L2 k kH4 

limit " ! 0 in [38] and [39] to obtain the corresponding equations with v" and v"0 replaced by v and v0 . Thus, v is a weak solution of the Euler equations with initial data v0 . This completes the outline of the proof of weak solutions to the 2D Euler equations.

R2

8 2 H 4 ðR2 Þ

½35

Thus, kp" ðtÞkH4  Ckv0 k2L2

8t 2 ½0; TÞ

This provides us with kp" ðtÞkH2  kD2 p" ðtÞkH4  Ckp" ðtÞkH4  Ckv0 k2L2 Combining [33] with [36], we obtain " dv ðtÞ 2 sup dt 2  Ckv0 kL2 0tT H Thus, from "

"

v ðt1 Þ  v ðt2 Þ ¼

Z

t1 t2

dv" ðtÞ dt dt

we have " dv ðtÞ kv" ðt1 Þ  v" ðt2 ÞkH2  sup dt 2 jt1  t2 j 0tT H  Ckv0 k2L2 jt1  t2 j Thus, [31] is proved as claimed. Thanks to the Aubin–Nitsche compactness lemma together with [30] and [31] we have a subsequence, denoted by the same notation, {v" } and v in L1 (0, T; W 1, p (R2 )) such that v" ! v weakly  in L1 ð0; T; W 1; p ðR2 ÞÞ

½36

and v" ! v in L2loc ðR2  ð0; TÞÞ

½37

as " ! 0. We know that as a classical solution each v" and v"0 satisfies Z ðx; 0Þv"0 ðxÞdx R2

þ

Z 0

T

Z R2

ðt  v" þ r : v" v" Þ dx dt ¼ 0

½38

2 for all  2 C1 0 (R  [0, T)) with div  = 0 and Z TZ r  v" dx dt ¼ 0 ½39 0

R2

2 for all 2 C1 0 (R  [0, T)). We can check easily that the convergence [36] and [37] is enough to pass to the

Notes on Further Results

The study of weak solutions of the 2D Euler equations was initiated by V Yudovich in 1963, where he proved the existence of weak solutions for initial data !0 2 L1 (R 2 ) \ L1 (R2 ). Subsequenthy, theory of weak solutions has been developed by studies of the vortex sheet problem due to DiPerna and Majda in 1987. For the existence of weak solutions to the vortex sheet initial data, namely the existence problem for initial vorticity !0 2 H 1 (R2 ) \ M(R 2 ), where M(R2 ) is the space of Radon measures on R2 , is still an outstanding open problem. The main physical motivation of this problem is to understand the dynamics of vortex sheets in the 3D turbulence. For this problem J M Delort proved existence assuming singlesignedness of the initial vortex sheet in 1991. The proof is simplified by A Majda in 1993, using the conservation of moment of impulse. The result is also reproved by L C Evans and S Mu¨ller in 1994, using the weak compactness of the Hardy space. Later in 2001, M C Lopes Filho, H J Nussenzveig Lopes and Z Xin allowed the change of sign for initial vortex sheet, but assumed special reflection symmetry to prove existence of global weak solutions. Related to this problem is the one of characterizing the precise borderline function space to which initial data belongs, and above which there is no concentration phenomenon for weakly approximating sequence of solutions; a recent analysis of this problem was done by E Tadmor in 2001. For the uniqueness problem of the weak solutions of the 2D Euler equations, there are remarkable works by V Scheffer (1993) and A Shnirelman (1997), where they constructed explicitly an L2loc (R2  R) weak solution starting from zero initial data. Also M Vishik (1999) extended the uniqueness class of the weak solutions of the 2D Euler equations, improving previous work by V Yudovich (1995). The class found by M Vishik, in particular, includes the BMO. There is another problem closely related to the weak solutions of the 2D Euler equations, called the vortex patch problem. The main question was if there is any singularity of the boundary of a patch (t) = {X(, t) j  2 0 }, where X(, t) is the particle trajectory mapping generated by a weak solution v(x, t), which is evolving from the initial data !0 (x) = 0 (x), the characteristic function of set 0

Indefinite Metric 17

with smooth boundary. The problem itself is well defined, due to the work of V Yudovich (1963), and there exists unique particle trajectories associated with such weak solutions. The problem was settled by J-Y Chemin in 1991. He proved the global-in-time preservation of the C1,  regularity of the boundary @(t), contrary to the previous numerical experiments. The proof of this result was later simplified by A Bertozzi and P Constantin in 1993. Another interesting problem related to the weak solutions of the Euler equations (2D or 3D) is whether or not the energy is preserved for the weak solutions, namely if there is any ‘‘intrinsic dissipation’’ to the singular solutions of the ideal fluids. In 1949, L Onsager conjectured that if the weak solution of 3D Euler equations belongs to certain Ho¨lder space, then the energy is conserved. This conjecture, in the setting of Besov space, was proved by P Constantin, W E and E S Titi in 1994. This question of possibility of dissipation of energy for weak solutions is further studied by J Duchon and R Robert in 2000. Later, in 2003 the present author considered the problem of helicity conservation for the weak solutions of the 3D Euler flows, which is related to the question of crossing/reconnections of the vortex tubes for weak solutions, and showed that for large class of weak solutions in certain Besov spaces the helicity is preserved. See also: Compressible Flows: Mathematical Theory; Evolution Equations: Linear and Nonlinear; Fluid Mechanics: Numerical Methods; Interfaces and Multicomponent Fluids; Intermittency in Turbulence; Inviscid Flows; Non-Newtonian Fluids; Partial Differential

Equations: Some Examples; Stability of Flows; Stochastic Hydrodynamics; Turbulence Theories; Viscous Incompressible Fluids: Mathematical Theory; Vortex Dynamics.

Further Reading Arnol’d VI and Khesin BA (1998) Topological Methods in Hydrodynamics. New York: Springer. Beale JT, Kato T, and Majda A (1984) Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Communications in Mathematical Physics 94: 61–66. Chemin J-Y (1998) Perfect Incompressible Fluids. New York: Oxford University Press. Chorin AJ and Marsden JE (1993) A Mathematical Introduction to Fluid Mechanics, 3rd edn. New York: Springer. Constantin P (1994) Geometric statistics in turbulence. SIAM Review 36(1): 73–98. Constantin P (1995) A few results and open problems regarding incompressible fluids. Notices of the American Mathematical Society 42(6): 658–663. Kato T (1972) Nonstationary flows of viscous and ideal fluids in R3 . Journal of Functional Analysis 9: 296–305. Lamb H (1932) Hydrodynamics. Cambridge: Cambridge University Press. Lions PL (1996) Mathematical Topics in Fluid Mechanics, Volume 1 (Incompressible Models). New York: Oxford University Press. Majda A and Bertozzi A (2002) Vorticity and Incompressible Flow. Cambridge: Cambridge University Press. Marchioro C and Pulvirenti M (1994) Mathematical Theory of Incompressible Nonviscous Fluids. New York: Springer. Temam R (1984) Navier–Stokes Equations, 3rd edn. New York: North-Holland. Triebel H (1983) Theory of Function Spaces. Boston: Birka¨user Verlag. Yudovich VI (1963) Non-stationary flow of an ideal incompressible liquid. Computational Mathematics and Mathematical Physics 3: 1407–1456.

Indefinite Metric H Gottschalk, Rheinische Friedrich-WilhelmsUniversita¨t Bonn, Bonn, Germany ª 2006 Elsevier Ltd. All rights reserved.

Introduction If, in a problem of quantization, state spaces with indefinite inner product are used instead of Hilbert spaces, one speaks of quantization with indefinite metric. The main domain of application is the quantization of gauge fields, like the electromagnetic vector potential A (x) or Yang–Mills fields in quantum chromodynamics (QCD) and the standard model. The conceptual problem with the indefinite metric is the occurrence of senseless negative probabilities in the formalism. Such negative probabilities,

however, only arise in expectation values of fields that are not gauge invariant and hence do not correspond to observable quantities. Equivalently, the inner product of vectors generated by application of such fields to the vacuum vector with itself can be negative or null. In order to extract the observable content of an indefinite-metric quantum theory, a subsidiary condition is needed to single out the physical subspace. Restricted to this subspace, the inner product is positive semidefinite. This subsidiary condition can be seen as the implementation of a gauge, as, for example, the Lorentz gauge @ A (x) = 0 in quantum electrodynamics (QED). This procedure is also known under the name Gupta–Bleuler formalism. The use of indefinite metric in the quantization of gauge theories like QED can be avoided entirely.

18

Indefinite Metric

This is called quantization in a physical gauge. The problem with such gauges is that they are not Lorentz invariant and that the vector potential A (x) is not a local field. An example is the Coulomb gauge defined by A0 (x) = 0 and @ i Ai (x) = 0 in QED. Furthermore, Dirac spinor fields (x) in such gauges do not anticommute when localized in spacelike separated regions. The Dirac fields therefore are also nonlocal quantities. Although not in contrast with special relativity, as Dirac spinors and the vector potential are not gauge invariant and hence are unobservable, this leads to severe technical problems in the formulation of interacting theories. In particular, the theory of renormalization heavily uses both locality and invariance. Therefore, the Gupta–Bleuler formalism generally is the preferred quantization procedure for a gauge theory. That a local and invariant quantization is not possible using a (positive-metric) Hilbert space has been proved by F Strocchi in a series of articles published between 1967 and 1970. If one wants to preserve locality and/or invariance of the quantized field theory, it is thus strictly necessary to give up the positivity of the state space. A short digression into the early history of the idea might be of interest. It dates back to 1941, where the use of indefinite metric in the quantization of relativistic equations was proposed by Paul Dirac in a lecture at the London Royal Society. The negative probabilities for the bosonic vector potential were thought to be connected with the problem of negative-energy solutions of relativistic equations as a type of surrogate of the ‘‘Dirac sea’’ in the quantization of fermions. Furthermore, Dirac proposed that negative-energy solutions and negative probabilities would jointly lead to the cancellation of divergences in QED. The latter idea was taken up by W Heisenberg in his lectures on the theory of elementary particles held in Munich in 1961, but the generally accepted solution to the problem of ultraviolet divergences was achieved without recourse to Dirac’s original motivation. In 1950 the consistent quantization of vector potential in the Lorentz gauge was formulated by S N Gupta and K Bleuler eliminating the use of negative-energy solutions. Since then the indefinite metric has become a building block of the standard theory of quantized gauge fields.

No-Go Theorems The strict necessity of the Gupta–Bleuler procedure for the local or covariant quantization of gauge fields has been demonstrated by F Strocchi in the form of no-go theorems for positive metric. Here we review their content for the case of the

electromagnetic field. Related statements can be obtained for nonabelian gauge theories. The main problem lies in the fact that standard assumptions on the quantization of relativistic fields are in conflict with Maxwell equations that should hold as operator identities in a positive-metric theory containing no unobservable states. Let F ðxÞ ¼ @ A ðxÞ  @ A ðxÞ

½1

be the quantized electromagnetic field strength tensor. Classically, the existence of A (x) is guaranteed from the first set of Maxwell equations  @ F (x) = 0. Here (and henceforth) indices are raised and lowered with respect to the Minkowski metric g and  is the completely antisymmetric tensor on Rd . Furthermore, we apply Einstein’s convention on summation over repeated upper and lower indices. Standard assumptions from axiomatic quantum field theory are: 1. The field strength tensor F (x) is an operatorvalued distribution acting on a (dense core of a) Hilbert space H with scalar product h., .i – in the indefinite-metric case, h., .i only needs to be an inner product. 2. F (x) transforms covariantly, that is, there is a strongly continuous unitary (with respect to h., .i) representation U of the orthochronous, proper Poincare´ group on H such that for translation a 2 Rd combined with a restricted Lorentz transformation , one has Uða; ÞF ðxÞUða; Þ1 ¼ ð1 Þ  ð1 Þ  F ðx þ aÞ

½2

3. There exists a unique (up to multiplication with C-numbers) translation invariant vector  2 H (the ‘‘vacuum’’), that is, U(a, 1) = 8a 2 Rd . 4. The representation of the translations fulfills the spectral condition Z R4

h; Uða; 1Þieipa da ¼ 0

½3

8,  2 H if p is not in the closed forward light þ cone V¯ = {p 2 R4 : p  p  0, p0  0}. Here the dot is the Minkowski inner product. So far the assumptions concerned only observable quantities. In the following, we also demand. 5. The vector potential A (x) is realized as an operator-valued distribution on H and transforms covariantly under translations Uða; 1ÞA ðxÞUða; 1Þ1 ¼ A ðx þ aÞ

½4

Indefinite Metric 19

The assumptions on the nature of the vector potential so far are rather weak. Strocchi’s no-go theorems show that one cannot add further desirable properties as Lorentz covariance and/or locality without getting into conflict with the Maxwell equations: Theorem 1 Suppose that the above assumptions (1)–(3) and (5) hold. If Maxwell’s equations in the absence of charges,  @ F ðxÞ ¼ 0;

@  F ðxÞ ¼ 0

½5

are valid as operator identities on H and the gauge potential transforms covariantly Uða; ÞA ðxÞUða; Þ1 ¼ ð1 Þ  A ðx þ aÞ

½6

the two-point function of the electromagnetic field tensor vanishes identically: h; F ðxÞF ðyÞi ¼ 0

8 x; y 2 R 4

½7

To gain a better understanding, where the difficulties in the quantization of the Maxwell equations arise from, here is a rough sketch of the proof: Maxwell equations and covariance imply that f (x  y) = h, A (x)F (y)i fulfills @  @ f (x) = 0 and hence its Fourier transform has support in the union of the forward and backward light cone. The Fourier transform thus can be split into a positive- and a negative-frequency part, and þ  f = f þ f accordingly. By the general analysis of axiomatic field theory (see Axiomatic Quantum  Field Theory), the functions f are boundary values of complex analytic functions on certain tubar domains T  transforming covariantly under a certain representation of the complex Lorentz group. By a theorem of Araki and Hepp giving a general representation of such functions and using the antisymmetry of the field tensor, the following formula can be derived:  f ðzÞ ¼ ðg @  g @ Þf  ðzÞ þ  @  h ðzÞ

z2T

½8

with f  , h invariant under complex Lorentz transformations. Taking boundary values in T , one obtains f = (g @  g @ )f þ  @  h, with þ  f = f¯ þ þ f¯  and h = h¯ þ h¯ , where the bar stands for the distributional boundary value. Maxwell’s equations imply @  f = (@  @ g  @ @ )f = 0 and   @ f = (@  @ g  @ @ )h = 0. The only Lorentz-invariant solutions to these equations are constant, which implies the statement of Theorem 1. The second no-go theorem eliminates the assumption that the vector potential A (x) is covariant;

however, a local gauge is assumed. The result is the same as in Theorem 1: Theorem 2 Suppose that the above assumptions (1)– (5) and Maxwell’s equations hold as operator identities on H. If, furthermore, the gauge is local, that is, ½A ðxÞ; A ðyÞ ¼ 0

if x  y is spacelike

½9

the two-point function of the field strength tensor vanishes again as in Theorem 1. Analyzing the interplay of the covariance properties of F (x) with the locality of A (x), Strocchi was able to show that the function f (x  y) must have the same covariance properties as in Theorem 1, which implies the assertion of Theorem 2. The first two no-go theorems deal with the free electromagnetic field that is not coupled to chargecarrying fields. This is, of course, already a real obstruction also for an interacting theory, since, by the LSZ formalism, one expects the asymptotic in=out incoming and outgoing fields Ain=out (x), F (x) to  be free. In fact, it has been proved by D Buchholz that, in the positive-metric case, such asymptotic fields can always be constructed. If one assumes a local and covariant gauge and positivity, the vanishing of the two-point function would also imply that the field F (x) = 0 identically by the Reeh–Schlieder theorem. The next no-go theorem shows that the problems connected to the quantization of the Maxwell equations are not connected only to the free electromagnetic fields. Let us assume that the second set of Maxwell equations is given by @  F ðxÞ ¼ j ðxÞ

½10

where j is the leptonic current, that is, j (x) = e : y (x)  (x): in the case of QED, where is the quantized Dirac field associated with electrons and positrons. Here :  : stands for Wick ordering and  are the Dirac matrices, y =  0 . The conservation of the current @  j (x) = 0 implies that the current charge Z Z QC ¼ lim ðx0 Þðx=RÞj0 ðx0 ; xÞ dx0 dx ½11 R!1

R3 R

is a constant of motion, where  and  are compactly supported infinitely differentiable funcR tions with R (x0 ) = 1 and (x) = 1 for jxj < 1. Now, an alternative definition of charge, called gauge charge (it generates the global U(1)-gauge transformation), is given by QG  ¼ 0; ½QC ; A ðxÞ ¼ 0 ½QG ; ðxÞ ¼ e ðxÞ

and ½12

20

Indefinite Metric

A third formulation of charge, the Maxwell charge QM , can also be given by replacing j0 (x) in [11] by @ F0 (x). Obviously, if Maxwell equations hold as operator identities, QC = QM . On observable states, all charges QM , QC , and QG ought to coincide. Strocchi’s third theorem shows that this cannot be achieved within a local gauge: Theorem 3 If the Maxwell equations [9] hold and the Dirac field (x) is local with respect to the electromagnetic field tensor F (x), that is, ½F ðxÞ; ðyÞ ¼ 0

if x  y is spacelike

½13

then [QM , (x)] = 0, hence QC = QM 6¼ QC . The proof is a simple consequence of the observation that j0 (x) = @  F0 (x) = @ i Fi0 (x) is a three-divergence as F00 (x) = 0 by antisymmetry of F (x). Hence, Z

½j0 ðxÞ; ðyÞðx0 Þðx=RÞ dx0 dx Z ½Fi0 ðxÞ; ðyÞðx0 Þ@ i ðx=RÞ ¼  lim

½QC ; ðyÞ ¼ lim

R!1

R4

R!1

R4

0

 dx dx ¼ 0

½14

since, for R sufficiently large, the support of (x0 )@i (x=R) becomes spacelike separated from y. It should be noted that the proof of none of the above theorems relies on the definiteness of the inner product. The main clue of the indefinite-metric formalism, therefore, is rather to give up Maxwell equations as operator identities. In the usual positive-metric formalism, where all states in H are physical states, this would not be legitimate. But in indefinite metrics, many states are unobservable – in particular, those with negative ‘‘norm’’ h, i < 0. On such states we can neglect the Maxwell equations.

Axiomatic Framework The formalism of axiomatic quantum field theory (see Axiomatic Quantum Field Theory) requires a revision in order to cover the case of gauge fields. The necessary adaptations have been elaborated by G Morchio and F Strocchi, but also earlier work of E Scheibe and J Yngvasson played a significant role in this development. Let (x) be a V 0 -valued quantum field, where V is a finite-dimensional C-vector space with involution . The prime stands for the (topological) dual. For the case of QED, V is eight dimensional,

containing four dimensions for the vector potential A (x) and another four for the Dirac spinors (x), y (x). Such a quantum field can be reconstructed from its vacuum expectation values (Wightman functions) as follows: let S1 = S(R4 , V) be the space of rapidly decreasing functions f : R 4 ! V endowed with the Schwarz topology. Then the Borcher’s algebra S be the free, unital, involutive tensor algebra over S 1 , that is, S = C1 n0 S n 1 with the multiplication induced by the tensor product and involution (f1   

fn ) = fn    f1 . S is endowed with the direct-sum topology. One can show that any linear, normalized, continuous functional W : S ! C, W(1) = 1, is determined by its restrictions Wn to S n 1 . By the Schwarz kernel theorem, Wn 2 S 0 (R4n , V n ). Conversely, any such sequence of Wightman distributions Wn determines a W. Given a Hermitian Wightman functional W such that W(f  ) = W(f ), 8f 2 S, LW = {f 2 S: W(h f ) = 0 8h 2 S} forms a left ideal and the inner product W(f  h) induces a nondegenerate inner product h., .i on H0 = S=LW . Furthermore, Borchers’ algebra S acts from the left on H0 . The quantum field (x) defined as the restriction of this canonical representation to the space S 1  S according to (f ) = R ‘‘ R4 a (x)fa (x)dx’’ where the index a runs over a basis of V. If the Wightman functional W has further properties from axiomatic QFT (see Axiomatic Quantum Field Theory) like invariance with respect to a given representation of the Lorentz group on V, translation invariance, locality, and the spectral property, the quantum field (x) fulfills the related requirements in analogy with the items (1)–(5) listed in the previous section for the case of the vector potential A (x). The Wightman distributions Wn as in the positive-metric case are related to the vacuum expectation values of the theory by Wna1 ;...;an ðx1 ; . . . ; xn Þ ¼ h; a1 ðx1 Þ    an ðxn Þi

½15

where  is the equivalence class of 1 in H0 . The state-space H0 produced by the Gelfand– Naimark–Segal (GNS) construction for innerproduct spaces might be too small to contain all states of physical interest. For example, in the QED case, it does not contain charged states (cf. Theorem 3). Depending on the physical problem, one might also be interested in constructing coherent or scattering states and translation-invariant states apart from the vacuum. Such states appear in problems related to symmetry breaking and confinement (the so-called -vacua) or in some problems of conformal QFT (see Boundary Conformal Field

Indefinite Metric 21

Theory) in two dimensions. It, therefore, has become the standard point of view that one needs to make a suitable closure of H0 such that this closure includes the states of interest (for an alternative point of view, see the last paragraph of the following section). Typically, larger closures are favorable, as they contain more states. One therefore focuses on maximal Hilbert closures of H0 . A Hilbert topology

is induced by an auxiliary scalar product (., .) on H0 . It is admissible, if it dominates the indefinite inner product jh, ij2 C(, )(, ) 8,  2 H0 for some C > 0. This guarantees that the inner product extends to the Hilbert space closure H of H0 with respect to . Furthermore, there exists a self-adjoint contraction  on H such that h, i = (, ) 8, 2 H. A Hilbert topology is maximal if there is no admissible Hilbert topology 0 that is strictly weaker than H0 . The classification of maximal admissible Hilbert topologies in terms of the metric operator  is given by the following theorem: Theorem 4 A Hilbert topology on H0 generated by a scalar product (., .) is maximal if and only if the metric operator  has a continuous inverse 1 on the Hilbert space closure H of H0 . In that case, one can replace (.,) by the scalar product (, )1 = (, jj) without changing the topology . The new metric operator 1 then fulfills 12 = 1H . For a proof of the first statement, see the original work of Morchio and Strocchi (1980). One can easily check that 1 = j1 j which implies the second assertion of the theorem. A Hilbert space (H, (., .)) with an indefinite inner product induced by a metric operator  with 2 = 1H is called a Krein space. For an extensive study of Krein spaces, see the monograph by Azizov and Iokhvidov (1989). Furthermore, one can show that given a nonmaximal admissible Hilbert space topology induced by some (., .), one obtains a maximal admissible Hilbert topology as follows: given the metric operator , we define a scalar product (, )1 = (, (1  P0 )) on H with P0 the null space projector of . Obviously, this scalar product is still admissible and it leads to a new metric operator 1 and a new closure H1 of H0 . Furthermore, it is easy to show that the scalar product (, )2 = (, j1 j)1 still induces an admissible Hilbert topology which is also maximal, as 2 = 1 j11 j clearly fulfills the Krein relation 22 = 1H2 . The question of the existence of a Krein space closure of H0 , therefore, reduces to the question of the existence of an admissible Hilbert topology on H0 . The following condition on the Wightman

functions Wn replaces the positivity axiom in the case of indefinite-metric quantum fields: Theorem 5 Given a Wightman functional W, there exists an admissible Hilbert space topology on H0 = S=LW if and only if there exists a family of Hilbert seminorms pn on S n such that jWnþn (f h)j pn (f )pm (h), 8n, m 2 N 0 , f 2 S n , h 2 S m . In some cases, covering also examples with nontrivial scattering in arbitrary dimension, the condition of Theorem 5 can be checked explicitly (see Non-trivial Models of Quantum Fields with Indefinite Metric). It should be mentioned that different choices of the Hilbert seminorms pn lead to potentially different maximal Hilbert space closures (Hoffmann 1998, Constantinescu and Gheondea 2001). In fact, often the topology is not even Poincare´ invariant and hence the states that can be approximated with local states depend on a chosen inertial frame. This fact, for the case of QED, has been interpreted in terms of physical gauges. Many results from axiomatic field theory (see Axiomatic Quantum Field Theory) with positive metric also hold in the case of QFT with indefinite metric, like the PCT and the Reeh–Schlieder theorem, the irreducibility of the field algebra (for massive theories) and the Bisoniano–Wichmann theorem (see Algebraic Approach to Quantum Field Theory). Other classical results, like the Haag– Ruelle scattering theory and the spin and statistics theorem definitively do not hold, as has been proved by counterexamples. This is, however, far from being a disadvantage, as, for example, it permits the introduction of various gauges in the scattering theory of the vector potential A (x) and fermionic scalar ‘‘ghost’’ fields in the BRST quantization (see BRST Quantization) formalism.

Gupta–Bleuler Gauge Procedure Here the Gupta–Bleuler gauge procedure is presented in a slightly generalized form following Steinmann’s monograph. Classically, the equations of motion for the vector potential A (x), @  @ A ðxÞ þ @ @  A ðxÞ ¼ j ðxÞ

½16

together with Lorentz gauge condition B(x) = @ A (x) = 0 imply the Maxwell equations [10]. Here,  2 R plays the role of a gauge parameter. As seen above, both equations, the socalled pseudo-Maxwell equations [16] and the Lorentz gauge condition B(x) = 0, cannot both hold as operator identities. The idea for the quantization

22

Indefinite Metric

of the theory therefore is to give up the Lorentz gauge condition as an operator identity on the entire state space H. Suppose one has constructed such a theory with an indefinite inner state space H. Already for the noninteracting theory, any invariant, spectral, local, and covariant solution requires indefinite metric, cf. the explicit formula [18] below. To complete the Gupta–Bleuler program, one needs to find a subspace of (equivalence classes of) physical states H0 of the inner-product space H0 such that the following conditions hold: 1. the vacuum is a physical state, that is,  2 H0 ; 2. observable fields like j (x) and F (x) map H0 to itself; 3. the inner product h.,.i restricted to H0 is positive semidefinite; 4. observable fields map H00 , the set of null vectors in H0 , to itself; and 5. the Maxwell equations hold on H0 in the sense h; @  F ðxÞi ¼ h; j ðxÞi;

8;  2 H0

½17

Then one obtains Hph as the completion of the quotient space H0 =H00 . The physical Hilbert space Hph contains the vacuum  (1), observable fields act on Hph (2) and (4), it is a Hilbert space (3) and the Maxwell equations hold on it (5). To see that such a construction is possible, consider the noninteracting case j (x) = 0, that is, the limit case of vanishing electrical charge e ! 0, first. By taking the divergence of [16], one obtains (1  )@  @ @  A (x) = 0. Excluding the Landau gauge ( = 1), this implies (@  @ )2 A (x) = 0. The most general solution for the two-point vacuum expectation values that is in agreement with [16] and the requirements of locality, translation invariance, the spectral condition, uniqueness of the vacuum, and the Lorentz covariance of A (x) is then h; A ðxÞA ðyÞi ¼ ðg þ @ @ ÞDþ ðx  yÞ  @ @ Eþ ðx  yÞ þ 1

½18

where Dþ and Eþ are the inverse Fourier transforms of (p0 ) (p2 ) and (p0 ) 0 (p2 ) respectively, p2 = p  p,  being the Heavyside function, the Dirac measure on R of mass one in zero and 0 its derivative.  and  are gauge parameters, for example, the Feynman gauge corresponds to  =  = 0. We have also omitted an overall factor corresponding to a field strength normalization (choice of numerical value of  h – here  h = 1).

Using Wick’s theorem and the GNS construction for inner-product spaces (cf. the preceding section), it is possible to realize a representation of the vector potential A (x) as operator-valued distribution on some indefinite-metric state space H with Fock structure, for example, a Krein closure of the GNS space with  the GNS vacuum and D H the canonical domain of definition. In the case of Feynman gauge, the metric operator  can be obtained P by a second quantization of the operator f ! 4 = 1 g f on the one-particle space S 1 . In particular, the field B(x) acts as an operatorvalued distribution on H and, by taking the divergence of [16], it follows that @  @ B(x) = 0. Thus, B(x) = Bþ (x) þ B (x) can be decomposed into a positive (‘‘annihilation’’) and a negative (‘‘creation’’) frequency part B (x). One obtains: Theorem 6 The space H0 = { 2 D: Bþ (x) = 0} fulfills all requirements (1)–(5) of the Gupta–Bleuler gauge procedure. Condition (1) is obvious and (2) follows from the fact that the fields F (x) and B(x) commute, which can be checked on the level of two-point functions [18]. In the same spirit, one can also use [18] to check (3) and (4) by explicit calculations on the oneparticle space and showing that H0 is the Fock space over the one-particle states annihilated by Bþ (x). Finally, by Hermiticity of A (x), Bþ (x) = B (x) and thus h, B(x)i = h, Bþ (x)i þ hBþ (x), i = 0. As the field B(x) stands for the obstruction to Maxwell equations, this implies condition (5). It should be noted that the physical state space Hph does not depend on the gauge parameters ,  and that it is spanned by repeated application of the field tensor F (x) to the vacuum. By current conservation, the divergence of [16] still yields @  @ B(x) = 0 also in the interacting case where e 6¼ 0. One can then choose the same gauge condition as in Theorem 6 to define H0 . One can then try to prove that this space fulfills all the requirements of the Gupta–Bleuler procedure, for example, in the sense of perturbation theory. Using more advanced formulations as, for example, BRST quantization and Bogoliubov’s local S-matrix formalism, this program has been completed up to a solution of the infrared problem (see Perturbative Renormalization Theory and BRST). A different procedure, motivated by the necessity of coincidence of all charges QC , QG , and QM on the physical state space, has been elaborated by Steinmann. It deviates from the standard procedure in the sense that the physical space H0 is not included in H, but Hph is directly obtained from the GNS procedure after taking certain limits of Wightman functions restricted to

Index Theorems

certain gauge-invariant algebras constructed from the Borchers algebra and a limiting procedure in a gauge parameter. The Wightman functional on this gaugeinvariant algebras are positive (in the sense of perturbation theory), the limiting procedure, however, implies that the so-obtained physical states are singular (i.e., have diverging inner product) to states in H, hence the so-defined state spaces corresponding to going to a physical gauge after solving the problem of a perturbative construction of an indefinite-metric solution, are not subspaces of H. See also: Algebraic Approach to Quantum Field Theory; Axiomatic Approach to Topological Quantum Field Theory; Axiomatic Quantum Field Theory; Boundary Conformal Field Theory; BRST Quantization; Perturbative Renormalization Theory and BRST; Quantum Fields with Indefinite Metric: Non-Trivial Models.

Further Reading Azizov TYa and Iokhvidov IS (1989) Linear Operators in Spaces with an Indefinite Metric. Chichester: Wiley-Interscience. Bleuler K (1950) Eine neue Methode zur Behandlung der longitudinalen und skalaren Photonen. Helvetica Physica Acta 23: 567.

23

Constantinescu T and Gheondea A (2001) Representations of Hermitian kernels by means of Krein spaces II: invariant kernels. Communications in Mathematical Physics 216: 409–430. Gottschalk H (2002) Complex velocity transformations and the Bisogniano–Wichmann theorem for quantum fields acting on Krein spaces. Journal of Mathematical Physics 43(10): 4753–4769. Gupta SN (1950) Theory of longitudinal photons in quantum electrodynamics. Proceedings of Physical Society A 63: 681. Hofmann G (1998) On GNS representations on inner product spaces: I. The structure of the representation space. Communications in Mathematical Physics 191: 299–323. Morchio G and Strocchi F (1980) Infrared singularities, vacuum structure and pure phases in local quantum field theory. Annals of the Institute Henry Poincare´ (Mathematical Physics). 33: 251–282. Morchio G and Strocchi F (1983) A nonperturbative approach to the infrared problem in QED: construction of charged states. Nuclear Physics B 211: 471–508. Morchio G, Pierotti D, and Strocchi F (1990) Infrared vacuum structure in two dimensional local quantum field theory models: the massless scalar field, SISSA Trieste. Journal of Mathematical Physics 31: 147. Steinmann O (2000) Perturbative Quantum Electrodynamics and Axiomatic Field Theory. Berlin: Springer. Strocchi F and Wightman A (1974) Proof of the charge superselection rule in local relativistic quantum field theory. Journal of Mathematical Physics 15(12): 2198–2224. Yngvason J (1973) On the algebra of test functions for field operators. Communications in Mathematical Physics 34: 315–333.

Index Theorems P B Gilkey, University of Oregon, Eugene, OR, USA K Kirsten, Baylor University, Waco, TX, USA R Ivanova, University of Hawaii Hilo, Hilo, HI, USA J H Park, Sungkyunkwan University, Suwon, South Korea

and sum over repeated indices. Relative to a local coordinate frame for V, D has the form n o D ¼  gij Id@ix @jx þ Ak @kx þ B

ª 2006 Elsevier Ltd. All rights reserved.

where Ak and B are endomorphisms (i.e., matrices) of V. We assume that V is equipped with a positivedefinite inner product and that D is self-adjoint. There is then a complete orthonormal basis { i } for L2 (V), where i 2 C1 (V) and D i = i i . The collection { i , i } is called a discrete spectral resolution of D. For example, if D = @2 on the circle, then the discrete spectral resolution is n pffiffiffiffiffi o e 1n ; n2

Introduction Let g be a Riemannian metric on a smooth compact manifold M of dimension m. We assume for the moment that the boundary of M is empty and postpone until later a discussion of the more general setting. If x = (x1 , . . . , xm ) is a local system of coordinates on M, let   gij :¼ g @ix ; @jx give the components of the metric tensor. Let D be an operator of Laplace type on a smooth vector bundle V over M. Adopt the Einstein convention

n2Z

If we order the eigenvalues 1 2    and repeat each eigenvalue according to multiplicity, then there is the following estimate due to Weyl: n  n2=m

as n ! 1

24

Index Theorems

We now suppose given a pair of vector bundles V1 and V2 over M and a kth-order partial differential elliptic operator A : C1 ðV1 Þ ! C1 ðV2 Þ Locally, we decompose A¼

X

aI @xI

jIjk

where I = (i1 , . . . , im ) is a multi-index and where  i  i @xI ¼ @1x 1 . . . @mx m The aI are linear maps from V1 to V2 . The leading symbol of A is then defined by setting pffiffiffiffiffiffiffi X L ðAÞðx; Þ :¼ ð 1Þk aI ðxÞI jIj¼k

where I = (1 )i1 . . . (m )im , and

cohomological information for general elliptic complexes. Further details appear later in the article. The primary focus here is on the complexes which are of Dirac type, that is, complexes where A is a first-order partial differential operator and where the associated second operators D1 := A A and D2 := AA are of Laplace type. Here is a brief outline of this article. The classical elliptic complexes (de Rham, signature, spin, Dolbeault, Yang–Mills) are discussed first. Next the characteristic classes are introduced, followed by the relevant formula for the index of the classical elliptic complexes, manifolds with boundary, and the equivariant index. Index theory is an enormous topic and here only classical features are emphasized as a complete treatment is beyond the scope of a short expository note such as this one. As some guide to various applications in mathematical physics, the reader is referred to the Further Reading section.

 ¼ ð1 ; . . . ; m Þ are local fiber coordinates on the cotangent bundle. The leading symbol is an invariantly defined map 

L : T M ! EndðV1 ; V2 Þ For example, if V1 = V2 and if D is an operator of Laplace type, then the leading symbol is given by the metric tensor, that is, ij

The Classical Elliptic Complexes The de Rham Complex

Let p M be the bundle of smooth p forms over M and let d : C1 ðp MÞ ! C1 ðpþ1 MÞ

2

L ðDÞ ¼ g i j Id ¼ jj Id If d is exterior differentiation, then the leading symbol is given by exterior multiplication, that is, pffiffiffiffiffiffiffi L ðdÞðÞ! ¼ 1 ^ ! The operator A is said to be elliptic if L (A) is an isomorphism from V1 to V2 for any  6¼ 0. If A is an elliptic partial differential operator, then indexðAÞ :¼ dim kerðAÞ  dim cokerðAÞ ¼ dim kerðA AÞ  dim kerðAA Þ is well defined. As the index vanishes if m is odd, we assume for the most part that m is even. If A" is a smooth one-parameter family of such operators, then index (A" ) is independent of ". The index depends only on the homotopy class of the leading symbol of A within the class of invertible symbols; it does not depend on the underlying metric of the manifold and it does not depend on the fiber metrics chosen for V1 and V2 . The Atiyah–Singer index theorem expresses the index as the integral of suitably chosen polynomials in the curvature tensor for the classical elliptic complexes and, more generally, in terms of

and  : C1 ðp MÞ ! C1 ðp1 MÞ be the exterior derivative and dually the interior derivative, respectively. We set  :¼ ðd þ Þ2

on

C1 ðMÞ

and the decompose  = p p , where p is an operator of Laplace type on C1 (p M). We have d2 = 0. The de Rham cohomology groups are given by taking the quotient of the closed forms by the exact forms: H p ðM; RÞ :¼

kerðd : C1 ðp MÞ ! C1 ðpþ1 MÞÞ imðd : C1 ðp1 MÞ ! C1 ðp MÞÞ

The Hodge–de Rham theorem identifies H p (M; R) with the kernel of the Laplacian kerðp Þ ¼ H p ðM; RÞ and with the topological cohomology groups. If  is a cotangent vector, let e() : ! !  ^ ! be exterior multiplication. Let i() be the dual operator, interior multiplication. If {ei } is a local

Index Theorems

ortho-normal frame for TM, let where I = {1  i1 <    < ip  m}.  0 1 I eðe Þe ¼ e1 ^ eI  i e 2 ^    ^ eip iðe1 ÞeI ¼ 0

eI = ei1 ^    ^ eip , Then we have

if i1 > 1

Define a Clifford module structure on M by ðÞ :¼ eðÞ  iðÞ If {ei } is a local orthonormal basis for TM, then ðei Þðej Þ þ ðej Þðei Þ ¼ 2ij Id so the usual Clifford commutation rules are satisfied. Let r be the Levi-Civita connection on M. We may then expand d ¼ eðei Þrei ;

 ¼ iðei Þrei

d þ  ¼ ðei Þrei The de Rham complex is then defined by taking even M :¼ k 2k M; 1

d þ  : C ð

even

where  M are the 1 eigenspaces of . The signature complex is then given by ðd þ Þ : C1 ðþ MÞ ! C1 ð MÞ

if i1 ¼ 1 if i1 > 1 if i1 ¼ 1

Twisted Signature Complex

Let V be an auxiliary complex vector bundle over M which is equipped with a unitary connection rV . We use the connection rV on V and the Levi-Civita connection on TM to covariantly differentiate tensors of all types. The twisted signature complex is defined by setting ðd þ ÞV :¼ ððei Þ  IdÞrei : C1 ðþ M  VÞ ! C1 ð M  VÞ Yang–Mills complex

This complex in dimension 4 arises from yet another decomposition of the exterior algebra. We use the discussion in the previous section to decompose

odd M :¼ k 2kþ1 M 1

MÞ ! C ð

odd

MÞ

2 M ¼ 2;þ M  2; M into the 1 eigenspaces of . Let : 2 M ! 2; M

The Signature Complex

The signature complex arises from a different decomposition of the exterior algebra. Let Clif M be the Clifford algebra of T  M; this is the universal unital algebra generated by T  M subject to the Clifford commutation relations given above: 1  2 þ 2  1 ¼ 2gð1 ; 2 Þ  Id We suppose M is orientable and let orn ¼ e1      em 2 Clif M be the orientation class. The map  ! () extends to a unital algebra homomorphism  : Clif M ! EndðMÞ (orn) defines an endomorphism of M which is, modulo suitable sign conventions, the Hodge ? operator. If m = 2k is even, then ðd þ ÞðornÞ ¼ ðornÞðd þ Þ Set pffiffiffiffiffiffiffi  :¼ ð 1Þk ðornÞ As 2 = Id, we can decompose M  C ¼ þ M   M

25

be orthogonal projection. The Yang–Mills complex is the 3-term sequence d : C1 ð0 MÞ ! C1 ð1 MÞ and d : C1 ð1 MÞ ! C1 ð2; MÞ We can wrap up this sequence to obtain an equivalent elliptic complex ðd þ Þ : C1 ðeven; MÞ ! C1 ðodd;þ MÞ As with the signature complex, this complex can be twisted by taking coefficients in an auxiliary vector bundle V. It is crucial to the study of fourdimensional geometry using Yang–Mills theory. Dolbeault Complex

Let z = (z1 , . . . , zk ) be a local system of holomorphic coordinates pffiffiffiffiffiffiffion a complex manifold M, where zi = xi þ 1yi . We define pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi dzi :¼ dxi þ 1dyi ; dzi :¼ dxi  1dyi pffiffiffiffiffiffiffi  pffiffiffiffiffiffiffi    @iz ¼ 12 @ix  1@iy ; @iz ¼ 12 @ix þ 1@iy

26

Index Theorems

 where and decompose d = @ þ @, @ :¼ eðdzi Þ@iz

and @ :¼ eðdzi Þ@iz

on the complexified exterior algebra. Let 0 be the  Let adjoint of @ and 00 be the adjoint of @. dzI :¼ dzi1 ^    ^ dzip ð0;evenÞ :¼ SpanfdzI gjIj is even ð0;oddÞ :¼ SpanfdzI gjIj is odd The Dolbeault complex is then defined by ð@ þ 00 Þ : C1 ðð0;evenÞ MÞ ! C1 ðð0;oddÞ MÞ This complex can be twisted by taking coefficients in a holomorphic bundle V over M. The Spin Complex

Let M be orientable. Let PSO be the principal SO bundle of orthonormal frames for the tangent bundle. A spin structure s on M is a principal Spin bundle PSp together with a double cover

: PSp ! PSO which respects the usual double cover : Spin ! SO of the structure groups. Equivalently, a spin structure is a lifting of the transition functions from SO to Spin which preserves the cocycle condition. One says that M is spin if it admits a spin structure. A manifold is orientable if and only if the first Stiefel–Whitney class of M vanishes; an orientable manifold is spin if and only if the second Stiefel– Whitney class of M vanishes as well; these are Z2 -valued cohomology classes. Inequivalent spin structures are parametrized by the cohomology group H 1 (M; Z2 ) or, equivalently, by real-line bundles on M. The spin representation S of Spin defines an associated spin bundle SM = S(M, s). There is a natural Clifford action c of TM on SM. The LeviCivita connection lifts to define the spin connection on S and the Dirac operator is defined by AðsÞ :¼ cðdxi Þr@ix

C1 ðSMÞ pffiffiffiffiffiffiffi k Let m = 2k and let  := ( 1) c(orn). Since c()2 = Id, one can decompose þ

on



SM ¼ S M  S M as the direct sum of the half-spin bundles to obtain the spin complex: AðsÞ : C1 ðS þ MÞ ! C1 ðS  MÞ As with the signature complex, the spin complex can be twisted by taking coefficients in an auxiliary vector bundle V.

Relating the Classic Elliptic Complexes

One has natural isomorphisms of virtual representations of the spinor group: þ   ¼ ðS þ  S  Þ  ðS þ þ S  Þ even  odd ¼ ð1Þm=2 ðS þ  S  Þ  ðS þ  S  Þ which show that the signature complex and de Rham complexes are the spin complexes with coefficients in the virtual bundles Sþ M þ S M

and

ð1Þm=2 ðS þ M  S  MÞ

respectively. If M is complex and spin, then the Dolbeault complex is the spin complex with coefficients in the square root of the canonical bundle. One can consider complex spinors to define the group Spinc (m). Any spin manifold admits a Spinc structure with trivial associated complex line bundle. Any complex manifold admits a Spinc structure with associated complex line bundle given by the canonical bundle. Thus, a complex manifold admits a Spinc structure if and only if it is possible to take a square root of the canonical line bundle; inequivalent Spin structures are parametrized by inequivalent square roots. If M is orientable, then M admits a Spinc structure if and only if the second Stiefel– Whitney class of M lifts from H 2 (M; Z2 ) to H 2 (M; Z); in the complex setting, this lifting is performed by the first Chern class. Inequivalent Spinc structures are parametrized by H 2 (M; Z) or, equivalently, by complex line bundles over M.

Characteristic Classes The Euler Form

Let r be the Levi-Civita connection on M. Let Rðx; yÞ :¼ rx ry  ry rx  r½x;y

be the curvature operator. Let {e1 , . . . , em } be a local orthonormal frame for TM and let Rijkl :¼ gðRðei ; ej Þek ; el Þ give the components of the curvature relative to a local orthonormal frame. Let "I;J :¼ gðei1 ^    ^ eim ; ej1 ^    ^ ejm Þ be the totally antisymmetric tensor; this is the sign of the permutation which sends i ! j . Let m = 2m. ¯ The Euler form is given by setting E m :¼

1  8m m m!

"I;J Ri1 i2 j1 j2 . . . Rim1 im jm1 jm

Index Theorems

Let ij := Rikkj and := ii be the Ricci tensor and the scalar curvature, respectively. Then, E2 ¼

1 4

and

E4 ¼

1 f 2  4j j2 þ jRj2 g 32 2

~ one As Li and Aˆ i are even symmetric functions of , can write Li = Li (p1 (A), . . . , pk (A)). For example,   1 L ¼ 1 þ 13 p1 þ 45 7p2  p21 þ    ^ ¼ 1  1 p1 þ 1 ð7p2  4p2 Þ þ    A 24

1

5760

Substituting (1=2 )R for A then permits one to define the Hirzebruch polynomial L(R) and the Aˆ ˆ genus A(R).

The Pontrjagin Forms

Since R(x, y) = R(y, x), we can regard R as a 2-form-valued endomorphism of the tangent bundle. We define the Pontrjagin forms pi 2 C1 (4i M) by expanding  1 R ¼ 1 þ p 1 þ p2 þ    det I þ 2 These differential forms are closed and the corresponding cohomology classes Pi ¼ ½pi 2 H 4i ðM; RÞ in the de Rham cohomology are independent of the particular Riemannian metric on M which was chosen. ˆ genus and the Hirzebruch L polynomial The A are expressed in terms of these classes using the splitting principle. Let A be a skew-symmetric matrix. One sets

The Chern Forms

Let V be a k-dimensional complex vector bundle over M. Let r be a Hermitian connection on V and let  be the associated curvature endomorphism. The Chern forms ci 2 C1 (2i M) are defined by expanding pffiffiffiffiffiffiffi ! 1  ¼ 1 þ c1 þ c2 þ    det I þ 2 As with the Hirzebruch polynomial and the Aˆ genus, the Chern character and Todd genus are expressed in terms of the generating functions: Y  ~ ¼ TdðÞ 1  e  and

pðAÞ :¼ detðI þ AÞ ¼ 1 þ p1 ðAÞ þ p2 ðAÞ þ    As A is skew symmetric, it decomposes as the direct sum of 2  2 blocks of the form  0 i i 0 We then have pðAÞ ¼

27

Y

1þ

2





~ ¼ chðÞ

X  

!

One has Td ¼ 1 þ Td1 þ Td2 þ     2  1 ¼ 1 þ 12 c1 þ 12 c1 þ c2 þ    Ch ¼ ch0 þ ch1 þ ch2 þ      ¼ k þ c1 þ 12 c21  2c2 þ   

so   pi ðAÞ ¼ si 21 ; 22 ; . . . where si is the ith symmetric function; X X 2i ; p2 ¼ 2i 2j p1 ¼ i

i 1;  E stable rank-r vector bundle over X. To give a concrete example, we will take r = 2 and fix det E ¼ OX :

Fact (Hitchin). Every such bundle E over X can be realized as the direct image of a line bundle over a r:I spectral curve  ! X. We introduce the moduli space M = SU X (2, OX ) = S-equivalence classes of E’s, E semistable rank-2 bundle over X, det E = OX . The dimension of M is 3g  3. Hitchin (1987) proved that T  M is ACI (generically, there exist 3g  3 regular functions in involution with respect to the standard symplectic structure, with invariant manifolds isomorphic to Prym , where  = spectral curve). To recognize the analog of the features highlighted above, we recall that Kodaira–Spencer deformation theory gives the following description of the cotangent bundle: since a rank-r vector bundle over X is determined by a 1-cocycle with values in GL(r, OX ), a first-order deformation of E is given by a 1-cocycle with values in the associated bundle of Lie algebras, hence by a class in H1 (X, End(E)), so the cotangent bundle has Serre-dual fiber H 0 (X, E  E  K). Hitchin map (E, ) 2 T  M (Higgs field, trace zero, 2 H 0 (X, End0 (E)  K)): H: 7! det (more generally for any r  2, tr ^i 2 H 0 (X, Ki )) i = 2, . . . , r;  7!  defines Prym , 2 = det 2 H 0 (X, K2 ) defines . Explicit Hamiltonians for the Hitchin System

The cases in which X is genus 0 and 1 were solved explicitly by Nekrasov (1996) using explicit parametrizations of the moduli spaces; this includes the case of insertions (singular curves), yielding (elliptic) Gaudin models. We report the solution for the genus-2 case (van Geemen and Previato 1996). Remark

The map H projectivizes,

 : PH 0 ðX; End0 ðEÞ  KÞ ! PH 0 ðX; K2 Þ H detðc Þ ¼ c2 det Coordinates on T  M can be given as follows :   Picg1 X = canonical theta divisor  : M ! j2j ¼ P2

g

1

E 7! DE ¼ f 2 Picg1 X : h0 ðE  Þ > 0g X hyperelliptic )  is 2:1 except for g = 2 (every point of M is fixed under the hyperelliptic

76

Integrable Systems and Algebraic Geometry

involution), where M ffi P3 . For a vector space V the Euler sequence gives 



PT PV ffi I ¼ fðx; hÞ 2 PV  PV : x 2 hg

Example

An example is constituted by

2

 ¼ ð2  1Þð2  4Þð2  9Þ ððx : y : z : 1Þ; ðu : v : w : ðxu þ yv þ zwÞÞÞ 2 A3  A3

In our case, PV  PV  ¼ j2j  j20 j

H1 ¼ uvð70xy  32x3 y  18xy3  10z  32x2 z

Define six polynomial functions Hi on P3  P3 by the requirement: for generic q 2 P3 , (Hi = 0) \ PTq P3 = ‘i [ ‘0i , the six pairs of bitangents to K \ PTq P3 , where K is the Kummer surface (the remaining 16 bitangents are cut out by the tropes.) Recall that the Grassmannian of Plines in 6 2 P3 , Gr(2, 4), is defined by an equation 1 Xi = 0 in Klein’s coordinates ðX1 : . . . : X6 Þ 2 P5 X1 ¼ p01 þ p23 ;

X2 ¼ iðp01  p23 Þ

X3 ¼ iðp02  p13 Þ;

X4 ¼ p02 þ p13

X5 ¼ p03 þ p12 ;

X6 ¼ iðp03  p12 Þ

where pij = Zi Wj  Wj Zi are Plu¨cker’s coordinates on the line hðZ0 : . . . : Z3 ÞðW0 : . . . : W3 Þi  P

3

Using coordinates on the incidence variety I given by the sections i of the bundle projection PT  P3 ! P3 , i : P3 ! PT  P3 = I  P3  P3 , q 7! (q, i (q)) = (q, Xi (q, )), explicitly given, for q = (x : y : z : t), by 1 ¼ ðy : x : t : zÞ;

2 ¼ ðy : x : t : zÞ

3 ¼ ðz : t : x : yÞ;

4 ¼ ðz : t : x : yÞ

5 ¼ ðt : z : y : xÞ;

6 ¼ ðt : z : y : xÞ

xj ¼ Xj ðhi ðqÞ; piÞ Fact For a point q 2 P3 , p 2 PTq P3 , p 62 i (q), the ith Klein coordinate of the line hi (q), pi is zero and p 2 ‘i [ ‘0i , Hi ðp; qÞ ¼

X

x2j

j6¼i

i  j

¼0

with xj = Xj (hi (q), pi). Conclusion In an affine patch C3  C3 3 (q, p) = ((x : y : z : 1), (u : v : w : (xu þ yv þ zw))) Hia ðp; qÞ ¼

X Xj ði ðqÞ; pÞ2 j6¼i

i  j

give six Hitchin Hamiltonians, any three of which are generically independent. The Hia have degree  4 in x, y, z and are homogeneous of degree 2 in u, v, w; they Poisson-commute with respect to dx ^ du þ dy ^ dv þ dz ^ dw.

þ 18y2 zÞ þ v2 ð9  30y2  16x2 y2  9y4  32xy2  16z2 Þ þ u2 ð16  40x2  16x4  9x2 y2 þ 18xyz  9z2 Þ þ vwð18x þ 10xy2 þ 10yz  32x2 yz  18y3 z  32xz2 Þ þ uwð32y þ 10x2 y  10xz  32x2 z  18xy2 z þ 18yz2 Þ þ w2 ð9x2  16y2 þ 10xyz  16x2 z2  9y2 z2 Þ

The concept of reduction and r-matrix have been generalized to Hitchin systems. Notably, Hitchin later showed that the Hamiltonians of the system appear as symbols of a heat operator that corresponds to a projectively flat connection, the quantization of the moduli space of bundles, obtained by changing the complex structure of the Riemann surface X.

Other Aspects Special Functions

Special functions have also been traditionally significant in both algebraic geometry and integrable systems. Within the examples presented, elliptic functions gave rise to surprisingly sophisticated theories. The 1-wave solution encountered in the introduction, u = 2} þ const. in the limit when one or both periods of the Weierstrass function go to zero, becomes exponential or rational, respectively. The higher-genus analogs give rise to solitons, or rational solutions. On the other hand, the KP solutions which are doubly periodic in the x variable (‘‘elliptic solitons’’) were classified by Krichever (cf. Dubrovin et al. (2001)), as forming an ACI Hamiltonian system (‘‘elliptic Calogero–Moser’’), which, 25 years later, is still generating important work, with Hamiltonian n X 1X H¼ p21 þ }ðqi  qj Þ 2 i6¼j i¼1 (where } is the Weierstrass function of a lattice L with associated elliptic curve X = C=L, q 2 X the P origin) and u = 2 ni= 1 }(x  xi (t2 , t3 , . . . )) is a solution of the KP hierarchy for suitable time flows tj of the system (t1 = x) and KP Baker function ðx; Þ ¼

ð  xÞ ððÞxÞ e

ðÞ  ðxÞ

The associated spectral curves have been classified in moduli by Treibich and Verdier (cf. Treibich (2001)); Krichever produced a two-field model as

Integrable Systems and Algebraic Geometry

77

well as a universal Poisson structure for the system; Donagi and Markman (1996) realized it as a generalized Hitchin system. More classically, elliptic potentials were the subject of much study, in particular by Lame´ and Hermite in the nineteenth century and Ince in the twentieth; a sample result due to Ince makes one feel like Alice in Wonderland, who ‘‘knelt down and looked along the passage into the loveliest garden you ever saw’’: the Lame´ operator L = @ 2 þ a(a þ 1)}(x  x0 ) with real, smooth potential is finite gap (namely, almost all the periodic eigenvalues are double) iff a 2 Z (if a is positive the number of gaps is a). A generalization to several variables (due to Chalykh and Veselov), X L ¼  þ g }ðh; xiÞ

common spectrum of a ring of commuting (g!  g!) matrix partial differential operators in g variables. The Fourier transform allowed him to extend Sato’s correspondence @ 1 $ z and give F a unique (free, rank-g!) DJac(X) -module structure, where F is a suitable coherent sheaf over Jac(X) generalizing the Baker function. In this model, the interchange of the x and z variables is known as bispectrality (cf. Gru¨nbaum (2001)): a somewhat narrower question is a characterization of the differential operators L in x for which there exists a differential operator B in k and a common eigenfunction: ( L ðx; kÞ ¼ f ðkÞ ðx; kÞ B ðx; kÞ ¼ ðxÞ ðx; kÞ

2Rþ

for some functions f , , typically polynomial. This question proved to be related with the KP hierarchy and isomonodromy deformations. When to a hierarchy there is associated an ACI Hamiltonian system (as in the Neumann case shown above), bispectrality may produce a dual system, in a sense related to the ones discussed, but somewhat mysteriously so.

where Rþ is the set of positive roots for a simple complex Lie algebra of rank n, h, i is some scalar product in Rn , invariant under the action of the Weyl group, and g = m (m þ 1)h, i for some m 2 Z, provides one of the few known examples of quantum completely integrable rings of differential operators in several variables. Roughly speaking, this means that the centralizer of L contains n operators with functionally independent symbols, where n is the number of variables. What is more, Chalykh et al. (2003) combine differential Galois theory and elliptic function theory to characterize (under some mild assumptions) the generalized Lame´ operators that are algebraically completely integrable: the differential Galois group of the solutions is abelian. Duality, Fourier–Mukai Transform, and Bispectrality

Duality is a concept imported from mathematical physics; as a mathematical phenomenon, it has not reached theoretical maturity. First observed in examples, as in Fock et al. (2000), where different definitions of dual ACI Hamiltonian systems were given (actionangle, action–action, and quantum), it resurfaced for the Hitchin system, in more than one guise, whether it be an interchange of position and momentum variables (Gawe¸ dzki and Tran-Ngoc-Bich 1998) or a duality between the Lagrangian tori that fiber two such systems, coming from a Fourier–Mukai transform, namely a twist by the (universal) Picard line bundle: P # 0 JacðXÞ  ðH ðX; KÞ ¼ T  JacðXÞÞ Notably, the Picard bundle was used by Nakayashiki to give a spectacular generalization of the Burchnall– Chaundy result for a genus-2 curve X (more generally, Jac(X) is replaced by a generic abelian variety in the statement): the coordinate ring of Jac(X)  X is the

Conclusion Many important mathematical topics and individual contributions regrettably have to go unmentioned in an article of this length. The aim was to illustrate by simplest examples the geometric nature of integrable systems and equations, in the areas of spectral curves, moduli of vector bundles over them, Grassmann manifolds, special functions, Poisson geometry, representation theory, as well as mention constructions that are not yet complete, such as spectral varieties of higher dimension, dualities sweeping vaster moduli spaces, and quantization. See also: Billiards in bounded convex domains;  @-Approach to Integrable Systems; Functional Equations and Integrable Systems; Integrable Systems and Discrete Geometry; Integrable Systems and Recursion Operators on Symplectic and Jacobi Manifolds; Integrable Systems and the Inverse Scattering Method; Integrable Systems in Random Matrix Theory; Integrable Systems: Overview; Multi-Hamiltonian Systems; Recursion Operators in Classical Mechanics; Riemann– Hilbert Methods in Integrable Systems; Solitons and Kac– Moody Lie Algebras.

Further Reading Adams MR, Harnad J, and Previato E (1988) Isospectral Hamiltonian flows in finite and infinite dimension. Communications in Mathematical Physics 117: 451–500. Adler M and van Moerbeke P (1980) Completely integrable systems, Euclidean Lie algebras and curves. Linearization of Hamiltonian systems, Jacobian varieties and representation theory. Advances in Mathematics 38: 267–379.

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Babelon O, Bernard D, and Talon M (2003) Introduction to Classical Integrable Systems. Cambridge: Cambridge University Press. Baker HF (1907) An Introduction to the Theory of MultiplyPeriodic Functions. Cambridge: Cambridge University Press. Chalykh O, Etingof P, and Oblomkov A (2003) Generalized Lame´ operators. Communications in Mathematical Physics 239(1–2): 115–153. Dickey LA (2003) Soliton Equations and Hamiltonian Systems, Advanced Series in Mathematical Physics, vol. 26, 2nd edn. River Edge, NJ: World Scientific. Donagi R and Markman E (1996) Spectral covers, algebraically completely integrable Hamiltonian systems, and moduli of bundles. In: Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, pp. 1–119. Berlin: Springer (Montecatini Terme, 1993). Dubrovin BA, Krichever IM, and Novikov SP (2001) Integrable Systems I, Dynamical Systems IV, Encyclopaedia of Mathematical Science, vol. 4, pp. 177–332. Berlin: Springer. Fock V, Gorsky A, Nekrasov N, and Rubtsov V (2000) Duality in integrable systems and gauge theories. Journal of High Energy Physics No. 7, pp. 40. Franc¸oise JP (1987) Monodromy and the Kovalevskaya top. Aste´rique 150–151; 87–108. Gawe¸ dzki K and Tran-Ngoc-Bich P (1998) Self-duality of the SL2 Hitchin integrable system at genus 2. Communications in Mathematical Physics 196(3): 641–670. Gru¨nbaum FA (2001) The bispectral problem: an overview. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), 129–140, NATO Science Series II: Mathematics, Physics, and Chemistry, vol. 30. Dordrecht: Kluwer Academic.

Hitchin N (1982) Monopoles and geodesics. Communications in Mathematical Physics 83(4): 579–602. Hitchin N (1987) Stable bundles and integrable systems. Duke Mathematical Journal 54(1): 91–114. McKean H and Moll V (1997) Elliptic Curves. Function Theory, Geometry, Arithmetic. Cambridge: Cambridge University Press. Moser J (1980) Geometry of quadrics and spectral theory. In: The Chern Symposium 1979, pp. 147–188. New York–Berlin: Springer. Mulase M (1984) Complete integrability of the Kadomtsev– Petviashvili equation. Advances in Mathematics 54(1): 57–66. Mumford D (1984) Tata Lectures on Theta II, Progr. Math., vol. 43. Boston: Birkha¨user. Nekrasov N (1996) Holomorphic bundles and many-body systems. Communications in Mathematical Physics 180(3): 587–603. Neumann C (1859) De problemat quodam mechanico quad ad primam integralium ultraellipticorum classem revocatur. J. reine angew Math. 56: 46–63. ˘ Ol’shanetskii˘ MA, Perelomov AM, Reiman AG, and SemenovTyan-Shanskii˘ MA (1987) Integrable systems, II. Current Problems in Mathematics. Fundamental Directions, vol. 16, pp. 86–226, 307; Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow. Siegel CL (1969) Topics in Complex Function Theory, vol. 1. New York: Wiley. Treibich A (2001) Hyperelliptic tangential covers, and finite-gap potentials. Uspekhi Matematicheskikh Nauk 56(6): 89–136. van Geemen B and Previato E (1996) On the Hitchin system. Duke Mathematical Journal 85(3): 659–683.

Integrable Systems and Discrete Geometry A Doliwa, University of Warmia and Mazury in Olsztyn, Olsztyn, Poland P M Santini, Universita` di Roma ‘‘La Sapienza,’’ Rome, Italy ª 2006 Elsevier Ltd. All rights reserved.

Introduction Although the main subject of this article is the connection between integrable discrete systems and geometry, we feel obliged to begin with the differential part of the relation. Classical Differential Geometry and Integrable Systems

The oldest (1840) integrable nonlinear partial differential equation recorded in literature is the Lame´ system @ 2 Hi 1 @Hj @Hi 1 @Hk @Hi   ¼ 0; @uj @uk Hj @uk @uj Hk @uj @uk

@ @uk

i; j; k distinct ½1    1 @Hj @ 1 @Hk 1 @Hj @Hk þ þ 2 ¼ 0 ½2 Hk @uk @uj Hj @uj Hi @ui @ui



describing orthogonal coordinates in the threedimensional Euclidean space E3 (indices i, j, k range from 1 to 3). Already in 1869, it was found by Ribaucour that the nonlinear Lame´ system possesses a discrete symmetry enabling to construct, in a linear way, new solutions of the system from the old ones. He gave also a geometric interpretation of this symmetry in terms of certain spheres tangent to the coordinate surfaces of the triply orthogonal system. In 1918, Bianchi showed that the result of superposition of the Ribaucour transformations is, in a certain sense, independent of the order of their composition. Such properties of a nonlinear equation are hallmarks of its integrability, and indeed, the Lame´ system was solved using soliton techniques in 1997–98. The above example illustrates the close connection between the modern theory of integrable partial differential equations and the differential geometry of the turn of the nineteenth and twentieth centuries. A remarkable property of certain parametrized submanifolds (and then of the corresponding equations) studied that time is that they allow for transformations which exhibit the so-called ‘‘Bianchi permutability property.’’ Such transformations called, depending on the context, the Darboux, Calapso, Christoffel, Bianchi, Ba¨cklund, Laplace,

Integrable Systems and Discrete Geometry

Koenigs, Moutard, Combescure, Le´vy, Goursat, Ribaucour, or the fundamental transformation of Jonas, can be geometrically described in terms of certain families of lines called line congruences. In the connection between integrable systems and differential geometry, a distinguished role is played by the multidimensional conjugate nets, described by the Darboux system, which is just the first part [1] of the Lame´ system with indices ranging form 1 to N  3. On the level of integrable systems, this dominant role has the following explanation: the Darboux system, together with equations describing isoconjugate deformations of the net, forms the multicomponent Kadomtsev–Petviashvilii (KP) hierarchy, which is viewed as a master system of equations in soliton theory. In fact, in appropriate variables, the whole multicomponent KP hierarchy can be rewritten as an infinite system of the Darboux equations.

Transition to the Discrete Domain

The recent progress in studying discrete integrable systems showed that, in many respects, they should be considered as more fundamental than their differential counterparts. Consequently, the natural problem of extending the geometric interpretation of integrable partial differential equations to the discrete domain arose, leading not only to the transition to the discrete domain of many results on the connection between the differential geometry and integrable systems, but also – and this seems to be even more important – to the description of integrability in a very elementary and purely geometric way. At the level of integrable equations, the transition ‘‘from differential to discrete’’ often makes formulas more complicated and longer. On the contrary, at the geometric level, in such a transition the properties of discrete submanifolds, relevant to their integrability, become simpler and more transparent. Indeed, the mathematics necessary to understand the basic ideas of the integrable discrete geometry does not exceed the ‘‘ruler and compass constructions,’’ and many proofs can be performed using elementary incidence geometry. We will concentrate our attention on the multidimensional lattice made from planar quadrilaterals, which is the discrete analog of a conjugate net. Together with the discussion of its properties, which are the core of the geometric integrability, we briefly present the analytic methods of construction of these lattices and we also describe some basic multidimensional integrable reductions of them. Then we discuss integrable discrete surfaces; some of them have been found in the early period of the ‘‘case-by-case’’ studies. We shall however try to present them, from a unifying perspective, as reductions of the quadrilateral lattice (QL).

79

Multidimensional Integrable Lattices The Quadrilateral Lattice

An N-dimensional lattice x : ZN ! RM is a lattice made from planar quadrilaterals, or a quadrilateral lattice (QL) in short, if its elementary quadrilaterals {x, Ti x, Tj x, Ti Tj x} are planar; that is, iff the following system of discrete Laplace equations is satisfied: i j x ¼ ðTi Aij Þi x þ ðTj Aji Þj x; i 6¼ j; i; j ¼ 1; . . . ; N

½3

where Aij : ZN ! R are functions of the discrete variable; here Ti is the translation operator in the ith direction, and i = Ti  1 is the corresponding difference operator. For simplicity, we work here in the affine setting neglecting projective geometric aspects of the theory. The geometric integrability scheme In the case N = 2 the definition [3] allows one to uniquely construct, given two discrete curves intersecting in a common vertex and two functions A12 , A21 : Z2 ! R, a quadrilateral surface. For N > 2 the planarity constraints [3] are instead compatible if and only if the geometric data Aij satisfy the nonlinear system k Aij þ ðTk Aij ÞAik ¼ ðTj Ajk ÞAij þ ðTk Akj ÞAik i; j; k

distinct

½4

This constraint has a very simple interpretation: in building the elementary cube (see Figure 1), the seven points x, Ti x, Tj x, Tk x, Ti Tj x, Ti Tk x, and Tj Tk x (i, j, k are distinct) determine the eighth point Ti Tj Tk x as the unique intersection of three planes in the three-dimensional space. The connection of this elementary geometric point of view with the classical theory of integrable systems is transparent: the planarity constraint corresponds to the set of linear spectral problems [3] and the resulting QL is characterized by the nonlinear equations [4], arising as the compatibility conditions for such spectral problems. Since the QL equations [4] are a master system in the theory of integrable equations, planarity can be viewed as the elementary geometric root of integrability. The idea

TiTjTk x

Tk x Tj x

TiTj x

x Ti x Figure 1 The geometric integrability scheme.

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Integrable Systems and Discrete Geometry

that integrability be associated with of a geometric (and/or algebraic) increasing the dimensionality of recurrent in the theory of integrable

the consistency property when the system is systems.

Other forms of the Darboux system The i $ j symmetry of the RHS of eqns [4] implies the existence of the potentials Hi : ZN ! R (the Lame´ coefficients) such that Aij ¼

j Hi ; Hi

i 6¼ j

½5

and then eqns [4] take the form   k Hj k j Hi  Tj j Hi Hj   j Hk  Tk k Hi ¼ 0; i; j; k Hk

distinct ½6

½7

i 6¼ j

½8

Then eqns [3] and [6] can be rewritten in the firstorder form i 6¼ j

i; j; k

½9

distinct

½10

The discrete Darboux system [10] implies the existence of other potentials i defined by the compatible equations T j i ¼ 1  ðTi Qji ÞðTj Qij Þ; i

i 6¼ j

½11

The i $ j symmetry of the RHS of eqns [11] implies the existence of yet another potential  : ZN ! R, Ti  i ¼ 

½12

which is called the -function of the QL. In terms of the -function, and of the functions ij ¼ Qij ;

i 6¼ j

½13

whose geometric interpretation will be given in a later section, the discrete Darboux equations take the following Hirota-type form: ðTi Tj Þ ¼ ðTi ÞTj   ðTi ji ÞTj ij ;

Consider the nonlocal

  ^ @ðzÞ þ ðRÞðzÞ ¼ @ðzÞ ½16

^ is the integral operator where @ = @=@z, R Z ^ ðRÞðzÞ ¼ Rðz; z0 Þðz0 Þ dz0 ^ dz0 C

and the functions Qij : ZN ! R, i 6¼ j, (the rotation coefficients) by equations

k Qij ¼ ðTk Qik ÞQkj ;

 The @-dressing method  @-problem

jzj!1

i x ¼ ðTi Hi ÞX i

j X i ¼ ðTj Qij ÞX j ;

We will show how one can construct large classes of solutions of the discrete Darboux equations and the corresponding QLs using two basic analytical  methods of the soliton theory: the @-dressing method and the algebro-geometric techniques.

lim ððzÞ  ðzÞÞ ¼ 0

which is the discrete version of the first part [1] of the Lame´ system. The Lame´ coefficients allow to define the suitably normalized tangent vectors X i : ZN ! RM by equations

i Hj ¼ ðTi Hi ÞQij ;

Analytic Methods

i 6¼ j

½14

ðTk ij Þ ¼ ðTk Þij þ ðTk ik Þkj ; i; j; k distinct ½15

and (z) is a given rational function of z. Let Q i 2 C, i = 1, . . . , N be pairs of distinct points of the complex plane, which define the dependence of the kernel R on the discrete variable n 2 ZN : ni N  Y z  Qþ i Rðz; z0 ; nÞ ¼ z  Q i i¼1 ni N  0 Y z  Q 0 i  R0 ðz; z Þ z0  Qþ i i¼1 We consider only kernels R0 (z, z0 ) such that the  nonlocal @-problem is uniquely solvable. If (z; n) is the unique solution with the canonical normalization  = 1, then the function ni N  Y z  Q i ðz; nÞ ¼ ðz; nÞ z  Qþ i i¼1 satisfies the system of the Laplace equations [3] with the Lame´ coefficients given by   ni z  Qþ i Hi ðnÞ ¼ limþ ðz; nÞ z  Q z!Qi i By construction, the system of such Laplace equations is compatible, therefore the Lame´ coefficients satisfy eqns [6]. To various n-independent measures da on C there correspond coordinates Z a ðz; nÞda ðzÞ x ðnÞ ¼ C

of a QL x, having Hi (n) as the Lame´ coefficients. To have real lattices, the kernel R0 , the points Q i , and the measures da should satisfy certain additional conditions. One can find a similar interpretation of the normalized tangent vectors X i and of the rotation

Integrable Systems and Discrete Geometry

coefficients Qij . If i (z; n) are the unique solutions of  the nonlocal @-problem [16] with the normalizations !  þ  Y N þ þ nk Q  Q Qi  Q i i k i ðz; nÞ ¼  z  Qþ Qþ i i  Qk k¼1;k6¼i then the functions

i (z; n),

defined by !nk N Y z  Q k i ðz; nÞ i ðz; nÞ ¼ z  Qþ k k¼1

satisfy the direct analog of the linear problem [9], j i ðz; nÞ ¼ ðTj Qij ðnÞÞ j ðz; nÞ;

i 6¼ j

½17

where z  Qþ j

Qij ðnÞ ¼ limþ

z  Q j

z!Qj

!

!nj

C

are coordinates of the normalized tangent vectors X i of the QL x constructed above. The algebro-geometric techniques Given a compact Riemann surface Pg R of genus g, consider a nonspecial divisor D =  = 1 P . Choose N pairs of points Q i 2 R and the normalization point Q1 . Given n 2 ZN , there exists a unique Baker–Akhiezer function (n), defined as a meromorphic function on R, with the following analytical properties: (1) as a function of P 2  R n [N i = 1 Qi , (n) may have as singularities only simple poles in the points of the divisor D; (2) in the points Q i function (n) has poles of the order ni ; and (3) in the point Q1 function (n) is normalized to 1. When z i (P) is a local coordinate on R centered at  Qi , then condition (2) implies that the function (n) in a neighborhood of the point Q i is of the form ! 1   ni X   s i s; ðnÞ z ðP; nÞ ¼ zi ðPÞ ½18 i ðPÞ s¼0

The Baker–Akhiezer function, as a function of the discrete variable n 2 ZN , satisfies the system of Laplace equations [3] with the Lame´ coefficients i Hi (n) = 0, þ (n). Again, by construction, the Lame´ coefficients satisfy eqns [6]. To various n-independent measures da on R there correspond coordinates Z a x ðnÞ ¼ ðP; nÞ da ðPÞ R

of a QL x.

We present the expression of the Baker–Akhiezer function and of the -function of the QL in terms of the Riemann theta functions. Let us choose on R the canonical basis of cycles {a1 , . . . , ag , b1 , . . . , bg } and the dual basis {!1 , H. . . , !g } of holomorphic differentials on R, that is, aj !k = jk H . Then the matrix B of b-periods defined as Bjk = bj !k is symmetric and has positively defined imaginary part. Denote by !PQ the unique differential holomorphic in Rn{P, Q} with poles of the first order in P, Q and residues, correspondingly,H 1 and 1, which is normalized by conditions aj !PQ = 0. The Riemann function (z; B), z 2 Cg , is defined by its Fourier expansion X

ðz; BÞ ¼ expfihm; Bmi þ 2ihm; zig m2Zg

i ðz; nÞ

Again, by construction, eqns [17] are compatible and the functions Qij (n) satisfy the discrete Darboux equations [10]. The functions Z Xai ðnÞ ¼ i ðz; nÞ da ðzÞ

81

where h , i denotes the standard bilinear form in Cg . Finally, theR Abel map A is given by A(P) = RP P ( P0 !1 , . . . , P0 !g ), where P0 2 R, and the Riemann constants vector K is given by I  1 þ Bjj X  !k ðPÞAj ðPÞ!j ; Kj ¼ 2 ak k6¼j j ¼ 1; . . . ; g The explicit form of the vacuum Baker–Akhiezer function can be written down with the help of the theta functions as follows:       þ  P þZ

AðPÞ þ N k¼1 nk A Qk  A Qk  ðn; PÞ ¼      þ  P

AðQ1 Þ þ N þZ k¼1 nk A Qk  A Qk ! Z P N X

ðAðQ1 Þ þ ZÞ nk !Q Qþ  exp k k

ðAðPÞ þ ZÞ Q1 k¼1

Pg where Z =  j = 1 A(Pj )  K.  Denote by r kj and skj the constants in the decomposition of the abelian integrals near the point Q j Z P   P!Q j   !Q Qþ ¼  kj log z j ðPÞ þ rkj þ O zj ðPÞ Z

P0 P

P0

k

k

  P!Q j   !Q1 Qþ ¼  kj þ log z j ðPÞ þ skj þ O zj ðPÞ k

Then the expression of the -function of the QL within the subclass of algebro-geometric solutions reads ðnÞ ¼

N X

!     þ  nk A Qk  A Qk þ AðQ1 Þ þ Z

k¼1



N Y k;j¼1

n nj

kjk

N Y k¼1

nkk

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Integrable Systems and Discrete Geometry

where kj ¼ exp

r kj

 2

rþ kj

l

! ¼ jk

      þZ 1 A Qþ þ k     exp s k ¼ kk  skk kk A Qk þ Z Finally, we remark that the geometric integrability scheme and the algebro-geometric methods work also in the finite fields setting, giving solutions of the corresponding integrable cellular automata. The Darboux-Type Transformations

We present the basic ideas and results of the theory of the Darboux-type transformations of the multidimensional QL. Line congruences and the fundamental transformation To define the transformations we need to define first N-dimensional line congruences (or, simply, congruences), which are families of lines in RM labeled by points of ZN with the property that any two neighboring lines l and Ti l, i = 1, . . . , N, are coplanar and therefore (eventually in the projective extension PM of RM ) intersect. The QL F (x) is a fundamental transform of the QL x if the lines connecting the corresponding points of the lattices form a congruence. The superposition of a number of fundamental transformations can be compactly formulated in the vectorial fundamental transformation. The data of the vectorial fundamental transformation are: (1) the solution Y i : ZN ! V, V being a linear space, of the linear system [9]; (2) the solution Y i : ZN ! V , V being the dual of V, of the linear system [8]. These allow to construct the linear operator-valued potential W(Y, Y ) : ZN ! L(V), defined by the following analog of eqn [7]:   ½19 i WðY; Y Þ ¼ Y i Ti Y i ; i ¼ 1; . . . ; N Similarly, one defines W(X, Y ) : ZN ! L(V, RM ) and W(Y, H) : ZN ! V. The transforms of the lattice x and other related functions are given by F ðxÞ ¼ x  WðX; Y ÞWðY; Y Þ1 WðY; HÞ F ðHi Þ ¼ Hi  Y i WðY; Y Þ1 WðY; HÞ; i ¼ 1; . . . ; N F ðX i Þ ¼ X i  WðX; Y ÞWðY; Y Þ1 Y i ; i ¼ 1; . . . ; N F ðQij Þ ¼ Qij  Y j WðY; Y Þ1 Y i ; i; j ¼ 1; . . . ; N; i 6¼ j     F ði Þ ¼ i 1 þ Ti Y i WðY; Y ÞY i ; i ¼ 1; . . . ; N F ðÞ ¼  det WðY; Y Þ

Ti

Ti l

* i (x)

TiTj (x )

(x) x

Ti x

i (x )

Figure 2 The fundamental transformation as the binary transformation.

Notice that, by the coplanarity of any two neighboring lines of the congruence, also the quadrilaterals {x, Ti x, F (x), F (Ti x)} are planar (see Figure 2). Then the construction of the transformed lattice mimics the geometric integrability scheme. In consequence, any quadrilateral {x; F 1 ðxÞ; F 2 ðxÞ; F 1 ðF 2 ðxÞÞ = F 2 ðF 1 ðxÞÞ} is planar as well. Therefore, on the discrete level, there is no difference between the lattice coordinate directions and the fundamental transformation directions. The distinction becomes visible in the limit from the QL to the conjugate net. Therefore, the vectorial description of the superposition of the fundamental transformations not only implies their permutability but also provides the explanation of the validity of the practical rule of ‘‘integrable discretization by Darboux transformations.’’ The Le´vy and Combescure transformations It is easy to see that the family ti of lines passing through the points x and Ti x of a QL forms a congruence, called the ith tangent congruence of the lattice. When the congruence of the transformation is the ith tangent congruence of the lattice x, then the corresponding reduction of the fundamental transformation is called the ‘‘Le´vy transformation’’ Li . It turns out that, for a generic congruence l, the lattice made from intersection points of the lines l and Ti1 l is a QL, called the ith focal lattice of the congruence. When the fundamental transform of the lattice x is the ith focal lattice of the transformation congruence, then the corresponding reduction of the fundamental transformation is called the ‘‘adjoint Le´vy transformation’’ L i . Both Le´vy transformations use only a half of the fundamental transformation data, and the corresponding reduction formulas (in the scalar case) for the lattice points read as follows: Li ðxÞ ¼ x  X i ðYi Þ1 WðY; HÞ L i ðxÞ ¼ x  WðX; Y Þ ðYi Þ1 Hi

Integrable Systems and Discrete Geometry

Notice that the composition of the Le´vy and the adjoint Le´vy transformations gives (see Figure 2) the fundamental transformation, also called, for this reason, the binary transformation. Another reduction of the fundamental transformation, important from a technical point of view, is the ‘‘Combescure transformation,’’ in which the tangent lines of the transformed lattice C(x) are parallel to those of the lattice x. The transformation formula reads CðxÞ ¼ x  WðX; Y Þ where only the solution Y of the adjoint linear system [8], necessary to build the transformation congruence, is needed. The Laplace transformations and the geometric meaning of the Hirota equation The Laplace transform Lij (x), i 6¼ j, of the QL x is the jth focal lattice of its ith tangent congruence (see Figure 3). It is uniquely determined once the lattice x is given. The transformation formulas of the lattice points and of the -function read as follows: Lij ðxÞ ¼ x 

1 i x Aji

½20

Lij ðÞ ¼ ij ¼ Qij

½21

The superpositions of Laplace transformations satisfy the following identities Lij  Lji ¼ id Ljk  Lij ¼ Lik Lki  Lij ¼ Lkj which allow to identify them with the Schlesinger transformations of the monodromy theory. In the simplest case N = 2 one obtains the so-called Laplace sequence of two-dimensional QLs x‘ ¼ L‘12 ðxÞ; L1 12 ¼ L21 ;

‘ ¼ L‘12 ðÞ

‘2Z

Equations [14] and [21] imply that the -functions of the Laplace sequence satisfy the celebrated Hirota equation (the fully discrete Toda system)

Distinguished Integrable Reductions

We will present here basic reductions of the multidimensional QL. The geometric criterion for their integrability is the compatibility with the geometric integrability scheme. The circular lattices and the Ribaucour congruences QLs ZN ! EM for which each quadrilateral is inscribed in a circle are called ‘‘circular’’ lattices. They are the integrable discrete analogs of submanifolds parametrized by curvature coordinates (e.g., the orthogonal coordinate systems described by the Lame´equations [1]–[2]). The integrability of circular lattices is the consequence of the fact that if the three ‘‘initial’’ quadrilaterals {x, Ti x, Tj x, Ti Tj x}, {x, Ti x, Tk x, Ti Tk x}, {x, Tj x, Tk x, Tj Tk x} are circular, then also the three new quadrilaterals constructed by adding the vertex Ti Tj Tk x are circular as well (see Figure 4). In fact, all the eight vertices belong to a sphere, and, in consequence, all the vertices of any K-dimensional, K = 2, . . . , N, elementary cell belong to a (K  1)-dimensional sphere. There are various equivalent algebraic descriptions of the circular lattices: 1. the normalized tangent vectors X i satisfy the constraint X i  Ti X j þ X j  Tj X i ¼ 0;

i 6¼ j

2. the scalar function x  x : ZN ! R satisfies the Laplace equations [3] of the lattice x; 3. the functions Xi = (x þ Ti x)  X i : ZN ! R satisfy the same linear system [9] as the normalized tangent vectors X i ; and 4. the functions X i  X i : ZN ! R satisfy eqns [11] and thus can serve as the potentials i . The Ribaucour transformation R is the restriction of the fundamental transformation to the class of circular lattices such that also the ‘‘side’’ quadrilaterals {x, Ti x, R(x), R(Ti x)} are circular. Again there is no geometric difference between the lattice directions and the Ribaucour transformation direction. Moreover, the quadrilaterals {x, R1 (x),

‘ T1 T2 ‘ ¼ ðT1 ‘ ÞðT2 ‘ Þ  ðT1 ‘1 ÞðT2 ‘þ1 Þ TiTjTk x Tj x T T x i j x

Ti x

Tk x Tj

ij (x )

ij (x)

Tj x Tj–1x Figure 3 The Laplace transformation Lij .

Ti

ij (x )

83

x

TiTj x Ti x

Figure 4 The geometric integrability of circular lattices.

84

Integrable Systems and Discrete Geometry

R2 (x), R1 (R2 (x)) = R2 (R1 (x))} are circular as well. In consequence, the vertices of the elementary K-cells, K = 2, . . . , N, of the circular lattice and the corresponding vertices of its Ribaucour transform are contained in a K-dimensional sphere. Finally, for K = N, one obtains a special ZN family of N-dimensional spheres, called the Ribaucour congruence of spheres. Algebraically, the Ribaucour transformation needs only a half of the data (necessary to build the congruence) of the fundamental transformation. The data of the vectorial Ribaucour transformation consists of the solution Y i : ZN ! V , of the linear system [8]. Then, because of the circularity constraint, Y i : ZN ! V given by Y i ¼ ðWðX; Y Þ þ Ti WðX; Y ÞÞT X i is a solution of the linear system [9], and the constraints WðY; HÞ þ WðX ; Y ÞT ¼ 2 WðX; Y ÞT x WðY; Y Þ þ WðY; Y ÞT ¼ 2 WðX; Y ÞT WðX; Y Þ are admissible. We remark that the above constraints have a simple geometric meaning when one considers the circular lattices in EM as the stereographic projections of QLs in the Mo¨bius sphere SM ; that is, as a special case of QLs subjected to quadratic constraints.

is the solution of the linear system [9]; notice that, equivalently, we could start from Y i . The constraint WðY; Y Þ ¼ WðY; Y ÞT is then admissible and gives a new symmetric lattice. There are other multidimensional reductions of the QL like, for example, the D-invariant and Egorov lattices or discrete versions of immersions of spaces of constant negative curvature. We remark that the transformations and reductions discussed above have also a clear interpretation on the level of the analytic methods.

Integrable Discrete Surfaces In this section we present some distinguished examples of discrete integrable surfaces. Notice that, although the geometric integrability scheme is meaningless for N = 2, it can be applied indirectly, by considering the discrete surfaces, together with their transformations, as sublattices of multidimensional lattices. We remark also that one can consider integrable evolutions of discrete curves, which give equations of the Ablowitz–Ladik hierarchy, and the corresponding integrable spin chains. Discrete Isothermic Nets

The symmetric lattice Given a QL x with rotation coefficients Qij and potentials i given by [11], then ~ ij , defined by equation the functions Q ~ ij ¼ i Ti Qji ; j T j Q

i 6¼ j

and called, because of their geometric interpretation, the backward rotation coefficients, satisfy the Darboux system [10] as well. A QL is called symmetric if its forward rotation coefficients Qij are also its backward rotation coefficients. Again the constraint is compatible with the geometric integrability scheme, that is, it propagates in the construction of the lattice. One can show that a QL is symmetric if and only if its rotation coefficients satisfy the following trilinear constraint:

An isothermic lattice is a two-dimensional circular lattice x : Z2 ! EM with harmonic quadrilaterals; that is, given x, T1 x and T2 x, then the point T1 T2 x is the intersection of the circle (passing through x, T1 x and T2 x) and the line passing through x and the meeting point of the tangents to the circle at T1 x and T2 x (see Figure 5). Therefore, given two discrete curves intersecting in the common vertex x0 , the unique isothermic lattice can be found using the above ‘‘ruler and compass’’ construction. Algebraically the reduction looks as follows. Any oriented plane in EM can be identified with the complex plane C. Given any four complex points z1 , z2 , z3 , and z4 , their complex cross-ratio is defined by qðz1 ; z2 ; z3 ; z4 Þ ¼

ðTi Qji ÞðTj Qkj ÞðTk Qik Þ ¼ ðTj Qij ÞðTi Qki ÞðTk Qjk Þ i; j; k distinct To obtain the corresponding reduction of the fundamental transformation we again need only half of the data. Given a solution Y i : ZN ! V , of the linear system [8], then, because of the symmetric constraint, Y i : ZN ! V, defined by Y i ¼ i ðTi Y ÞT

ðz1  z2 Þðz3  z4 Þ ðz2  z3 Þðz4  z1 Þ

T2x

T1T2x

x

T1x Figure 5 Elementary quadrilaterals of the isothermic lattice.

Integrable Systems and Discrete Geometry

85

One can show that the cross-ratio is real if and only if the four points are cocircular or collinear. In particular, a harmonic quadrilateral with vertices numbered anticlockwise has cross-ratio equal to 1. Therefore, abusing the notation (it can be formalized using Clifford algebras), the isothermic lattice is defined by the condition

the tangent plane) field N : Z2 ! R3 via the discrete analog of the Lelieuvre formulas

qðx; T1 x; T1 T2 x; T2 xÞ ¼ 1

T1 T2 N þ N ¼ FðT1 N þ T2 NÞ

We remark that the definition of isothermic lattices can be slightly generalized allowing for the above cross-ratio to be a ratio of two real functions of single discrete variables. The restriction of the Ribaucour transformation to the class of isothermic lattices, named after Darboux who constructed it for isothermic surfaces, has as its data a real parameter and the starting point D(x0 ), and can be described as follows. Given the elementary quadrilateral {x, T1 x, T2 x, T1 T2 x} of the isothermic lattice, and given the point D(x), then the points D(T1 x) and D(T2 x) belong to the corresponding planes and are constructed from equations

1 x ¼ ðT1 NÞ  N;

It turns out that the point D(T1 T2 x), constructed by the application of the geometric integrability scheme, is such that the quadrilateral {D(x), D(T1 x), D(T2 x), D(T1 T2 x)} is harmonic. Moreover, the construction of the Darboux transformation is compatible; that is, the new side quadrilaterals have the correct cross-ratios and  . There are various integrable reductions of the isothermic lattice, for example, the constant mean curvature lattice and the minimal lattice. Asymptotic Lattices and Their Reductions

An asymptotic lattice is a mapping x : Z2 ! R 3 such that any point x of the lattice is coplanar with its four nearest neighbors T1 x, T2 x, T11 x, T21 x (see Figure 6). Such a plane is called the tangent plane of the asymptotic lattice in the point x. It can be shown that any asymptotic lattice x can be recovered from its suitably rescaled normal (to

T2x T1–1x

Figure 6 Asymptotic lattices.

½23

2

for some potential F : Z ! R. Given a scalar solution of the Moutard equation [23], a new solution M(N) of the Moutard equation, with the new potential MðFÞ ¼

ðT1 ÞðT2 Þ F ðT1 T2 Þ

can be found via the Moutard transformation equations MðT1 NÞ  N ¼

ðMðNÞ  T1 NÞ T1

½24

MðT2 NÞ  N ¼

ðMðNÞ  T2 NÞ T2

½25

Now, via the Lelieuvre formulas [22], one can construct a new asymptotic lattice M(x) = x  M(N)  N. The lines connecting corresponding points of the asymptotic lattices x and M(x) are tangent to both lattices. Such a Z2 -family of lines in R 3 is called Weingarten (or W for short) congruence. Notice that this is not a congruence as considered earlier. Various integrable reductions of asymptotic lattices are known in the literature: pseudospherical lattices, asymptotic Bianchi lattices and isothermally asymptotic (or Fubini–Ragazzi) lattices, and discrete (proper and improper) affine spheres. Formally, the Moutard transformation is a reduction of the (projective version of the) fundamental transformation for the Moutard reduction of the Laplace equation. However, the geometric relation between asymptotic lattices and QLs is more subtle and the geometric scenery of this connection is the line geometry of Plu¨cker. Straight lines in R3 P3 are considered there as points of the so-called Plu¨cker quadric QP P5 . A discrete asymptotic net in P3 , viewed as the envelope of its tangent planes, corresponds to a congruence of isotropic lines in QP , whose focal lattices represent the asymptotic directions. The discrete W-congruences are represented by twodimensional QLs in the Plu¨cker quadric. The Koenigs Lattice

x T2–1x

½22

By the compatibility of the Lelieuvre formulas, the normal field N satisfies the discrete Moutard equation

qðx; DðxÞ; DðT1 xÞ; T1 xÞ ¼ qðx; DðxÞ; DðT2 xÞ; T2 xÞ ¼ 

2 x ¼ N  ðT2 NÞ

T1x

A two-dimensional QL x : Z2 ! PM is called a Koenigs lattice if, for every point x of the lattice,

86

Integrable Systems and Discrete Geometry Discrete Two-Dimensional Schro¨dinger Equation

T 12 x –1

In the previous sections we have discussed examples of integrable discrete geometries described by equations of hyperbolic type. Below we present some results associated with the elliptic case; it is remarkable that the QL provides a way to connect these two subjects. Consider a solution N : Z2 ! R3 of the general selfadjoint five-point scheme on the star of the Z2 lattice

T1x –1 x –1 T1–1x

T2x x T 1x

T2–1x

x1 T2x1

aT1 N þ T11 ðaNÞ þ bT2 N þ T21 ðbNÞ  cN ¼ 0 ½27

T 22x1 Figure 7 The Koenigs lattice.

the six points x1 , Ti x1 , Ti2 x1 , i = 1, 2, of its Laplace transforms belong to a conic (see Figure 7). The nonlinear constraint in definition of the Koenigs lattice can be linearized, with the help of the Pascal ‘‘mystic hexagon’’ theorem, to the form that the line passing through x and T1 T2 x, the line passing through x1 and T12 x1 , and the line passing through x1 and T22 x1 intersect in a point. Algebraically, the geometric Koenigs lattice condition means that the Laplace equation of the lattice in homogeneous coordinates x : Z2 ! RMþ1 can be gauged into the form T1 T2 x þ x ¼ T1 ðFxÞ þ T2 ðFxÞ

½26

It turns out that, if N is a solution of the Moutard equation [23], then x = T1 N þ T2 N satisfies the Koenigs lattice equation. Therefore, the algebraic theory of the discrete Koenigs lattice equation [26], its (Koenigs) transformation, and the permutability of the superpositions of such transformations is based on the corresponding theory for the Moutard equation [23]. Geometrically, the Koenigs lattices are selected from the QLs as follows. Given a two-dimensional QL x : Z2 ! PM and given a congruence l with lines passing through the corresponding points of the lattice. Denote by yi = Ti1 l \ l, i = 1, 2, points of the focal lattices of the congruence. For every line l, denote by { the unique projective involution exchanging yi with Ti yi . If, for every congruence l, the lattice K(x) : Z2 ! PM , with points K(x) = {(x), is a QL, then the lattice x is a Koenigs lattice. The above construction gives also the corresponding reduction of the fundamental transformation. A distinguished reduction of the Koenigs lattice is the quadrilateral Bianchi lattice. The natural continuous limit of the corresponding equation is equivalent to the Bianchi (or hyperbolic Ernst) system describing the interaction of planar gravitational waves.

then the lattice x : Z2 ! R3 obtained by the Lelieuvre type formulas   1 x ¼  T21 b N  T21 N ½28   2 x ¼ T11 a N  T11 N is a QL having N as normal (to the planes of elementary quadrilaterals) vector field. The following gauge-equivalent form of eqn 27, namely      T1 þ T11 þ T2 T1  T1  T2      q ¼0 ½29 þ T21 T2  an integrable discretization of the Schro¨dinger equation @2 @2 þ  Q ¼0 @x21 @x22 is also the Lax operator associated with an integrable generalization of the Toda law to the square lattice. The five-point scheme [27] is also a distinguished illustrative example of the sublattice theory. Indeed, it can be obtained restricting to the even sublattice Z2e the discrete Cauchy–Riemann equations T1 T2  ¼ iGðT1  T2 Þ

½30

Because of the equivalence (on the discrete level!) between eqn [30] and the discrete Moutard equation [23], the five-point scheme [27] inherits integrability properties (Darboux-type transformations, superposition formulas, analytic methods of solution) from the corresponding (and simpler) integrability properties of the discrete Moutard equation.  See also: Ba¨cklund Transformations; @-Approach to Integrable Systems; Integrable Discrete Systems; Integrable Systems and Algebraic Geometry; Integrable Systems and the Inverse Scattering Method; Integrable Systems: Overview; Nonlinear Schro¨dinger Equations; Sine-Gordon Equation; Stability Theory and KAM; Toda Lattices.

Integrable Systems and Recursion Operators on Symplectic and Jacobi Manifolds

Further Reading Akhmetshin AA, Krichever IM, and Volvovski YS (1999) Discrete analogs of the Darboux–Egoroff metrics. Proceedings of the Steklov Institute of Mathematics 225: 16–39. Białecki M and Doliwa A (2005) Algebro-geometric solution of the discrete KP equation over a finite field out of a hyperelliptic curve. Communications in Mathematical Physics 253: 157–170. Bobenko AI (2004) Discrete differential geometry. Integrability as consistency. In: Grammaticos B, Kosmann–Schwarzbach Y, and Tamizhmani T (eds.) Discrete Integrable Systems, pp. 85–110. Berlin: Springer. Bobenko AI and Seiler R (eds.) (1999) Discrete Integrable Geometry and Physics. Oxford: Clarendon. Bogdanov LV and Konopelchenko BG (1995) Lattice and q-difference Darboux–Zakharov–Manakov systems via @ method. Journal of Physics A 28: L173–L178. Cies´lin´ski J (1997) The spectral interpretation of N-spaces of constant negative curvature immersed in R 2N1 . Physics Letters A 236: 425–430. Doliwa A, Grinevich PG, Nieszporski M, and Santini PM (2004) Integrable lattices and their sublattices: from the discrete Moutard (discrete Cauchy–Riemann) 4-point equation to the self-adjoint 5-point scheme, nlin.SI/0410046. Doliwa A, Man˜as M, Martı´nez Alonso L, Medina E, and Santini PM (1999) Charged free fermions, vertex operators and transformation theory of conjugate nets. Journal of Physics A 32: 1197–1216. Doliwa A, Nieszporski M, and Santini PM (2001) Asymptotic lattices and their integrable reductions. I. The Bianchi and the

87

Fubini–Ragazzi lattices. Journal of Physics A 34: 10423–10439. Doliwa A, Nieszporski M, and Santini PM (2004) Geometric discretization of the Bianchi system. Journal of Geometry and Physics 52: 217–240. Doliwa A and Santini PM (2000) The symmetric, D-invariant and Egorov reductions of the quadrilateral latice. Journal of Geometry and Physics 36: 60–102. Doliwa A, Santini PM, and Man˜as M (2000) Transformations of quadrilateral lattices. Journal of Mathematical Physics 41: 944–990. Klimczewski P, Nieszporski M, and Sym A (2000) Luigi Bianchi, Pasquale Calapso and solitons. Rend. Sem. Mat. Messina, Atti del Congresso Internazionale in Onore di Pasquale Calapso, Messina, 12–14 October 1998, pp. 223–240. Man˜as M (2001) Fundamental transformation for quadrilateral lattices: first potentials and -functions, symmetric and pseudo-Egorov reductions. Journal of Physics A 34: 10413–10421. Matsuura N and Urakawa H (2003) Discrete improper affine spheres. Journal of Geometry and Physics 45: 164–183. Rogers C and Schief WK (2002) Ba¨cklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory.Cambridge: Cambridge University Press. Schief WK (2003a) Lattice geometry of the discrete Darboux, KP, BKP and CKP equations. Menelaus’ and Carnot’s theorems. Journal of Nonlinear Mathematical Physics 10(suppl. 2): 194–208. Schief WK (2003b) On the unification of classical and novel integrable surfaces. II. Difference geometry. Proceedings of the Royal Society of London A 459: 373–391.

Integrable Systems and Recursion Operators on Symplectic and Jacobi Manifolds R Caseiro and J M Nunes da Costa, Universidade de Coimbra, Coimbra, Portugal ª 2006 Elsevier Ltd. All rights reserved.

fF; Gg! ¼ 0

Introduction Let (M, !) be a symplectic manifold of dimension 2n. We denote by ] the natural isomorphism between T M and TM, defined by the equation i] ! ¼ ;



2T M

We say that two smooth functions F, G : M ! R are in involution if

½1

We say that ] df is the Hamiltonian vector field defined by the Hamiltonian f : M ! R. Associated with the nondegenerated closed 2-form ! there is also a Poisson bracket on C1 (M), the space of real differentiable functions on M, defined by f:; :g! : C1 ðMÞ  C1 ðMÞ ! C1 ðMÞ ðf ; gÞ 7! ff ; gg! ¼ !ð] df ; ] dgÞ

½2

Suppose we have n independent smooth functions in involution H1 , . . . , Hn , such that the associated Hamiltonian vector fields X1 , . . . , Xn are complete on the level manifold Ma ¼ fx 2 M : Hj ðxÞ ¼ aj ; j ¼ 1; . . . ; ng

½3

The classical theorem of Arnol’d–Liouville states that 1. the submanifold Ma is invariant with respect to each one of the Hamiltonian commuting flows generated by H1 , . . . , Hn ; 2. every connected component of Ma is diffeomorphic to a product of a Euclidean space by a torus, Rnk  Tk ; 3. there exist coordinates f1 , . . . , fnk , ’1 , . . . , ’k in Ma such that the Hamiltonian systems in Ma , associated with the Hamiltonians Hj , have the form f_s ¼ cjs ’_ m ¼ ! jm

ð! ! ðaÞ; c ¼ const:Þ

½4

88

Integrable Systems and Recursion Operators on Symplectic and Jacobi Manifolds

4. if Ma is compact then it is diffeomorphic to Tn and there exists a neighborhood of Ma on M, symplectically diffeomorphic to Bn  Tn . A completely integrable Hamiltonian system is a Hamiltonian vector field X, that admits n integrals H1 , . . . , Hn satisfying the hypothesis of Arnol’d– Liouville theorem. It may happen that a system has more than n independent integrals of motion. In this case it is called superintegrable and not all the integrals are in involution. Supposing that Ma ¼ fx 2 M : Hj ðxÞ ¼ aj ; j ¼ 1; . . . ; n þ kg is compact and connected and that H1 , . . . , Hnk commute with all the n þ k integrals, then Ma is diffeomorphic to the torus Tn  k . In particular, if the system is maximally superintegrable, that is, k = n  1, Ma is diffeomorphic to T 1 = S1 and all the trajectories are closed. To prove that a system is completely integrable, we have to find a sufficient number of integrals of the system in involution. The Lax pair is an extremely powerful tool in this task, although it does not guarantee the involution of the integrals found. A Lax pair of a vector field X on a smooth manifold M is a pair of operators (L, M) such that L_ ¼ ½M; L ¼ ML  LM

½5

This equation is equivalent to U

1

LU ¼ L0

Uð0Þ ¼ I

Let X be a vector field on a smooth manifold M. A recursion operator of X is a (1, 1)-tensor R invariant of X: LX R ¼ 0

½8

The (1, 1)-tensors, and in particular the recursion operators, may be regarded as fiber endomorphisms of TM. So, given a (1, 1)-tensor R, we denote by t R : T  M ! T  M the transpose of R : TM ! TM, that is, ht RðÞ; Xi ¼ h; RðXÞi;

 2 T  M; X 2 TM

½6

½7

So, the eigenvalues of L are integrals of X. Notice that all the pairs (Lk , M), k 2 N, are Lax pairs of the system and we may conclude that the functions tr Lk , k 2 N, are integrals of X. The first goal of this article is to relate integrable Hamiltonian systems and recursion operators, where some of the most important properties of the latter are exhibited. Very naturally, the Poisson–Nijenhuis manifolds appear in this context and the Toda lattice is the example chosen in order to show the whole theory working in practice. Also, we see how recursion operators can help in the construction of quadratic algebras of integrals of motion and, in the last section, we present the generalization to Jacobi manifolds of the Nijenhuis structures defined for Poisson manifolds.

½9

where h. , .i denotes the canonical pairing between T  M and TM. Recursion operators also generate symmetries. If R is a recursion operator and Y is a symmetry of X, that is, [X, Y] = 0, then RY is also a symmetry of X. So, given a recursion operator R of X, we may construct a sequence of symmetries of X, Rk Y, k 2 N. The Nijenhuis torsion of a (1, 1)-tensor R is the (1, 2)-tensor T (R) defined by T ðRÞðX; YÞ ¼ ½RX; RY  Rð½X; RY þ ½RX; Y R½X; YÞ; X; Y 2 XðMÞ ½10 A Nijenhuis operator is a (1, 1)-tensor, R, with vanishing Nijenhuis torsion, that is, LRX R ¼ RLX R

where U is the solution operator of the Cauchy problem _ ¼ MU; U

Integrable Systems on Poisson–Nijenhuis Manifolds

½11

These operators can generate sequences of closed 1-forms. If R is a Nijenhuis operator and  is a closed 1-form such that dt R() = 0, then dt Rk () = 0, k 2 N. In the particular case of  being exact, that is,  = df and the first cohomology group being trivial, then we have a sequence of local integrals of motion dfk = t Rk (df ). A Nijenhuis recursion operator R and a symmetry Y of a vector field X lead to a sequence of commuting symmetries Rk Y, k 2 N, ½Ri Y; Rj Y ¼ 0;

i; j 2 N

½12

To define the integrability in terms of a (1, 1)tensor is of special relevance when we try to extend everything to the infinite-dimensional case. Notice that in coordinates (q1 , . . . , qn ), the condition [8] is equivalent to R_ ¼ ½A; R where A is the n  n matrix defined by  j @X Aij ¼ @qi

½13

Integrable Systems and Recursion Operators on Symplectic and Jacobi Manifolds

and Xj = X(qj ) = q_ j , j = 1, . . . , n. So, the pair (R, A) is a local Lax pair of the system and the eigenvalues of R are integrals of X. If a recursion operator R of a vector field X on a manifold M has vanishing Nijenhuis torsion and n doubly degenerated eigenvalues i , with nowherevanishing differentials, (di )p 6¼ 0, then X defines a completely integrable Hamiltonian system. Now suppose X defines a completely integrable Hamiltonian system with Hamiltonian H on a symplectic manifold (M, !). Let (I1 , . . . , In , ’1 , . . . , ’n ) be the action-angle variables in a neighborhood of an invariant torus. Two cases may happen: 1. The Hamiltonian H is separable in the action variable, that is, X H¼ Hk ðIk Þ ½14 k

In this case, the (1, 1)-tensor   X @ @ k ðIk Þ dIk  þ d’k  R¼ @Ik @’k k

 2  @ H det 6¼ 0 @Ik @Ij

½16

In this case we may define new coordinates k ¼

@H ; @Ik

k ¼ 1; . . . ; n

½17

and a new symplectic structure !1 ¼

X

dk ^ d’k ¼

k

X @2H dIk ^ d’j @Ik @Ij k;j

½18

The vector field X is Hamiltonian with respect to !1 , with Hamiltonian 1X 2  2 k k

½19

  @ @ k ðIk Þ dk  þ d’k  @k @’k

½20

H¼ and the (1, 1)-tensor R¼

X k

is a recursion operator of X.

Nijenhuis operators also allow the construction of master symmetries from conformal ones. A conformal symmetry of a tensor field T is a vector field Z such that LZ T ¼ T;

for some constant 

A master symmetry of a vector field X is a vector field Y such that ½X; ½X; Y ¼ 0;

but ½X; Y 6¼ 0

Let R be a recursion operator of X0 and Z0 be a conformal symmetry of X0 and R such that LZ0 X0 ¼ X0

and

LZ0 R ¼ R

½21

for some constants , . If R is also a Nijenhuis operator, then defining the sequences of commuting symmetries Xk = Rk X0 and of conformal symmetries Zk = Rk Z0 , k 2 N, we have, for all k, j 2 N0 , LZk R ¼ Rkþ1

½22

½Zk ; Zj  ¼ ðj  kÞZjþk

½23

½Zk ; Xj  ¼ ð þ jÞXkþj

½24

½15

where k are functions with nowhere-vanishing differentials, is a recursion operator of X, and has vanishing Nijenhuis torsion and doubly degenerated eigenvalues. 2. The Hamiltonian has nonvanishing Hessian

89

A bi-Hamiltonian manifold is a smooth manifold M endowed with two linearly independent Poisson tensors 0 , 1 , compatible in the sense that their Schouten bracket vanishes, [0 , 1 ] = 0. A vector field is said to be bi-Hamiltonian if it is Hamiltonian with respect to both Poisson structures. The equation that rules the flow of this vector field is said to be a bi-Hamiltonian system. When one of the Poisson structures is obtained from the other by means of a Nijenhuis operator, we obtain a Poisson–Nijenhuis manifold. Hence, a Poisson–Nijenhuis manifold is a differentiable manifold M endowed with a Poisson tensor  and a (1, 1)-tensor R such that R] ¼ ]t R;

½R;  ¼ 0 and ½R; R ¼ 0

A classical example is the one of a bi-Hamiltonian manifold (M, 0 , 1 ) where 0 is nondegenerated. In this case we may define the Nijenhuis operator R = ]1 ]1 and the manifold M is a Poisson– 0 Nijenhuis one. The characteristics of the Poisson–Nijenhuis manifold guarantee that all the bivectors k = Rk  are compatible Poisson tensors and the manifold is not just bi-Hamiltonian but multi-Hamiltonian. From what we saw, a Hamiltonian system is completely integrable if and only if it is bi-Hamiltonian

90

Integrable Systems and Recursion Operators on Symplectic and Jacobi Manifolds

in a neighborhood of an invariant torus with the eigenvalues of the existing recursion operator providing its complete integrability. These Poisson–Nijenhuis manifolds appear quite frequently in dynamics and allow us to obtain some interesting properties easily. We finish this section with the Toda lattice. This system is a good illustration of what has been said until now. Consider R2 n1 with coordinates (a1 , . . . , an1 , b1 , . . . , bn ) equipped with the following compatible Poisson tensors:   n1 1X @ @ @ 0 ¼ ai ^  4 i¼1 @ai @bi @biþ1

1 ¼

n1 X i¼1



a2i

n1 @ @ 1X @ ^  ai @b iþ1 @bi 4 i¼1 @ai

@ @ @ ^ aiþ1 þ 2biþ1  2bi @aiþ1 @biþ1 @bi

Consider the Flaschka transformation  : R2n ! R2n1 ðq1 ; . . . ; qn ; p1 ; . . . ; pn Þ 7! ða1 ; . . . ; an1 ; b1 ; . . . ; bn Þ where q  q 1 1 i iþ1 ai ¼ exp ; bj ¼  pj 2 2 2 i ¼ 1; . . . ; n  1; j ¼ 1; . . . ; n

This application is a Poisson morphism between e 0,  e 1 ) and (R2n1 , 0 , 1 ), where (R2n , 

½25

e0 ¼ 

e1 ¼ 



½29

n1 X

eqi qiþ1

i¼1

½26

þ

n X i¼1

n X @ @ ^ @p @q i i i¼1

@ @ ^ @piþ1 @pi

X @ @ @ @ pi ^ þ ^ @qi @pi @q @q j i i 0. In summary, in the ‘‘direct problem,’’ we have found particular solutions of eqn [5a] which are sectionally holomorphic:

Mðk; xÞ Nðk; xÞ



and

Mðk; xÞ Nðk; xÞ

The Inverse Problem

Equation [28] expresses N in terms of q. Is it possible to find an alternative expression for N in terms of some appropriate ‘‘spectral data’’? The answer is positive and is a direct consequence of the fact that eqn [29] defines the ‘‘jump condition’’ of an RH problem. Indeed, it can be shown that a(k) may have simple zeros k1 , . . . , kn in the positive imaginary axis of the k-complex plane. Hence, in general, M=a can be expressed in the form n X Aj ðxÞ Mðk; xÞ ¼ Mðk; xÞ þ ; aðkÞ k  ipj j¼1

pj > 0

where M(k, x) as a function of k is holomorphic for Im k > 0. It can also be shown that Aj (x) = Cj exp[2pj , x]N(kj , x). Hence eqn [29] becomes Mðk; xÞ  Nðk; xÞ n X Cj e2pj x Nðipj ; xÞ ¼ þ ðkÞe2ikx Nðk; xÞ; k 2 R k  ip j j¼1

½34

j¼1

Indeed, eqn [28] implies



are holomorphic for Im k > 0 and Im k < 0, respectively. These solutions, which are characterized in terms of q by eqns [27] and [28], are simply related by eqn [29].

½33

i lim Nðk; xÞ ¼ 1  2k k!1

Z

1

dqðÞ

x

Comparing this expression with the large-k behavior of eqn [33], we find [34]. Time Dependence of the Scattering Data

We now use eqn [5b] to compute the time dependence of the scattering data by evaluating eqn [5b] as x ! 1 we find  = 4ik3 . Then, evaluating it as x ! 1 and using

 aeikx þ beikx ;

x ! þ1

we find at ¼ 0;

bt ¼ 8ik3 b

Hence, aðt; kÞ ¼ að0; kÞ;

ðt; kÞ ¼ ð0; kÞe8ik

3

t

½35

Thus, pj ðtÞ ¼ pj ð0Þ;

3

Cj ðtÞ ¼ Cj ð0Þe8pj t

½36

The above formal results motivate the following definitions (for simplicity, we assume that a(k) has no zeros). Given a decaying real function

100 Integrable Systems and the Inverse Scattering Method

q0 (x), x 2 R, define M0 (k, x) as the solution of the linear Volterra integral equation i M0 ðk; xÞ ¼ 1 þ 2k Im k  0

Z

x

dð1 þ e2ikðxÞ qðÞM0 ðk; ÞÞ

1

Given M0 (k, x), define a0 (k) and b0 (k) by M0 ðk; xÞ ! a0 ðkÞ þ b0 ðkÞe2ikx ;

x ! 1;

k2R

Given a0 and b0 , define N(k, x, t) by the solution of the linear integral equation 1 Nðk; x; tÞ  2

Z

1

dl 1

b0 ðlÞ 8il3 tþ2ilx Nðl; x; tÞ ¼1 e a0 ðlÞ l þ k þ i0

A theorem of Gohberg and Krein implies that this equation has a unique global solution. Given a0 , b0 , N, define q(x, t) by qðx; tÞ ¼ 

1 @  @x

Z

1

1

dk

b0 ðkÞ 8ik3 tþ2ikx e Nðk; x; tÞ a0 ðkÞ

Then it can be shown that q(x, t) satisfies the KdV equation and q(x, 0) = q0 (x).

A Unification After the emergence of a method for solving the initial-value problem for nonlinear integrable evolution equations in one and two space variables, the most outstanding open problem in the analysis of these equations became the solution of initial boundary-value problems. A general approach for solving such problems for evolution equations in one space dimension was provided by Fokas (1997). This approach has already been used for the study of nonlinear integrable evolution PDEs on the half-line (Fokas 2002, 2005), on the interval, and in a timedependent domain. An important advantage of this new method is that it yields the formulation of a matrix RH problem (or a @ problem in the case of a convex time-dependent domain), which although has more complicated jump matrices than the analogous problem on the infinite line, it still has an explicit exponential (x, t) dependence. This fact allows one to describe effectively the asymptotic properties of the solution, using the powerful Deift–Zhou method (Deift and Zhou 1993). For example, the long-time asymptotics of boundary-value problems on the half line are discussed in Fokas and Its (1996). It is remarkable that the above results have motivated the discovery of a new method for solving

boundary-value problems, not only for linear evolution PDEs, but also for linear elliptic PDEs in two dimensions. This includes the Laplace, the biharmonic and the Helmholtz equations in a convex polygon (Dassios and Fokas 2005). In a most recent development, this method has also been applied to certain classes of linear PDEs with variable coefficients. This highly unexpected development unifies and extends several classical branches of mathematics. In particular, it unifies the classical transform methods for simple linear PDEs as well as the method of images, the treatment of linear PDEs via certain ingenious techniques such as the Wiener–Hopf technique, the formulation of Ehrenpreis type integral representations, and the solution of integrable nonlinear PDEs via the inverse-scattering transform. Furthermore, it extends these results to arbitrary domains and to certain classes of PDEs with variable coefficients. Regarding linear equations we note the following: Almost as soon as linear two-dimensional PDEs made their appearance, d’Alembert and Euler discovered a general approach for constructing large classes of their solutions. This approach involved separating variables and superimposing solutions of the resulting ODEs. The method of separation of variables naturally led to the solution of PDEs by a transform pair. The prototypical such pair is the direct and the inverse Fourier transforms; variations of this fundamental transform include the Laplace, Mellin, sine, cosine transforms, and their discrete analogs. The proper transform for a given boundary-value problem is specified by the PDE, by the domain, and by the given boundary conditions. For some simple boundary-value problems, there exists an algorithmic procedure for deriving the associated transform. This procedure involves constructing the Green’s function of a single eigenvalue equation, and integrating this Green’s function in the k-complex plane, where k denotes the eigenvalue. The transform method has been enormously successful for solving a great variety of initial- and boundary-value problems. However, for sufficiently complicated problems the classical transform method fails. For example, there does not exist a proper analog of the sine transform for solving a third-order evolution equation on the half-line. Similarly, there do not exist proper transforms for solving boundary-value problems for elliptic equations even of second order and in simple domains. The failure of the transform method led to the development of several ingenious but ad hoc techniques, which include: conformal mappings for the Laplace and the biharmonic equations; the Jones method and the formulation of the Wiener–Hopf factorization problem; the use of some integral representation, such as that of Sommerfeld; the

Integrable Systems and the Inverse Scattering Method

formulation of a difference equation, such as the Malyuzhinet’s equation. The use of these techniques has led to the solution of several classical problems in acoustics, diffraction, electromagnetism, fluid mechanics, etc. The Wiener–Hopf technique played a central role in the solution of many of these problems. A crucial role in the new method is played by the global equation satisfied by the boundary values of q and of its derivatives. For evolution equations and for elliptic equations with simple boundary conditions, this involves the solution of a system of algebraic equations, while for elliptic equations with arbitrary boundary conditions, it involves the solution of an RH problem. For simple polygons, this RH problem is formulated on the infinite line, thus it is equivalent to a Wiener–Hopf problem. This explains the central role played by the Wiener–Hopf technique in many earlier works. For linear PDEs, the explicit x1 , x2 dependence of q(x1 , x2 ) is consistent with the Ehrenpreis formulation of the solution. Thus, this method provides the concrete implementation as well as the generalization to concave domains of this fundamental principle. For nonlinear equations, it provides the extension of the Ehrenpreis principle to integrable nonlinear PDEs. See also: Boundary value Problems for Integrable  Equations; @-Approach to Integrable Systems; Integrable Systems and Algebraic Geometry; Integrable Discrete Systems; Integrable Systems and Discrete Geometry; Integrable Systems in Random Matrix Theory; Integrable Systems: Overview; Korteweg–de Vries Equation and Other Modulation Equations; Partial Differential Equations: Some Examples; Riemann–Hilbert Methods in Integrable Systems; Sine-Gordon Equation; Toda lattices; Twistor Theory: Some Applications [in Integrable Systems, Complex Geometry and String Theory].

Further Reading Ablowitz MJ and Segur H (1977) Exact linearization of a Painleve´ transcendent. Physical Review Letters 38: 1103–1106. Ablowitz MJ, Yaakov DB, and Fokas AS (1983) On the inverse scattering transform for the Kadomtsev–Petviashvili equation. Studies in Applied Mathematics 69: 135–142. Beals R and Coifman RR (1982) Scattering, transformations spectrales, et equations d’evolution nonlineaire. I. In: Seminaire

101

Goulaouic–Meyer–Schwartz, Expose` 21. E´cole Polytechnique, Palaiseau. Calogero F (1991) In: Zakharov VE (ed.) What Is Integrability?, Springer. Dassios G and Fokas AS (2005) The basic elliptic equations in an equilateral triangle. Proceedings of the Royal Society of London, Series A 461: 2721–2748. Deift PA and Zhou X (1993) A steepest descent method for oscillatory Riemann–Hilbert problems. Annals of Mathematics 137: 245–338. Flaschka H and Newell AC (1980) Monodromy and spectrumpreserving deformation I. Communications in Mathematical Physics 76: 65–116. Fokas AS (1997) A unified transform method for solving linear and certain nonlinear PDE’s. Proceedings of the Royal Society of London, Series A 453: 1411–1443. Fokas AS (2002) Integrable nonlinear evolution equations on the half-line. Communications in Mathematical Physics 230: 1–39. Fokas AS (2005) A generalised Dirichlet to Neumann map for certain nonlinear evolution PDEs. Communications on Pure and Applied Mathematics 58: 639–670. Fokas AS and Ablowitz MJ (1983a) On the initial value problem of the second Painleve´ transcendent. Communications in Mathematical Physics 91: 381–403. Fokas AS and Ablowitz MJ (1983b) On the inverse scattering of the time dependent Schro¨dinger equation and the associated KPI equation. Studies in Applied Mathematics 69: 211–228. Fokas AS and Its AR (1996) The linearization of the initialboundary value problem of the nonlinear Schro¨dinger equation. SIAM Journal on Mathematical Analysis 27: 738–764. Fokas AS and Santini PM (1990) Dromions and a boundary value problem for the Davey–Stewartson I equation. Physica D 44: 99–130. Fokas AS and Zhou X (1992) On the solvability of Painleve´ II and IV. Communications in Mathematical Physics 144: 601–622. Fokas AS, Its AR, Kapaev AA, and Novokshenov VY (2006) Painleve´ Transcendents: The Riemann–Hilbert Approach. Providence, RI: American Mathematical Society. Gardner CS, Greene JM, Kruskal MD, and Miura RM (1967) Method for solving the Korteweg–de Vries equation. Physical Review Letters 19: 1095–1097. Hietarinta J (2002) Scattering of solitons and dromions. In: Sabatier P and Pike E (eds.) Scattering. San Diego: Academic Press. Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Communications on Pure and Applied Mathematics 21: 467–490. Zakharov V and Shabat A (1972) Exact theory of twodimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics – JEPT 34: 62–69.

102 Integrable Systems in Random Matrix Theory

Integrable Systems in Random Matrix Theory C A Tracy, University of California at Davis, Davis, CA, USA H Widom, University of California at Santa Cruz, Santa Cruz, CA, USA ª 2006 Higher Education Press. Published by Elsevier Ltd. All rights reserved. An earlier version of this article was originally published as ‘‘Distribution functions for largest eigenvalues and their applications’’. Proceedings of the International Congress of Mathematics, Volume 1 (2002), pp. 587–596. Beijing, China: Higher Education Press, with permission.

exist and are given explicitly by   F2 ðsÞ ¼ det I  KAiry  Z 1  ¼ exp  ðx  sÞq2 ðxÞ dx s

where 0 0 : AiðxÞAi ðyÞ  Ai ðxÞAiðyÞ KAiry ¼ xy

Random Matrix Models

acting on L2 ðs; 1ÞðAiry kernelÞ

A random matrix model is a probability space (, P, F ) where the sample space  is a set of matrices. There are three classic finite N random matrix models (see, e.g., Mehta (1991)): 1. Gaussian orthogonal ensemble ( = 1): (a)  = N  N real symmetric matrices; (b) P = ‘‘unique’’ measure that is invariant under orthogonal transformations and the matrix elements are i.i.d. random variables; explicitly, the density is   cN exp trðA2 Þ dA

½1

whereQcN is Q a normalization constant and dA = i dAii i0

and subsequently a large class of potentials V was analyzed by Deift et al. (1999). These analyses require proving new Plancherel–Rotach type formulas for nonclassical orthogonal polynomials. The proofs use Riemann–Hilbert methods. It was shown that the generic behavior is GUE; hence, the limit law for the largest eigenvalue is F2 . However, by finely tuning the potential new universality classes will emerge at the edge of the spectrum. For  = 1, 4 a universality theorem was proved by Stojanovic (2000) for the quartic potential. In the case of noninvariant measures, Soshnikov (1999) proved that for real symmetric Wigner matrices

A major breakthrough occurred with the work of Baik, Deift, and Johansson (see Baik et al. (2000) and references therein) when they proved that the limiting distribution of the length of the longest increasing subsequence in a random permutation is F2 . Precisely, if ‘N () is the length of the longest increasing subsequence in the permutation  2 SN , then ! pffiffiffiffiffi ‘N  2 N < s ! F2 ðsÞ P N 1=6 as N ! 1. Here the probability measure on the permutation group SN is the uniform measure. Further discussion of this result can be found in Johansson (2000b). Baik and Rains (2001) showed by restricting the set of permutations (and these restrictions have natural symmetry interpretations) that F1 and F4 also appear. Even the distributions F12 and F22 (Tracy and Widom 1999) arise. By the Robinson–Schensted–Knuth correspondence, the Baik–Deift–Johansson result is equivalent to the limiting distribution on the number of boxes in the first row of random standard Young tableaux. (The measure is the push-forward of the uniform measure on SN .) These same authors conjectured that the limiting distributions of the number of boxes in the second, third, etc., rows were the same as the limiting distributions of the next-largest, next-next-largest, etc., eigenvalues in GUE. Since these eigenvalue distributions were also found in Tracy and Widom (1996), they were able to compare the then unpublished numerical work of Odlyzko and Rains (2000) with the predicted results of random matrix theory. Subsequently, Baik et al. (2000) proved the conjecture for the second row. The full conjecture was proved by Okounkov (2000) using topological methods and by,

104 Integrable Systems in Random Matrix Theory

among others, Johansson (2001) using analytical methods. For an interpretation of the Baik–Deift– Johansson result in terms of the card game patience sorting, see the very readable review paper by Aldous and Diaconis (1999). Growth Processes

Growth processes have an extensive history both in the probability literature and the physics literature (see, e.g., Meakin (1998) and references therein), but it was only recently that Johansson (2002b) proved that the fluctuations about the limiting shape in a certain growth model (‘‘corner growth model’’) are F2 . Johansson further pointed out that certain symmetry constraints (inspired from the Baik and Rains (2001) work) lead to F1 fluctuations (see Growth Processes in Random Matrix Theory). Subsequently, Baik and Rains (2000) and Gravner et al. (2002) have shown the same distribution functions appearing in closely related lattice growth models. Pra¨hofer and Spohn (2000) reinterpreted the work of Baik et al. in terms of the physicists’ polynuclear growth (PNG) model thereby clarifying the role of the symmetry parameter . For example,  = 2 describes growth from a single droplet, whereas  = 1 describes growth from a flat substrate. They also related the distribution functions F to fluctuations of the height function in the KPZ equation (Kardar et al. 1986, Meakin 1998). (The connection with the KPZ equation is heuristic.) Thus, one expects on physical grounds that the fluctuations of any growth process falling into the 1 þ 1KPZ universality class will be described by the distribution functions F or one of the generalizations by Baik and Rains (2000). Such a physical conjecture can be tested experimentally. Earlier Myllys et al. established experimentally that a slow, flameless burning process in a random medium (paper!) is in the 1 þ 1KPZ universality class. This sequence of events is a rare instance in which new results in mathematics inspire new experiments in physics. In the context of the PNG model, Pra¨hofer and Spohn have given a process interpretation, the Airy process, of F2 . There is an extension of the growth model in Gravner et al. (2002) to growth in a random environment. In Gravner et al. (2002) the following model of interface growth in two dimensions is considered by introducing a height function on the sites of a one-dimensional integer lattice with the following update rule: the height above the site x increases to the height above x  1, if the latter height is larger; otherwise, the height above x increases by 1 with probability px . It is assumed that the px are chosen independently at random with

a common distribution function F, and that the initial state is such that the origin is far above the other sites. In the pure regime, Gravner–Tracy–Widom identify an asymptotic shape and prove that the fluctuations about that shape, normalized by the square root of the time, are asymptotically normal. This contrasts with the quenched version: conditioned on the environment and normalized by the cube root of time, the fluctuations almost surely approach the distribution function F2 . We mention that these same authors find, under some conditions on F at the right edge, a composite regime where now the interface fluctuations are governed by the extremal statistics of px in the annealed case while the fluctuations are asymptotically normal in the quenched case. Random Tilings

The Aztec diamond of order n is a tiling by dominoes of the lattice squares [m, m þ 1]  [‘, ‘ þ 1], m, n 2 Z, that lie inside the region {(x, y) : jxj þ jyj  n þ 1}. A domino is a closed 1  2 or 2  1 rectangle in R 2 with corners in Z2 . A typical tiling is shown in Figure 2. One observes that near the center the tiling appears random, called the temperate zone, whereas near the edges the tiling is frozen, called the polar zones. As n ! 1 the boundary between the temperate zone and the polar zones (appropriately scaled) converges to a circle (‘‘arctic circle theorem’’). Johansson (2002a) proved that the fluctuations about this limiting circle are F2 . Statistics

Johnstone (2001) considers the largest principal component of the covariance matrix Xt X where X is an n  p data matrix all of whose entries are independent standard Gaussian variables and proves that for appropriate centering and scaling, the limiting distribution equals F1 in the limit n, p ! 1 with n=p !  2 Rþ . Soshnikov has removed the Gaussian assumption but requires that n  p = O(p1=3 ). Thus, we can anticipate applications of the distributions F (and particularly F1 ) to the statistical analysis of large data sets.

Figure 2 Random tilings.

Integrable Systems in Random Matrix Theory

105

Queuing Theory

Further Reading

Glynn and Whitt (1991) consider a series of n singleserver queues each with unlimited waiting space with a first-in and first-out service. Service times are i.i.d. with mean one and variance 2 with distribution V. The quantity of interest is D(k, n), the departure time of customer k (the last customer to be served) from the last queue n. For a fixed number of customers, k, they prove that

Aldous D and Diaconis P (1999) Longest increasing subsequences: from patience sorting to the Baik–Deift–Johansson theory. Bulletin of American Mathematical Society 36: 413–432. Baik J, Deift P, and Johansson K (2000) On the distribution of the length of the second row of a Young diagram under Plancherel measure. Geometry of Functional Analysis 10: 702–731. Baik J and Rains EM (2000) Limiting distributions for a polynuclear growth model. Journal of Statistical Physics 100: 523–541. Baik J and Rains EM (2001) Symmetrized random permutations. In: Bleher P and Its A (eds.) Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publications, vol. 40, pp. 1–19. Cambridge: Cambridge University Press. Bleher P and Its A (1999) Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and universality in the matrix model. Annals of Mathematics 150: 185–266. Deift P, Kriecherbauer T, McLauglin K.T-R, Venakides S, and Zhou X (1999) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weight and applications to universality questions in random matrix theory. Communications on Pure and Applied Mathematics 52: 1335–1425. Glynn PW and Whitt W (1991) Departure times from many queues. Annals of Applied Probability 1: 546–572. Gravner J, Tracy CA, and Widom H (2002) A growth model in a random environment. Annals of Probability 30: 1340–1368. Jimbo M, Miwa T, Moˆri Y, and Sato M (1980) Density matrix of an impenetrable Bose gas and the fifth Painleve´ transcendent. Physica D 1: 80–158. Johansson K (2000) Shape fluctuations and random matrices. Communications in Mathematical Physics 209: 437–476. Johansson K (2001) Discrete orthogonal polynomial ensembles and the Plancherel measure. Annals of Mathematics 153: 259–296. Johansson K (2002a) Non-intersecting paths, random tilings and random matrices. Probability Theory and Related Fields 123: 225–280. Johansson K (2002b) Toeplitz determinants, random growth and determinantal processes. In: Proceedings of the ICM, Beijing, ICM vol. 3, pp. 53–62. Johnstone I (2001) On the distribution of the largest principal component. Annals of Statistics 29: 295–327. Kardar M, Parisi G, and Zhang Y-C (1986) Dynamical scaling of growing interfaces. Physical Review Letters 56: 889–892. Meakin P (1998) Fractals, Scaling and Growth Far from Equilibrium. Cambridge: Cambridge University Press. Mehta ML (1991) Random Matrices, 2nd edn. Academic Press. Myllys M, Maunuksela J, Alava M, Ala-Nissila T, Merikoski J, and Timonen J Kinetic roughening in slow combustion of paper. Physical Review E 64: 036101-1–036101-12. Odlyzko AM and Rains EM On the longest increasing subsequences in random permutations. In: Grinberg EL, Berhanu S, Knopp M, Mendoza G, and Quinto ET (eds.) Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis, American Mathematical Society. pp. 439–451. O’Connell N and Yor M (2002) A representation for non-colliding random walks. Electronic Communications in Probability 7: 1–12. Okounkov A (2000) Random matrices and random permutations. International Mathematics Research Notices 20: 1043–1095. Porter CE (1965) Statistical Theories of Spectra: Fluctuations. Academic Press. Pra¨hofer M and Spohn H (2000) Universal distributions for growth processes in 1 þ 1 dimensions and random matrices. Physical Review Letters 84: 4882–4885. Soshnikov A (1999) Universality at the edge of the spectrum in Wigner random matrices. Communications in Mathematical Physics 207: 697–733.

Dðk; nÞ  n pffiffiffi  n ^k converges in distribution to a certain functional D of k-dimensional Brownian motion. They show that ^ k is independent of the service time distribution V. D It was shown in, for example, Gravner et al. (2002) ^ k is equal in distribution to the largest that D eigenvalue of a k  k GUE random matrix. This fascinating connection has been greatly clarified in recent work of O’Connell and Yor (2002). From Johansson (2002), it follows for V Poisson that   Dðbxnc; nÞ  c1 n P < s ! F2 ðsÞ c2 n1=3 as n ! 1 for some explicitly known constants c1 and c2 (depending upon x). Superconductors

Vavilov et al. (2001) have conjectured (based upon certain physical assumptions supported by numerical work) that the fluctuation of the excitation gap in a metal grain or quantum dot induced by the proximity to a superconductor is described by F1 for zero magnetic field and by F2 for nonzero magnetic field. They conclude their paper with the remark: The universality of our prediction should offer ample opportunities for experimental observation.

Acknowledgments This work was supported by the National Science Foundation through grants DMS-9802122 and DMS-9732687. See also: Determinantal Random Fields; Growth Processes in Random Matrix Theory; Integrable Systems and Algebraic Geometry; Integrable Systems and the Inverse Scattering Method; Integrable Systems: Overview; Quantum Calogero–Moser Systems; Random Partitions; Random Matrix Theory in Physics; Symmetry Classes in Random Matrix Theory; Toeplitz Determinants and Statistical Mechanics.

106 Integrable Systems: Overview Stojanovic A (2000) Universality in orthogonal and symplectic invariant matrix models with quartic potential. Mathematical Physics Analysis and Geometry 3: 339–373. Tracy CA and Widom H (1994) Fredholm determinants, differential equations and matrix models. Communications in Mathematical Physics 163: 151–174. Tracy CA and Widom H (1996) On orthogonal and symplectic matrix ensembles. Communications in Mathematical Physics 177: 727–754. Tracy CA and Widom H (1999) Random unitary matrices, permutations and Painleve´. Communications in Mathematical Physics 207: 665–685.

van Moerbeke P (2001) Integrable lattices: random matrices and random permutations. In: Blcher P and Its A (eds.) Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publications vol. 40, pp. 321–406. Cambridge: Cambridge Univ. Press. Vavilov MG, Brouwer PW, Ambegaokar V, and Beenakker CWJ (2001) Universal gap fluctuations in the superconductor proximity effect. Physical Review Letters 86: 874–877. Wu TT, McCoy BM, Tracy CA, and Barouch E (1976) Spin–spin correlation functions of the two-dimensional Ising model: exact theory in the scaling region. Physical Review B 13: 316–374.

Integrable Systems: Overview Francesco Calogero, University of Rome, Rome, Italy and Institute Nazionale di Fisica Nucleare, Rome, Italy

we shall consider is the N-body problem characterized by the Newtonian equations of motion

ª 2006 Elsevier Ltd. All rights reserved.

€ n ¼  !2 qn q þ 2g2

Introduction

ðqn  qm Þ3 ; n ¼ 1; 2; . . . N

½2

m¼1; m6¼n

This section introduces some elementary notions and sets the (mathematically low brow) tone of this presentation. A dynamical system is characterized by an evolution equation the general structure of which reads Qt ¼ F

½1

Here Q  Qðx, tÞ is the dependent variable, and it might be a scalar, a vector, a matrix, you name it. The focus of interest is on its evolution as function of the (real, scalar) ‘‘time’’ variable t. The a priori unknown quantity Q might moreover depend on another independent ‘‘space’’ variable (scalar or vector) x, Q  Qðx, tÞ. The appended variable t in the left-hand side of the above equation denotes partial differentiation, and this notation will be used throughout, although when t is the only independent variable differentiation with respect to it might be instead denoted by a superimposed dot:

Qt 

N X

@Qðx; tÞ @Qðx; tÞ _ dQðtÞ ; Qx  ;Q @t @x dt

The quantity in the right-hand side of the evolution equation (1), which has of course the same (scalar, vector, matrix) character as Q, is an assigned function of t, x and Q, F  ðx, t, QÞ (more generally, its dependence on Q might be functional, see below). A typical example of the dynamical systems

where the dependent variable is the N-vector ~ q ðq1 , . . . qN Þ, the components of which are the ‘‘particle coordinates’’ qn  qn ðtÞ. Note however that these equations of motion are of second-order in time (contrary to (1)); but they can of course be reformulated as first-order ODEs indeed their Hamiltonian version, derived in the standard manner from the Hamiltonian H¼

N   1X p2n þ !2 q2n 2 n¼1 N g2 X ðqn  qm Þ2 þ 2 m; n¼1;m6¼n

½3a

reads q_ n ¼ pn

½3b

p_ n ¼ !2 qn þ 2g2

N X

ðqn  qm Þ3 ; n ¼ 1; 2; . . . N

½3c

m¼1; m6¼n

Other typical examples are the (‘‘Korteweg-de Vries’’, ‘‘Burgers’’, ‘‘Nonlinear Schro¨dinger’’, ‘‘sine Gordon’’) PDEs satisfied by the scalar dependent variable q  qðx, tÞ,   ½4 qt ¼ qxxx þ 2qx q ¼ qxx þ q2 x   qt ¼ qxx þ 2qx q ¼ qx þ q2 x ½5 h i qt ¼ i qxx þ sjqj2 q ; s ¼  ½6 qt  qx ¼ s; st þ sx ¼ sin q

½7

Integrable Systems: Overview

as well as the integrodifferential (‘‘Benjamin–Ono’’) equation Z 1 qyy ðyÞ þ qx q qt ¼ P dy ½8 xy 1 and the (‘‘Kadomtsev–Petviashvili’’) PDE satisfied by the scalar dependent variable q  qðx, y, tÞ, qtx ¼ ðqxxx þ qx qÞx þ sqyy ; s ¼ 

½9

This last equation should of course be reformulated as an integrodifferential equation to fit with (1). These are all examples of integrable systems (see below). In this presentation we restrict attention to dynamical systems of these general types, without considering evolutions in which the space variable, and/or the time variable, and/or the dependent variable, only take discrete values, forsaking thereby the discussion of discrete evolution equations, cellular automata and functional equations, see other entries of this Encyclopedia. We shall consider mainly the ‘‘initial-value problem’’ in which the solution is assigned at the initial time, say at t = 0, Qðx; 0Þ ¼ Q0 ðxÞ and the subsequent evolution of the dependent variable, namely the values taken by Qðx, tÞ for t > 0, is the focus of attention. Note however that, except when there is no dependence at all on the space variable x (see for instance (2)), the functional class to which Qðx, tÞ belongs as regards its x-dependence should be specified (and the assigned initial-value Q0 ðxÞ should of course belong to this functional class). A typical class of functions are those vanishing (adequately fast) at (spatial) infinity; another typical class are those characterized by periodicity properties as functions of x; and still another class are those restricted to a finite spatial domain (for instance, the positive x-axis, x > 0, or a finite interval, a  x  bÞ, in which cases the initialvalue problem must be supplemented by assigning boundary conditions. These latter class of problems, called initial/boundary-value problems, are generally more difficult; even the identification of which boundary conditions are adequate to identify uniquely the solution may be a nontrivial task. In the following we will always focus on the simpler class of problems characterized by solutions defined in the entire space region and vanishing (sufficiently fast) asymptotically (far away). Thus, in the spirit of the initial-value problem, a dynamical system is generally characterized by assigning its evolution equation, the functional class to which its solutions are required to belong, and possibly in addition some (additional) restriction on the set of initial data.

107

Let us finally mention that, aside from considering the initial-value problem, the study of dynamical systems may focus on the identification of special (classes of) solutions, for instance those obtained by using symmetry properties of the evolution equation under consideration (yielding, say, ‘‘similarity solutions’’), and, in the integrable case, ‘‘solitonic’’ and ‘‘multisolitonic’’ solutions (see below).

Integrable dynamical systems The solution of a dynamical system, however simple the equation that defines its time evolution, see (1), may be extremely complicated, indeed its time-dependence might feature one or more of the characteristics of deterministic chaos, such as a sensitive dependence on the initial data. But there are ‘‘exceptional’’ dynamical systems, the behavior of which is instead, in some sense, simple. Such systems are termed – in the least technical sense of the word – ‘‘integrable’’. This characterization can be made precise for Hamiltonian systems with a finite number N of degrees of freedom, the equations of motion of which read     @H ~ p;~ q @H ~ p;~ q q_ n ¼ ; p_ n ¼  ; n ¼ 1;. . . N @pn @qn Such a system is integrable if there exist,   in addition to the Hamiltonian H ~ p, ~ q  H ð1Þ ~ p, ~ q itself, N  1 other (nontrivial and functionally indepen  dent) constants of motion H ðmÞ ~ p,~ q in involution, namely such that their Poisson brackets vanish: n o H ðnÞ ; H ðmÞ "     N X @H ðnÞ ~ p; ~ q @H ðmÞ ~ p; ~ q  @q‘ @p‘ ‘¼1    # @H ðmÞ ~ p;~ q @H ðnÞ ~ p;~ q  ¼ 0; @q‘ @p‘ n; m ¼ 1; . . . ; N Let us however emphasize the crucial role of the words ‘‘there exist’’, as used just above. For definiteness  let  us require that the constants of motion H ðnÞ ~ p, ~ q be analytic functions of their 2N arguments, and not excessively multivalued: they might feature some branch points, but not so many to vanify their effectiveness in constraining the time evolution of the dynamical variables qn ðtÞ, pn ðtÞ sufficiently to avoid their behavior from being too complicated. On the other hand it is of course not necessary that these functions H ðnÞ ~ p, ~ q be explicitly known. When these conditions hold it is in principle possible (‘‘Liouville theorem’’) to identify a

108 Integrable Systems: Overview

canonical transformation from the canonical coordinates and momenta qn and pn to action-angle variables n and In such that   In ¼ H ðnÞ ~ p;~ q ½10 Then these action variables evolve trivially, In ðtÞ ¼ In ð0Þ; n ðtÞ ¼ n ð0Þ þ In ð0Þt; n ¼ 1; . . . N Note that, once these new canonical variables are identified, the solution of the initial-value problem for the original Hamiltonian problem is provided directly by the expressions of the action-angle variables n and In in terms of the original variables qn and pn , as well as the expressions of the latter in terms of the former. The second step of this procedure requires inverting the expressions (10), and the corresponding expressions of the angle variables n in terms of the original variables qn and pn ; a necessary condition in order that this step allow to identify uniquely, at least in principle, the original canonical variables qn and pn in terms of the action-angle variables In and n – hence imply a simple time-evolution of these original variables – is the requirement, as mentioned above, that the expressions of the constants of the motion  H ðnÞ ~ p, ~ q in terms of their arguments qn and pn not be excessively multivalued. The statements outlined above can be rigorously formulated for finite-dimensional Hamiltonian systems, and they can be heuristically extended to all analogous dynamical systems with a finite number of degrees of freedom, even if they are not Hamiltonian. A system with N degrees of freedom might possess more than N constants of motion. Such a system that possesses 2N  1 (nontrivial and functionally independent) constants of motion (the maximal number, to avoid the evolution being frozen) is called superintegrable, and its evolution is in some sense analogous to that of a system with a single degree of freedom, in particular all its confined and nonsingular motions are then completely periodic, qn ðt þ TÞ ¼ qn ðtÞ; pn ðt þ TÞ ¼ pn ðtÞ; n ¼ 1; . . . ; N The period T depends generally on the initial data. If it does not, at least for an open set of such data having full dimensionality in phase space, the system is called isochronous: all its motions in that phase space region are then completely periodic with the same period. A dynamical system might be integrable in a region of its ‘‘natural’’ phase space, and nonintegrable in another region. Sometimes such systems are referred to as partially integrable. There even are systems which are isochronous (hence superintegrable) in a region of their phase space, and behave instead chaotically in another region. These regions are generally separated by boundaries where the evolution of the system runs

into singularities, and the constants of motion associated with the integrable behavior become excessively multivalued in the regions where the behavior is chaotic. (see Isochronous Systems). Dynamical systems featuring an additional space variable x (see Section 1) can be interpreted as infinitedimensional dynamical systems (by considering the variable x as a continuous label for the dependent variable Q). Accordingly, a necessary condition in order that such systems be considered integrable is the requirement that they possess an infinite number of constants of the motion. But – even for such systems that allow a Hamiltonian formulation – this condition cannot be considered sufficient (due to the inherent ambiguities in the counting of infinities), and in fact a completely cogent, universally accepted definition of integrability for infinite-dimensional dynamical systems is still lacking (various definitions can of course be given in special contexts). It is nevertheless rather well understood by practitioners what is meant by such a term at least for integrable equations such as those indicated at the end of the previous section, which generally give rise to the solitonic phenomenology – as explained below. The study of integrable systems has an illustrious history, to which many eminent mathematicians and mathematical physicists contributed after the Newtonian revolution: Euler, Jacobi, Poincare´, Painleve´, Kowalewskaya, Kolmogorov, Moser . . . Below we report – most tersely – on the bloom that this topic has witnessed over the last 3–4 decades, without being generally able, due to space constraints, to attribute the appropriate credit to the many colleagues, most of them still living, who contributed to this endeavor. For more detailed treatments of the topics outlined below, of related developments not mentioned here, and of such credits, the interested reader is referred to the bibliography given below, including the additional references traceable from there. Integrable many-body problems

An important class of integrable dynamical systems is provided by N-body problems characterized by Hamiltonians such as N   1X H ~ p; ~ q ¼ p2 þ Vð~ qÞ 2 n¼1 n

½11

with a potential energy Vð~ qÞ that includes ‘‘external’’ and ‘‘two-body’’ forces, Vð~ qÞ ¼

N X

V ð1Þ ðqn Þ þ

n¼1

V ð2Þ ðqÞ ¼ V ð2Þ ðqÞ

N 1 X V ð2Þ ðqn  qm Þ; 2 m; n¼1;m6¼n

½12

Integrable Systems: Overview

The corresponding Hamiltonian and Newtonian equations of motion read q_ n ¼ pn ; p_ n ¼  €n ¼  q

N X @V ð1Þ ðqn Þ @V ð2Þ ðqn  qm Þ  ; @qn @qn m¼1; m6¼n

N X @V ð1Þ ðqn Þ @V ð2Þ ðqn  qm Þ  @qn @ ¼ qn m¼1; m6¼n

½13

½14

be equivalent to the Hamiltonian equations of motion (13). Here and throughout the notation [A, B] denotes the commutator: ½A; B  A B  B A Because this matrix equation clearly entails that the N traces Tn ¼ trace ½Ln ; n ¼ 1; . . . ; N are constants of the motion, T_ n ¼ 0; n ¼ 1; . . . ; N the possibility to write the Hamiltonian equations (13) in the Lax form (14) yields as a bonus N constants of the motion, namely it entails that the Hamiltonian system under consideration is integrable. (One must moreover show that these constants of motion are in involution; this is usually the case). Hence a route to identify integrable N-body problems is via the search of Lax pairs L, M of matrices such that (14) correspond to (13), with an appropriate assignment of the potential energy (12). For N > 2 this is a nontrivial task, because (13) is a system of 2N ODEs in 2N unknowns, while the matrix Lax equation (14) amounts to a system of N2 ODEs. Functional equations and the identification of integrable many-body problems A convenient ansatz to identify a Lax pair suitable for the purpose outlined above reads as follows: Lnm ¼ pn for n ¼ m; Lnm ¼ ðqn  qm Þ for m 6¼ n; Mnm ¼

N X

may be assigned so that the corresponding Lax equation (14) be equivalent to the Hamiltonian equations (13) with V ð1Þ ðqÞ ¼ 0

½15a

V ð2Þ ðqÞ ¼ ðqÞðqÞ

½15b

provided the function ðxÞ satisfies the functional equation

The Lax pair and the constants of motion   Suppose ~ that two N  N matrices L  L p ,~ q and M    M~ p,~ q could be found such that the matrix ‘‘Lax equation’’ L_ ¼ ½L; M

109

ðqn  q‘ Þ for n ¼ m;

‘¼1;‘6¼n

Mnm ¼ ðqn  qm Þ; for m 6¼ n where ðqÞ, ðqÞ and ðqÞ are 3 functions to be determined. It is then easily seen that these functions

ðxÞ0 ðyÞ  ðyÞ0 ðxÞ ¼ ðxÞ  ðyÞ; ðxÞ ¼ ðxÞ ðx þ yÞ The general solution of this functional equation yields via [15b] the two-body potential V ð2Þ ðqÞ ¼ g2 a2 }ða qj!; !0 Þ

½16

where g and a are two arbitrary constants and }ðxj!, !0 Þ is the Weierstrass elliptic function (with semiperiods ! and !0 , as well arbitrary). One concludes therefore that the N-body problem characterized by the Hamiltonian (11) with (12), (15a) and (16) is integrable. This Hamiltonian system has played, since the midseventies, a seminal role in the developments of finitedimensional integrable systems that occurred over the last few decades. However, since the Weierstrass function is doubly-periodic, from a ‘‘physical’’ point of view this N–body problem is rather unrealistic, or perhaps rather suited for the study of crystalline configurations, including their statistical mechanics. But there are two special cases, obtained by assigning an infinite value to one or both of the semiperiods of the Weierstrass function in (16), that qualify V ð2Þ ðqÞ as a physical two-body potential: V ð2Þ ðqÞ ¼

g2 a2 sinh2 ða qÞ

V ð2Þ ðqÞ ¼

g2 q2

½17a

½17b

(Of course the second of these two-body potentials, (17b), is merely the special case of the first, (17b), corresponding to a = 0). These Hamiltonian models are then naturally interpretable as one-dimensional many-body problems with repulsive two-body forces singular at zero separation and vanishing at large distances. Actually the fact that these systems are integrable is far from remarkable, since it is generally true that any many-body problem characterized by repulsive forces vanishing at large distances (hence causing unconfined motions) is integrable: indeed in such models the particles eventually separate and move freely, so that their trajectories cannot display the extreme complication

110 Integrable Systems: Overview

characterizing a chaotic (i.e., nonintegrable) behavior. But these models are in fact superintegrable and they (as well as various integrable extensions of them) feature many (physically and mathematically) interesting properties. For instance the asymptotic behavior of their trajectories, ðÞ ðÞ qn ðtÞ ¼ pðÞ n t þ qn þ oð1Þ; pn ðtÞ ¼ pn þ oð1Þ as t ! 1; n ¼ 1; . . . ; N ½18

is characterized by the simple rules ðÞ

pðþÞ n ¼ pNþ1n ; n ¼ 1; . . . ; N; N X

ðÞ qðþÞ n ¼ qn þ

  ðÞ  pðÞ m  pn ; g; a

½19

m¼1; m6¼n

n ¼ 1; . . . ; N with h i log 1 þ ðga=pÞ2 ðp; g; aÞ ¼ signðpÞ

2a ðþÞ

ðÞ

The formula (19) indicates that the shift qn  qn among the asymptotic positions of the particles (see (18)) is merely a sum of two-body shifts  (which incidentally vanish altogether if a = 0, namely in the (17b) case), and it only depends on the velocities ðÞ pn of the particles in the remote past (not on the ðÞ corresponding asymptotic positions qn , in spite of their relevance in determining the order in which the different particles approach each other through the motion). A generalization of the above model in the (17b) case – nontrivial inasmuch as it yields confined motions – is characterized by the additional presence in the potential (12) of the one-body potential V ð1Þ ðqÞ ¼ 12 !2 q2

½20

yielding the Hamiltonian (3a). This model is integrable, indeed superintegrable, indeed isochronous, all its (real) solutions being completely periodic with period T¼

2 !

½21

A neat way to understand this result is by noting ~ðtÞ is a (possibly complex) solution of the that, if q model discussed above (in this subsection, with the two-body potential (17b) and no one-body potential, see (15a)), then qn ðtÞ ¼ expði!tÞ~ qn ðÞ;  ¼

expð2 i!tÞ  1 2 i!

provides a (possibly real) solution of the Newtonian equations of motion (2), namely of the same model

but with the additional one-body potential (20). Remarkably this model was solved firstly in the quantal case (at the beginning of the seventies), and only a few years later in the classical case considered here (by J. Moser, who, for the ! = 0 case, introduced the special version of the Lax matrix appropriate for this case). Another class of many-body problems, introduced in the mid-sixties by M. Toda, played a seminal role in the study of integrable dynamical systems, indeed the first application (independently by H. Flaschka and S. Manakov) of the Lax approach to integrable many-body problems occurred in that context. This model is often referred to as the Toda lattice, because its (two-body) interaction (of exponential type) is only assumed to act among ‘‘nearest neighbors’’. A particularly interesting, and just as integrable, generalization of this class of Hamiltonian manybody problems features an extra parameter, say c, which might be considered to play the role of ‘‘speed of light’’. These models reduce to those considered above for c = 1, and for finite c they are invariant under the Poincare´ group of coordinate transformations (while of course the many-body problems described above are invariant under the Galilei group). They are sometimes termed RS models, to recognize those who first introduced them (S. Ruijsenaars and H. Schneider) as well as the possibility to interpret them in some sense as ‘‘relativistic’’ generalizations of the ‘‘nonrelativistic’’ models described above. Reduction of the solution to algebraic operations The solution of the models described above can actually be reduced to purely algebraic operations. For instance for the model characterized by the Newtonian equations of motion (2) such a solution of the initial-value problem is provided by the following prescription: the particle coordinates qn ðtÞ coincide with the N eigenvalues of the N  N matrix: ~ nm ðtÞ ¼ qn ð0Þ cosð!tÞ þ q_ n ð0Þ sinð!tÞ for n ¼ m; Q ! ig sinð!tÞ ~ nm ðtÞ ¼ Q for n 6¼ m !½qn ð0Þ  qm ð0Þ

Many-body problems related to the motion of the zeros of linear PDEs Another convenient approach to manufacture and investigate integrable manybody problems is by identifying the motion of the particles with that of the zeros of (polynomial)

Integrable Systems: Overview

solutions of linear (hence solvable) evolution PDEs. Assume for instance that the monic polynomial ðz; tÞ ¼ xN þ

N X

cm ðtÞxNm ¼

m¼1

N Y

½z  zn ðtÞ

½22

n¼1

z

t

½23

C€zn þ E_zn ¼ B0 þ B1 zn  2ðN  1ÞA3 z2n ðzn  zm Þ

ðzn  zm Þ1

are invariant under rescaling of the dependent variables ðzn ¼) czn Þ. Let us then assume to work in the complex rather than the real, and let us set

where the letters A0 , A1 , A2 , A3 , B0 , B1 , C, D0 , D1 , D2 , E denote 11 arbitrary constants. Then the zeros zn ðtÞ evolve according to the system of ODEs

þ

N X

€zn þ E_zn ¼ B1 zn þ

   2_zn z_ m  D1 ðz_ n þ z_ m Þzn þ 2A2 z2n ½25

 ½NðN  1ÞðA2  A3 zÞ þ NB1  ¼ 0

N X

model, (24), with C = 1 and with A0 = A1 = A3 = B0 = D0 = D2 = 0 so that its equations of motion,

m¼1; m6¼n

satisfies the (compatible) linear PDE   A0 þ A1 z þ A2 z2 þ A3 z3 zz   þ B0 þ B1 z  2ðN  1ÞA3 z2 þ C tt þ ½E  ðN  1ÞD2 z   þ D0 þ D1 z þ D2 z2 zt

111

1

m¼1; m6¼n

 ½2C_zn z_ m  ðz_ n þ z_ m ÞðD0 þ D1 zn Þ  D2 zn ðz_ n zm þ z_ m zn Þ   þ 2 A0 þ A1 zn þ A2 z2n þ A3 z3n ½24 interpretable as the Newtonian equations of motion of an N-body problem with one- and two-body (velocity-dependent) forces. This problem is integrable, indeed its solution can be reduced to the algebraic problem of finding the zeros of the polynomial ðz, tÞ, see (22), whose time evolution can be ascertained by solving the linear PDE (23), itself a purely algebraic problem as it amounts to solving the system of (constant coefficients, linear) ODEs implied via (22) by this PDE (23) for the N coefficients cm ðtÞ. This class of many-body problems is rather rich, thanks to the arbitrariness of the 11 constants it features. Several subcases, characterized by special choices of these constants, are suitable to display a gamut of different phenomenological behaviors: confined and nonconfined motions, periodic and nonperiodic evolutions, limit cycles, Hamiltonian cases, . . . .

Solvable many-body problems in the plane The many-body problems considered above were all essentially one-dimensional. But via a simple trick it is possible to obtain from some of them manybody problems in the plane (which should of course be rotation-invariant to be certified as such). Consider for instance the special case of the above

~ ~ B1 ¼  þ i; E ¼  þ i!; A2 ¼  þ i; D1 ¼  þ i~ where the Greek letter indicate now real constants, and let us moreover relate the N complex coordinates zn to N two-vectors ~ rn in the horizontal plane via the self-evident positions ^ ¼ ð0; 0; 1Þ zn ¼ xn þ iyn ;~ rn ¼ ðxn ; yn ; 0Þ; k

½26

It is then easily seen that the integrable equations of motion (25) become the following rotation-invariant Newtonian equations of motion identifying a (no less integrable) N-body problem in the plane:   ^ ~ ~ rn þ  þ !k^ rn   ^ ~ ¼  þ ~k^ rn þ

N X

r2 nm

m¼1; m6¼n 

h n     o 2 ~ rn ~ rnm þ~ rnm ~ rm rm ~ rnm ~ rm ~ rn ~ rn ~  n   ^ ~   þ ~k^ rn þ~ rm r2n  ð~ rn ~ rm Þ h  h  i io ~ rn ~ rn þ~ rm þ~ rn þ~ rm rm ~ rn ~ rm ~    i  ^ ~ ~k^ þ2 þ rn r2n  2ð~ rn ~ rm Þ þ~ rm r2n ½27 Here and below we use the short-hand notation ~ rn ~ rm entailing r2nm = r2n þ r2m  2~ rn ~ rm , the rnm =~ symbol ^ denotes the three-dimensional vector product so that kˆ ^~ rn = ðyn , xn , 0Þ (see (26)), and the rest of the notation is self-evident. Note that these rotationinvariant Newtonian equations of motion are also ~ = 0. translation-invariant if  = ~ =  = ~=  = 

The ‘‘goldfish’’ model The attribute of ‘‘goldfish’’ has been attributed to the special case of the above model with all ‘‘coupling constants’’ vanishing, thanks to the neatness of its equations of motion, which in their complex version read €zn ¼ 2

N X m¼1; m6¼n

z_ n z_ m ; n ¼ 1; . . . ; N zn  zm

112 Integrable Systems: Overview

and in their real (‘‘physical’’) version as Newtonian equations of motion of an N-body problem in the horizontal plane read       N ~ r_ m ~ r_ n ~ rnm þ~ rnm ~ rm rn ~ rn ~ r_ m ~ rnm ~ X ~ rn ¼2 2 rnm m¼1;m6¼n n ¼ 1;...;N (This name has also been attributed to some extensions of this model, see the entry Isochronous Systems in this Encyclopedia). This model is invariant under time rescaling ðt ) ctÞ, in its physical version it is translation- and rotationinvariant, it only features two-body forces and in spite of their velocity-dependence it is Hamiltonian (it is in fact a simple instance of the RS models mentioned above). The solution of its initial-value problem (in its complex version) is given by a remarkably neat rule: the N coordinates zn ðtÞ are the N roots of the following algebraic equations in z: N X 1 z_ n ð0Þ ¼ z  z t ð0Þ n n¼1

½28

The phenomenology of its generic solution is also remarkable, corresponding to the ‘‘game of musical chairs’’: in the remote past all particles but one are almost at rest in N  1 positions (‘‘sitting in N  1 chairs’’) and one particle comes in from infinity, moving initially as a free particle; as it approaches, all the particles begin to move around (‘‘dancing’’); in the remote future one particle goes away (moving eventually with the same speed as the incoming particle), and all the others settle down in the same N  1 positions (‘‘of the N  1 chairs’’), but with the possibility that the outgoing particle be different from the incoming one, and that the other particles have reshuffled their ‘‘seating’’. Another remarkable version (also translation- and rotation-invariant, as well as Hamiltonian) of the N-body model in the plane (27) obtains if all the ‘‘coupling constants’’ vanish except !. Then all its nonsingular solutions – which are given by the same prescription indicated just above, except for the i! replacement of 1t with expði!tÞ1 in the right-hand side of (28) – are completely periodic with periods which are an integer multiple – no larger than a number depending on N, generally (much) smaller than N! – of T (see (21)), the domains of phase space that give rise to solutions with different periodicity being separated from each other by boundaries characterized by lower-dimensional sets of initial data yielding trajectories that run into singularities corresponding to particle collisions (note that when

two or more particles collide their individuality gets lost, and their velocities diverge).

Integrable many-body problems in spaces with arbitrary dimensions Integrable, or even solvable, many-body problems in spaces with more than two dimensions – with rotation-invariant equations of motion of Newtonian type – can be manufactured by starting from an appropriate integrable, or solvable, second-order matrix evolution equation, and by then parametrizing the evolving matrix in terms of multidimensional vectors so as to transform the matrix evolution equation into a covariant – hence rotation-invariant – system of evolution equations for these vectors, interpretable as Newtonian equations of motion of a many-body problem in multidimensional space. For instance the matrix equation _ ¼ AM þ MA þ M3 M is integrable. Here M  MðtÞ is a square matrix of arbitrary order and A is an arbitrary constant matrix. By parametrizing appropriately these two matrices one concludes that either one of the following two Newtonian systems of ODEs is integrable:

~ rnm ¼

N X

n ~ r m þ

¼1

M X N X

  ~ r m rn ~ r ~

¼1 ¼1

n ¼ 1; . . . ; N; m ¼ 1; . . . ; M;

~ rnm ¼

N X ¼1

n ~ r m þ

M X N X

  ~ rnm r ~ r ~

¼1 ¼1

n ¼ 1; . . . ; N; m ¼ 1; . . . ; M: Here N and M are arbitrary positive integers, the NM constants nm are also arbitrary, the NM ‘‘particle coordinates’’ ~ rnm  ~ rnm ðtÞ are S-vectors, with S an arbitrary positive integer, and the dots sandwiched among these S-vectors denote the standard scalar product in S-dimensional space. Let us emphasize the physical relevance of this class of many-body problems, characterized by linear and cubic forces. This is reinforced by the fact that these models are Hamiltonian. Nonlinear harmonic oscillators Two classes of integrable systems obtain from the classes written above by first setting to zero all the constants nm and by then performing the change of variables ~ nm ðtÞ ¼ expði!tÞ~ rnm ðÞ;  ¼ w

expði!tÞ  1 i!

½29

Integrable Systems: Overview

with ! > 0. The corresponding Newtonian equations of motion read



~ nm  3i!~ wnm  2~ wnm ¼ w

M X N X

The class of linear dispersive evolution PDEs reads

 @ ut ðx; tÞ ¼ i! i uðx; tÞ; 1 < x < 1 ½30 @x

  ~ nm ~ w ~ w w

where the ‘‘dispersion function’’ !ðzÞ is, say, a (real) polynomial (which must be odd to guarantee that this PDE be real). The solution of this PDE is achieved via the introduction of the Fourier transform uðk, tÞ, ˆ Z 1 1 ^ðk; tÞ uðx; tÞ ¼ ð2Þ dk expði kxÞ u ½31a

n ¼ 1; . . . ; N; m ¼ 1; . . . ; M;



M X N X

Identification and investigation of integrable PDEs via the inverse spectral transform technique

  ~ m ~ n w ~ w w

¼1 ¼1

~ nm  3i!~ w wnm  2~ wnm ¼

113

¼1 ¼1

n ¼ 1; . . . ; N; m ¼ 1; . . . ; M These equations of motion cause the N M evolving ~ nm  w ~ nm ðtÞ to be complex (see the S-vectors w second term in their left-hand sides), but a real system (with double the number of dependent variables) can be easily obtained by setting

1

^ðk; tÞ ¼ u

Z

1

dx expði kxÞuðx; tÞ

½31b

1

~ nm ¼ ~ w unm þ i~ vnm Remarkably (but clearly suggested by (29)), all the nonsingular solutions of each of these two manybody problems are completely periodic, with a period which is an integer multiple of the period T, see (21). This justifies the title given to this subsection. It also shows that these are isochronous systems (see Isochronous Systems).

Integrable nonlinear PDEs As indicated in Section 1 another class of integrable systems are nonlinear evolution PDEs. In this section we outline (some of) their properties, focussing mainly on the Korteweg-de Vries PDE (4), the solution of which by C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura in the mid-sixties was the opening shot of a major scientific development which is still blooming. Other important early steps of this development were, in the late sixties, the introduction by P. D. Lax of what is now called the Lax pair technique, and at the beginning of the seventies the solution by V. E. Zakharov and A. B. Shabat of the Nonlinear Schro¨dinger equation (6) – an evolution PDE of great applicative importance. Subsequently many researchers developed various techniques to identify, classify and investigate integrable nonlinear PDEs, a continuing activity for an overall appraisal of which the interested reader is referred to the bibliography reported below. Here we outline one of the approaches to obtaining these results; other approaches are tersely mentioned below.

whose evolution corresponding to (30) is then given by the simple linear ODE ^t ðk; tÞ ¼ i!ðkÞ^ uðk; tÞ; 1 < k < 1 u

½32a

which can be immediately integrated: ^ðk; tÞ ¼ u ^ðk; 0Þ exp½i!ðkÞt u

½32b

Thus the solution of the initial-value problem of (30) is achieved via three steps: (i) at the initial time one obtains the initial value of the Fourier transform, uðk, 0Þ, from the initial datum uðx, 0Þ (via (31b)); (ii) ˆ one then obtains uðk, tÞ (via (32b)); (iii) one finally ˆ obtains uðx, tÞ (via (31a)). From these formulas the main features of the resulting phenomenology are easily evinced (even when the above integrals cannot be explicitly performed). A class of integrable nonlinear evolution PDEs reads ut ðx; tÞ ¼ ðRÞux ðx; tÞ

½33

where the assigned function ðzÞ is again, say, a (real) polynomial, while R is now the integrodifferential ‘‘recursion operator’’ defined by the following formula that specifies its action on a generic function f ðx, tÞ (vanishing asymptotically so as to allow all integrations to converge): R f ðx; tÞ ¼ fxx ðx; tÞ  4uðx; tÞf ðx; tÞ Z 1 þ 2ux ðx; tÞ dy f ðy; tÞ

½34

x

Note that the presence of the time variable t plays no relevant role (it is merely parametric). A remarkable property of this operator – which depends on uðx, tÞ – is that any power of it acting

114 Integrable Systems: Overview

on ux ðx, tÞ yields a nonlinear combination of uðx, tÞ and its x-derivatives – without any left-over integration, in fact yielding a result which is itself an exact x-derivative, ready for exact integration in case of a further application of R, see the last term in the right-hand side of (34). For instance   Rux ¼ uxxx  6ux u ¼ uxx  3u2 x ; R2 ux ¼ uxxxxx  10uxxx u  20uxx ux þ 30ux u2   ¼ uxxxx  10uxx u  5u2x þ 10u3 x and so on. Hence the simplest nonlinear evolution equation contained in the class (33) is the Kortewegde Vries (KdV) equation ut þ uxxx ¼ 6ux u

½35

(corresponding to ðzÞ = z; and note the identity with (4), via the trivial rescaling qðx,tÞ = 3 uðx, tÞ). Note that, if one neglects all nonlinear contributions, the class (33) reduces to (30) with   !ðzÞ ¼ z z 2 The solution of this class of nonlinear PDEs, (33), is given by a somewhat analogous procedure to that described above for the class of linear dispersive PDEs (30). Firstly, one introduces the spectral transform, a nonlinear generalization of the Fourier transform which indeed reduces to it if nonlinear effects are altogether neglected. That relevant for the class of PDEs (33) is based on the spectral problem associated with the linear Schro¨dinger operator

2 @ L¼ þ uðx; tÞ; 1 < x < 1 ½36 @x Via it, the spectral transform S½uðx; tÞ ¼ fRðk; tÞ; 1 < k < 1; pn ; n ðtÞ; n ¼ 1; . . . ; Ng

½37

is introduced. Here the function Rðk, tÞ is the ‘‘reflection coefficient’’ associated to the eigenvalue k2 of the continuous spectrum of L, while the nonnegative number N gives the number of discrete eigenvalues of L, and the positive quantities pn and n ðtÞ are associated to these discrete eigenvalues, specifically p2n are the ‘‘binding energies’’, and n ðtÞ the ‘‘normalization coefficients’’, associated to the ‘‘bound states’’ possessed by the ‘‘potential’’ uðx, tÞ. (All this terminology comes from the interpretation of the above spectral problem in quantummechanical terms). And it can be shown not only that there is a one-to-one correspondence among a function uðx, tÞ and its spectral transform S[u(x, t)],

but moreover that both the direct spectral problem to compute S[u(x, t)] from u(x, t) (arbitrarily assigned within an appropriate class), and the inverse spectral problem to compute u(x, t) from S[u(x, t)] (arbitrarily assigned within an appropriate class), only entail solving linear equations (an ODE in the former case, a Fredholm integral equation in the latter case). Note that, in the above definition of the spectral transform, the time variable t plays merely a parametric role. But the usefulness of this spectral transform to solve the PDE (33) resides in the fact that, if u(x, t) evolves in time according to this PDE, the corresponding evolution of the spectral transform is quite simple: the number N and the positive numbers pn are time-independent (as already implied by our notation), while the time evolution of the reflection coefficient R(k, t) and of the normalization coefficients n ðtÞ is given by the simple linear ODEs   Rt ðk; tÞ ¼ 2ik 4k2 Rðk; tÞ; 1 < k < 1

½38a

_ n ðtÞ ¼ 2pn ð4p2n Þn ðtÞ; n ¼ 1; . . . ; N

½38b

which can be readily integrated:    Rðk; tÞ ¼ Rðk; 0Þ exp 2ik 4k2 t   n ðtÞ ¼ n ð0Þ exp 2pn ð4p2n Þt

½39a ½39b

Hence the solution of the initial-value problem for the class of nonlinear PDEs (33) can now be achieved via the following three steps: (i) at the initial time, via the solution of the direct spectral problem, the spectral transform S[uðx, 0Þ] (see (37)) is obtained (from u(x, 0), arbitrarily assigned within an appropriate class); (ii) the spectral transform at time t is then obtained via (39); (iii) by solving the inverse spectral problem, u(x, t) is obtained from S[u(x, t)] (see (37)). The analogy of this procedure to that outlined above for the class of linear dispersive PDEs (30) is clear, and the fact that in this manner the solution of the initial-value problem for the nonlinear PDEs (33) can be achieved via a sequence of steps involving only the solution of linear problems is an indication of the integrable character of this class of nonlinear evolution PDEs. And it allows to gain thereby a lot of insight on the behavior of these solutions, and also to construct classes of explicit solutions of these equations, as we now indicate.

Integrable Systems: Overview Solitons

The integrable nonlinear PDE (33) possesses the single-soliton solution uðx; tÞ ¼

2p2 2

cosh fp½x  ðtÞg

ðtÞ ðtÞ ¼ ð2pÞ1 log ¼ ð0Þ þ vt; 2p   v ¼  4p2

½40a

½40b

to which corresponds the simple spectral transform

½42

thus all solitons of the KdV equation move from left to right, and taller and thinner solitons move faster than less tall and more fat ones. More generally, every PDE of the class (33) possesses the N-soliton solution

2 @ uðx; tÞ ¼ 2 log det½I þ Cðx; tÞ ½43a @x Here I is the N  N unit matrix and CðtÞ is the N  N matrix Cmn ðx;tÞ ¼ ½m ðtÞn ðtÞ

1=2

exp½ðpm þ pn Þx pm þ pn

p1 < p2 < < pN so that the corresponding soliton velocities, vn = 4p2n , are as well ordered in increasing order: The N-soliton solution (43) is not so transparent, especially if N is large, but it becomes quite simple in the remote past and future: uðx; tÞ

n ðtÞ

½41

This solution, (40), describes a localized wave of constant shape moving with the constant speed v: the ‘‘soliton’’. It is characterized by two (real) parameters, ð0Þ and p. The first identifies the initial location of the soliton; its arbitrariness corresponds to the translation invariant character of (33). The second, p, the spectral significance of which is clear from (41), determines the shape of the soliton (both its ‘‘height’’ 2p2 and its ‘‘width’’ 1p) as well as its speed v (see (40b)); note that the shape is identical for all the nonlinear evolution PDEs of the class (33), while the speed depends on the function ðzÞ, see (40b), namely it depends on which specific equation of the class (33) one is considering. For instance for the KdV equation (35), corresponding to ðzÞ = z, the speed of the soliton is v ¼ 4p2

and let us order the N positive numbers pn in increasing order,

v1 < v2 < < vN



S½uðx; tÞ ¼ fRðk; tÞ ¼ 0; p1 ¼ p;   1 ðtÞ ¼ ðtÞ ¼ ð0Þ exp 2pð4p2 Þt ; N ¼ 1g

115

N X

n¼1 ¼ nðÞ

2p2n cosh2 fpn ½x  n ðtÞg

;

þ vn t; t ! 1 ðÞ

with the 2N (real) constants n related to one another (see below). It is thus seen that, both in the remote past and future, the N-soliton solution (43) splits into the sum of N separated solitons. In the remote past the solitons are ranged, from left to right, in order of decreasing amplitude, and they move to the right with speeds ordered in decreasing magnitude; then the taller and faster solitons gradually catch up and eventually ‘‘overtake’’ the fatter and slower ones (the quotation marks underscore the fact that whenever two, or possibly more, solitons get together, their individuality is in fact lost: for a while the solution might have just one peak, or instead the ‘‘overtaking’’ of two solitons may rather appear as an ‘‘exchange of identity’’, with the taller soliton becoming fatter and the fatter becoming taller as they get close together until they separate again because the one in front, having become taller, speeds up while the one behind, having become fatter, slows down). The final outcome is of course that the order of the solitons gets altogether reversed, with the taller and faster heading the escape to the right. The most remarkable aspect of this phenomenology is that precisely the same solitons that existed in the remote past are found in the remote future, the only effect of their ‘‘interaction’’ having been to shift the position of the n-th soliton, relative to what it would have been if it had been moving in isolation, by the amount n ¼ nðþÞ  nðÞ

½43b

where the time-evolution of the n ðtÞ’s is given by (39b). Indeed the spectral transform of this solution is given by (37) with Rðk, tÞ= 0 and n ðtÞ given by (39b). To discuss the multisolitonic phenomenology, let us focus on the KdV equation, so that the speed of each soliton is given by the simple formula (42)

These N shifts are moreover determined (while ðÞ ðþÞ either the N quantities n or the N quantities n can be arbitrarily assigned), being given by the simple rule n ¼

n1 X m¼1

ðpn ; pm Þ 

N X m¼nþ1

ðpn ; pm Þ

½44a

116 Integrable Systems: Overview

 1 pn þ pm ðpn ; pm Þ ¼ log pn jpn  pm j

½44b

Of course in (44a) a sum vanishes if its lower limit exceeds its upper limit. This formula (44), has a simple phenomenological significance. From the two-soliton case ðN = 2Þ it is seen that in a two-body encounter the taller and faster soliton gets advanced by the amount ðp2 , p1 Þ, while the slower and fatter one gets delayed by the amount ðp1 , p2 Þ. Hence the overall shift (44) experienced by the n-th soliton in the N-soliton case is the sum of the n  1 positive shifts derived from its ‘‘overtaking’’ n  1 slower solitons and the N  n negative shifts derived from its being ‘‘overtaken’’ by N  n faster solitons. This outcome is obvious when each two-soliton encounter occurs separately, but is quite nontrivial in the general case when, at some intermediate time, several solitons might all encounter simultaneously. This soliton phenomenology strongly suggest ascribing to each soliton an individuality, even though in configuration space it only shows up as a separate entity in the remote past and future. The separated identity of each soliton is instead quite clear in the spectral transform context, since each of them corresponds to a (time-independent) discrete eigenvalue of the spectral problem. Indeed in the spectral context this identity is clear also for the generic solution of the class of integrable nonlinear PDEs (33) which, in contrast to the purely solitonic solution (43), is not characterized by a vanishing reflection coefficient Rðk, tÞ. And indeed, even in configuration space, the soliton phenomenology described above is still featured by a generic solution (each of which is characterized, via its spectral transform (37), by the number N of its solitons), up to the additional presence of a ‘‘background’’ component of this solution (corresponding to the nonvanishing reflection coefficient Rðk, tÞÞ, which however behaves in a manner analogous to the solution of the linear, dispersive part of the PDE under consideration, becoming eventually locally small due to its dispersive character. Kinks, breathers, boomerons and trappons, dromions The solitonic phenomenology described above for the class of integrable PDEs (33), and in particular for the KdV equation (35), is more or less common to all integrable nonlinear evolution PDEs – of which many other classes exist besides (33). But there also are some significant differences, some of which we now review tersely. For certain integrable PDEs the typical shape of the soliton is not localized, but it rather has the form

of a ‘‘kink’’. Some integrable PDEs also feature additional kinds of localized ‘‘solitons’’ which, in isolation, move overall with constant speed as ordinary solitons, but feature in addition a timedependent amplitude modulation and are therefore called ‘‘breathers’’. For integrable matrix nonlinear evolution PDEs – or, equivalently, for integrable systems of coupled PDEs – the new phenomenology may emerge of solitons that, even in isolation, move with a variable speed, the change of which over time is correlated with the variable interplay of the amplitudes of the different components of the solution: typically such solitons come in from one side in the remote past and boomerang back to that side in the remote future (‘‘boomerons’’), or they may be trapped to oscillate around some fixed position (‘‘trappons’’); and there are integrable evolution equations in which both these types of solitons are simultaneously present in a generic solution. All these phenomenologies refer to the simpler class of integrable evolution PDEs in 1 þ 1 (one space and one time) variables, with asymptotically vanishing boundary conditions (at large space distances; or perhaps asymptotically constant, as in the case of kinks). There also exist integrable evolution PDEs in 2 þ 1 dimensions (such as the KP equation (9)) the generic solution of which may feature localized soliton-like components, although in this case appropriate boundary conditions play a crucial role (for this reason such solitons have been called ‘‘dromions’’, hinting at their being to some extent driven by the boundary conditions, as objects moving in a stadium). While there are quite many (classes of) integrable PDEs in 1 þ 1 dimensions, there are only a few in 2 þ 1 dimensions, and there is a widespread belief that no integrable PDEs exist in D þ 1 dimensions with D > 2. But already in the early days of soliton theory it was pointed out that there do exist quite many (classes of) integrable PDEs in 1 þ D dimensions (namely, one space and D time variables) and that it is quite possible via a different formulation of the initial-value problem to interpret such equations as (no less integrable) PDEs in D þ 1 dimensions (D space and one time variables); and integrable PDEs in D þ 1 dimensions have also been identified and investigated in the context of (the simpler class of) C-integrable PDEs (see below). Other properties of integrable PDEs

For the linear evolution equations (30) the main message implied by their solvability via the Fourier transform is, that the time-evolution is much simpler in Fourier space (see (32)) than in configuration

Integrable Systems: Overview

space. This has a profound impact on the understanding of all phenomena describable by such equations, to the extent of determining the kind of experimental tools better suited to understand the underlining physics (for instance, the use of monochromatic beams of light, the use of high-energy particle accelerators, and so on). The same kind of message is as well relevant for the class of integrable nonlinear PDEs solvable via the spectral transform technique – even more so inasmuch as the timeevolution is in this case so much simpler in the spectral space (being actually linear there, see (38) and (39)) than in configuration space (where the evolution is nonlinear, see (33)). It is indeed the basis for the possession by the class of integrable nonlinear PDEs (33) of several other remarkable properties as outlined tersely in the following subsections. Ba¨cklund transformations A Ba¨cklund transformation is a formula relating two functions, say uð0Þ ðx, tÞ and uð1Þ ðx, tÞ, so that, if one of them satisfies a (generally nonlinear) PDE, the other one satisfies the same PDE. In the context of the class (33) of integrable PDEs, such a (class of) Ba¨cklund transformations is provided by the formula h i gðLÞ uð0Þ ðx; tÞ  uð1Þ ðx; tÞ þ hðLÞG 1 ¼ 0 ½45 where gðzÞ and hðzÞ are two (a priori arbitrary) entire functions (say, two polynomials), while L and G are two integrodifferential operators the effect of which on a function f ðx, tÞ (such that all relevant integrations are convergent) reads h i ð1Þ Gf ðx; tÞ ¼ uð0Þ x ðx; tÞ þ ux ðx; tÞ f ðx; tÞ h i þ uð0Þ ðx; tÞ  uð1Þ ðx; tÞ Z 1 h i  dy uð0Þ ðy; tÞ  uð1Þ ðy; tÞ f ðy; tÞ ½46a x

h i Lf ðx; tÞ ¼ fxx ðx; tÞ  2 uð0Þ ðx; tÞ þ uð1Þ ðx; tÞ f ðx; tÞ Z 1 dyf ðy; tÞ ½46b þG x

Note that here the variable t plays no relevant role (its presence is merely parametric), and that G and L depend (in a symmetrical way) on uð0Þ ðx, tÞ and uð1Þ ðx, tÞ, whose presence causes the Ba¨cklund transformation (45) to be nonlinear in these functions. Also important is the observation that, for uð0Þ ðx,tÞ = uð1Þ ðx,tÞ = uðx, tÞ, the operator L becomes the recursion operator R, see (34).

117

The reason why the formulas (45) constitute a class of Ba¨cklund transformations is because – as a property of the spectral transform based on the linear Schro¨dinger operator L, see (36) – if two ‘‘potentials’’ uð0Þ ðx, tÞ and uð1Þ ðx, tÞ are related by (45), the corresponding ‘‘reflection coefficients’’ Rð0Þ ðk, tÞ and Rð1Þ ðk, tÞ are related algebraically, as follows: h i gð4k2 Þ Rð0Þ ðk; tÞ  Rð1Þ ðk; tÞ h i þ 2ikhð4k2 Þ Rð0Þ ðk; tÞ þ Rð1Þ ðk; tÞ ¼ 0 ½47a entailing Rð1Þ ðk; tÞ ¼ Rð0Þ ðk; tÞ

gð4k2 Þ þ 2ikhð4k2 Þ ½47b gð4k2 Þ  2ikhð4k2 Þ

Clearly this formula entails that, if Rð0Þ ðk, tÞ satisfies (38a), so does Rð1Þ ðk, tÞ. Hence, as the fact that Rð0Þ ðk, tÞ satisfies (38a) is a consequence of the fact that uð0Þ ðx, tÞ satisfies (33), likewise the fact that Rð1Þ ðk, tÞ satisfies (38a) provides the basis for concluding that uð1Þ ðx, tÞ also satisfies (33). The simpler version of the Ba¨cklund transformation (45) obtains by setting gðzÞ = 2phðzÞ with p an arbitrary constant, hence it reads ð1Þ wð0Þ x ðx; tÞ þ wx ðx; tÞ h i ¼ 2p wð0Þ ðx; tÞ  wð1Þ ðx; tÞ i2 1h  wð0Þ ðx; tÞ  wð1Þ ðx; tÞ 2

½48

Here and below we use for convenience the functions wðjÞ ðx, tÞ related to uðjÞ ðx, tÞ as follows: Z 1 dy uðjÞ ðy; tÞ; wðjÞ ðx; tÞ ¼ ½49 x ðjÞ ðjÞ wx ðx; tÞ ¼  u ðx; tÞ A convenient application of Ba¨cklund transformations is to yield new solutions of (33) from known solutions; for instance from the trivial solution uð0Þ ðx,tÞ = wð0Þ ðx, tÞ = 0 the single-soliton solution (40) can be readily obtained via (48) and (49) (of course an appropriate time-dependence must be attributed to the x-independent ‘‘integration constant’’ that obtains from the integration of (48), which is an ODE in the independent variable x). Another important property of Ba¨cklund transformations is their commutativity. Consider two sets of two polynomials, gðmÞ ðzÞ and hðmÞ ðzÞ, m = 1, 2, and the two Ba¨cklund transformations (45) they generate, say BT1 and BT2. Take as starting point some function uð0Þ ðxÞ and associate to it two functions,

118 Integrable Systems: Overview

uð1Þ ðxÞ respectively uð2Þ ðxÞ, obtained from uð0Þ ðxÞ via these two Ba¨cklund transformations, BT1 respectively BT2. Then obtain a new function, say uð12Þ ðxÞ, from uð1Þ ðxÞ via BT2; and likewise obtain uð21Þ ðxÞ from uð2Þ ðxÞ via BT1. The property of commutativity entails that, provided an appropriate choice is made of integration constants (see (45)), uð12Þ ðxÞ ¼ uð21Þ ðxÞ

½50

This property is highly nontrivial when viewed, as we just did, in configuration space; it is instead rather obvious in the spectral space, indeed the corresponding property for the ‘‘reflection coefficients’’ reads (in self-evident notation, see (47b)) Rð12Þ ðkÞ ¼ Rð21Þ ðkÞ ¼ Rð0Þ ðkÞ Bð1Þ ðkÞ Bð2Þ ðkÞ

BðmÞ ðkÞ ¼

½51a

gðmÞ ð4 k2 Þ þ 2ikhðmÞ ð4 k2 Þ ; gðmÞ ð4 k2 Þ  2ikhðmÞ ð4 k2 Þ

m ¼ 1; 2

½51b

hence it corresponds simply to the commutativity of the ordinary product. Nonlinear superposition principle Another remarkable property of the class of evolution equations (33) is a straightforward consequence of the commutativity property, (50), of Ba¨cklund transformations. It reads (hereafter with a slight abuse of language we refer to ‘‘solutions’’ wðjÞ even though the actual solutions are the functions uðjÞ related to the wðjÞ by (49))   2ðp1 þ p2 Þ wð1Þ  wð2Þ ð12Þ ð21Þ ð0Þ w ¼w ¼w  ½52 2ðp1  p2 Þ þ wð1Þ  wð2Þ where wð0Þ  wð0Þ ðx, tÞ is an arbitrary solution of (33), wð1Þ  wð1Þ ðx, tÞ respectively wð2Þ  wð2Þ ðx, tÞ are likewise the solutions of the same PDE related to wð0Þ by the Ba¨cklund transformation (48) with p = p1 respectively p = p2 , and wð12Þ ðx, tÞ = wð21Þ ðx, tÞ is another solution of the same PDE. Note that this formula, for which the title of this subsection seems appropriate, provides a completely explicit, rational expression of a new solution of (33) in terms of three other solutions of the same equation: an arbitrary solution wð0Þ , and the two solutions wð1Þ and wð2Þ related to it by a simple Ba¨cklund transformation, see (48). Soliton ladder A simple application of the preceding formula is to start from the trivial solution wð0Þ ¼ 0

½53

so that (see (48)) h n h ioi ðjÞ ; wðjÞ ðx; tÞ ¼ 2pj 1  tan pj x  x0 þ ð4p2 Þt j ¼ 1; 2

½54a

where, in order that this function be real, either h i ðjÞ ½54b Im x0 ¼ 0 or h i  ðjÞ Im x0 ¼ 2pj

½54c

Via (49), the expression (54a) with (54b) yields, for each value of j, a version of the single-soliton solution (40). Insertion of (53) and (54a) in (52) yields, via (49), the two-soliton solution of (33), ð1Þ provided 0 < p1 < p2 and x0 satisfies (54b) while ð2Þ x0 satisfies (54c) (otherwise the solution produced by (52) is complex or singular). Having thus obtained the two-soliton solution, one can apply the nonlinear superposition formula (52) to get the three-soliton solutions, by inserting in place of wð0Þ the single-soliton expression (54a) (with parameter, say, p1 ) and in place of wð1Þ and wð2Þ the two-soliton expression (with parameters p1 and p2 respectively p1 and p3 ); and the process can be continued, as suggested by the title of this subsection. In this manner the multisolitonic solution can be constructed by a sequence of purely algebraic operations: and simple rules can be given, detailing the restrictions on the soliton parameter pn ðnÞ and the reality properties of the constants x0 ((54b) or (54c)) to insure that the solution so arrived at be real and nonsingular, and thus coincide with (43).

Conservation laws As mentioned above, integrable evolution PDEs are interpretable as infinitedimensional dynamical systems. It is therefore natural that they possess an infinite number of conserved quantities. For instance every PDE of the class [33] possesses the following infinite sequence of conserved quantities: Z ð1Þn 1 Cn ¼ dx Rn ½xux ðx; tÞ þ 2uðx; tÞ; 2n þ 1 1 n ¼ 0; 1; 2; . . . ; ½55a where R is the recursion operator (34). An alternative definition for this sequence is Z ð1Þn 1 ~ n Cn ¼ dxR uðx; tÞ; 2n þ 1 1 n ¼ 0; 1; 2; . . . ; ½55b

Integrable Systems: Overview

~ is in some where the integrodifferential operator R sense the adjoint of R, being defined by the formula ~ ðx; tÞ ¼ fxx ðx; tÞ  4uðx; tÞf ðx; tÞ Rf Z 1

þ2

dy uðy; tÞfy ðy; tÞ

½55c

x

that specifies its action on a generic function f ðx, tÞ (such that the integration converge). The first 3 of these conserved quantities read as follows: Z 1 dx uðx; tÞ; C0 ¼ 1

C1 ¼

Z

1

^ ðx; tÞ ¼ fx ðx; tÞ  Rf

C2 ¼

1

1

x

  dx 2 u3 ðx; tÞ þ u2x ðx; tÞ

n¼0

fC n ; C m g ¼ 0 Note that, in this context, the KdV PDE (35) coincides with the Hamiltonian equation

 @ H ut ðx; tÞ ¼ fuðx; tÞ; H g ¼ @ x  uðx; tÞ

" cn z2nþ1 ¼ sin

1 1 H ¼ C2 ¼ 2 2

1

1

X

# Cn z2nþ1

n¼0

which is to be understood by expanding the righthand side in powers of z and then equating the coefficients of equal powers of z: c0 ¼ C0 ; c1 ¼ C1  16 C30 ; 1 c2 ¼ C2  12 C20 C1 þ 120 C50

and so on. Of course all these conservation laws are applicable to the class of solutions of (33) defined for all (real) values of x and vanishing asymptotically (as x ! 1). But they can also be reformulated as local ‘‘continuity equations’’. And – rather remarkably – all these results hold as well for the explicitly timedependent class of PDEs that obtains if one allows the polynomial ðzÞ in the right-hand side of (33) to feature an arbitrary time-dependence, say

with Z

y

Note that the integrodifferential operator L0 is just L, see (46), with uð0Þ ðx, tÞ = 0 and uð1Þ ðx, tÞ = uðx, tÞ. The constants cn are also all independent of each other, but there is a relationship between the constants of the two sequences, (55) and (56), X

(where A and B are functionals of uðxÞ and = uðxÞ denotes the functional derivative), they are in involution,

dy uðy; tÞ f ðy; tÞ;

L 0 f ðx; tÞ ¼ fxx ðx; tÞ  2 uðx; tÞ f ðx; tÞ Z 1 þ ux ðx; tÞ dy f ðy; tÞ x Z 1 Z 1 þ uðx; tÞ dy uðy; tÞ dz f ðz; tÞ

dx u2 ðx; tÞ;

These constants of the motion (55) are functionally independent and, in the context of a Hamiltonian formulation characterized by the Poisson bracket Z 1 A @ B dx fA; Bg ¼  uðxÞ @ x  uðxÞ 1

x

1

1

Z

Z

119

ðz; tÞ ¼

  dx 2 u3 ðx; tÞ þ u2x ðx; tÞ

M X

m ðtÞzm

½57

m¼0

Several alternative sequences of constants of motion also exist. For instance another infinite sequence is provided by the two equivalent formulas Z 1 ^ 2n 1 cn ¼ ð1Þn dx R ½56a

Finally let us note that there is an additional conserved quantity for this (generalized) class of PDEs, C¼

Z

1

1



Z t ~ t0 Þuðx; tÞ dx xuðx; tÞ þ dt0 ðR; 0

1

cn ¼ ð1Þn

Z

1

1

dx Ln0 uðx; tÞ

½56b

^ and L0 with the integrodifferential operators R defined by the formulas

~ defined by (55c). This implies that, for the with R generic solution of this (generalized) class of PDEs the center of mass R1 dx x uðx; tÞ XðtÞ ¼ R1 1 1 dx uðx; tÞ

120 Integrable Systems: Overview

moves according to the formula XðtÞ ¼ X0 þ 

M X

entailing the linear PDE

ð1Þmþ1 ð2 m þ 1Þ

m¼0

Z

t

dt0 m ðt0 Þ; X0 ¼

0



Cm C0



C C0

Hence for all the autonomous evolution PDEs of the class (33) (with ðz, tÞ = ðzÞ, m ðtÞ = m , see (57)) the center of mass of the generic solution moves uniformly, XðtÞ ¼ X0 þ Vt with the (constant) speed V¼

M X m¼0

ð1Þ

mþ1



 Cm ð2 m þ 1Þ m C0

Other techniques to identify, classify and investigate integrable PDEs

The spectral transform approach on which we focussed above is just one of the various techniques used to identify and investigate integrable nonlinear evolution PDEs. (Incidentally; because the less standard aspect of this approach is the inverse transformation to reconstruct, in the framework of the spectral problem, the ‘‘potential’’ u(x) from its spectral transform, this approach is often called the Inverse Spectral, or Scattering, Transform method – abbreviated as IST). In this subsection we tersely mention some other approaches, referring to the literature indicated below for more adequate treatments. An approach starts from a trivially integrable PDE – say, linear and autonomous, see for instance (30) – and performs a nonlinear change of dependent, and possibly as well of independent, variables. The PDE thus obtained is generally integrable, indeed the term C-integrable is used to denote such equations (to distinguish them from the S-integrable equations solvable via IST: the letter C refers to the Change of variables, the letter S to the Spectral, or Scattering, transform). A simple instance of C-integrable equations is the Burgers equation (5), which is linearized via the change of dependent variable Z x

~ðx; tÞ ¼ qðx; tÞ exp  q dyqðy; tÞ 1

~ðx; tÞ q Rx qðx; tÞ ¼ ~ðy; tÞ 1  1 dy q

~xx ¼ 0 ~t þ q q A second example is the ‘‘Liouville equation’’ uxt ¼ expðuÞ

½58a

or equivalently, in ‘‘light-cone coordinates’’ ð = x þ t,  = x þ tÞ u  u ¼ expðuÞ

½58b

the general solution of which reads  Z x dx0 exp½f ðx0 Þ uðx; tÞ ¼ f ðxÞ  gðtÞ  2 log a x0  Z t þ ð2aÞ1 dt0 exp½gðt0 Þ t0

with f(x) and g(t) arbitrary functions and x0 , t0 , a arbitrary constants. And a third example is the Eckhaus equation n h   i o qt ¼ i qxx þ 2 jqj2 þ jqj4 q ½59 x

which is linearized by the transformation Z x

2 ^ðx; tÞ ¼ qðx; tÞ exp q dyjqðy; tÞj 1

^ðx; tÞ q ffi qðx; tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rx ^ðy; tÞj2 1 þ 2 1 dyjq entailing the linear PDE ^t ¼ i^ q qxx Thanks to the simplicity of the technique to solve them, C-integrable PDEs provide a convenient tool to investigate the phenomenology associated with nonlinear PDEs. For instance the Burgers equation (5), which possesses kink-like solitons, is a simple nonlinear generalization of the heat equation; and the ‘‘relativistic invariance’’ of the Liouville equation, see (58b), makes it a convenient ‘‘toy model’’ in the context of relativistic field theory. The Eckhaus equation, (59), provides an interesting theoretical tool because of its similarity with the phenomenologically important NLS equation (6), as well as the fact that, thanks to its C-integrability, the structure of its solutions – which feature a remarkable solitonic zoology, including the possibility of ‘‘anelastic’’ solitonic reactions – can be studied in considerable detail, entailing an understanding of why such anelastic reactions are unlikely to be featured by solutions obtained in the context of the initialvalue problem.

Integrable Systems: Overview

C-integrable PDEs are generally as well S-integrable, being generally associable with a spectral problem that can be explicitly solved; the converse, instead, is not generally true. Hence C-integrability represents a higher level of integrability than S-integrability; a ranking that is quite useful in spite of its lack of strict cogency caused by the possibility to consider also the transformation from a function to its spectral transform as a change of (dependent) variable. The Lax approach, described in some detail above in the context of finite-dimensional integrable dynamical systems, was in fact originally invented in the context of integrable PDEs. For instance the KdV equation (35) corresponds to the (operator) Lax equation (to be compared with the matrix Lax equation (14)) Lt ¼ ½L; M where now the Schro¨dinger operator L is defined by (36) (so that Lt = ut ðx, tÞÞ and the operator M is defined as follows:

3 @ @ þ 3ux ðx; tÞ M ¼ 4 þ 6uðx; tÞ @x @x Closely connected with this approach is the AKNS method (due to M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur), based on the observation that the KdV equation (35) coincides with the integrability condition xxt

¼

½60

txx

for the following pair of linear PDEs (the first of which is just the eigenvalue equation for the Schro¨dinger operator L, see (36)) satisfied by the function ðx, k, tÞ : xx

t

  ¼ uðx; tÞ  k2

  ¼ ux ðx; tÞ þ 4ik3   þ 2 uðx; tÞ þ 2 k2

½61a

½61b

x

and, more generally, that every equation of the class (33) coincides with the integrability condition (60) for the eigenvalue equation (61a) and the equation t

¼ aðx; k; tÞ

þ bðx; k; tÞ

x

½61c

with an appropriate choice of the two functions aðx, k, tÞ and bðx, k, tÞ. Indeed this ansatz, (61c), with aðx, k, tÞ and bðx, k, tÞ low-order polynomials in k, provides a quite straightforward technique to identify the simpler equations of the class (33); ditto

121

for the extension of this approach based on more general eigenvalue problems than (61a). Another powerful approach suitable to identify and investigate integrable PDEs is the so-called ‘‘dressing method’’ (introduced by V. E. Zakharov and A. B. Shabat and pursued by many others), in which one starts again (as in the approach leading to C-integrable equations) from an easily solvable evolution equation and then performs transformations (less elementary than just a change of variables) that modify (‘‘dress’’) the original equation, obtaining thereby new (nontrivial and interesting) evolution equations, the integrability of which hinges on the control one has on the (dressing) transformation relating (both ways) the solutions of the new equations with those of the original equation. Of course many specific techniques are accommodated within this (admittedly vague) description; we must confine our remarks here to noting the crucial role that the Riemann-Hilbert problem generally plays in this context (indeed the Riemann-Hilbert problem also lies at the core of the solvability of the inverse spectral problem, although techniques not explicitly relying on it are also available). Algorithmic approaches, particularly suitable to manufacture multisolitonic solutions and to identify nonlinear PDEs that are integrable inasmuch as they feature such solutions, were developed already at the beginning of the 70’s. The pioneer of this approach was R. Hirota; less than a decade later a more sophisticated and general development – the so-called ‘‘tau-function’’ method – was invented by M. Sato and his pupils/collaborators. Finally let us mention that many remarkable connections exist among integrable PDEs and integrable finite-dimensional dynamical systems such as those discussed above; for instance the time-evolution (taking generally place in the complex plane) of the poles of rational solutions of certain integrable PDEs obey the equations of motion of integrable dynamical systems interpretable as many-body problems. Why are certain nonlinear PDEs both integrable and widely applicable?

Several integrable PDEs play a key role in various applicative contexts, justifying the question figuring as title of this subsection. A metamathematical but enlightening, and heuristically quite useful, reply to this question reads as follows. Consider as starting point a large class of nonlinear PDEs, and associate to it via some kind of asymptotic limit procedure a single nonlinear

122 Integrable Systems: Overview

PDE – to which it is then justified to attribute a certain universal character. If this procedure corresponds to a physically (or, more generally, applicatively) significant limit, it stands to reason that this universal PDE play a role in several applicative contexts (because the original class of PDEs, being large, certainly contains several equations of applicative relevance). And if the limit procedure is in some sense asymptotically exact, and it therefore preserves the property of integrability, it is also likely that this universal PDE be integrable, because for this it is sufficient that the original, large class of PDEs contain just one integrable PDE. For instance most phenomena characterized by a dominant dispersive plane wave in a weakly nonlinear context can be shown, via an asymptotically exact multiscale expansion, to be modeled by the Nonlinear Schroedinger equation (6), the solution of which provides then the evolution, in appropriately rescaled ‘‘slow’’ and ‘‘coarse-grained’’ time and space variables, of the amplitude modulation of the dominant dispersive wave. This explains why this nonlinear PDE plays a key role in so many, disparate applicative contexts, and it also implies, in the light of the above argument, its integrability. The reasoning outlined above is quite robust, and it allows to infer that, if instead the universal limit equation is not integrable, then the large class of PDEs from which it originates cannot contain any integrable equation, providing thereby the point of departure to obtain (quite useful) necessary conditions for integrability. Indeed these conditions are adequate to distinguish among different levels of integrability, for instance among C-integrability and S-integrability; with the Eckhaus equation (59) playing in this context a somewhat analogous role for C-integrable PDEs to that played by the Nonlinear Schro¨dinger equation (6) for S-integrable PDEs.

Outlook Many more important developments than could be covered in this overview have occurred in the last few decades; for these we refer to the books listed below (and there are many more), and to the literature cited there. Let us end this entry by emphasizing that both the study of integrable systems, and its application to phenomenologically interesting situation – including technological innovations, for instance in nonlinear optics and telecommunications – are still in the forefront of current research; although perhaps the ‘‘heroic era’’ of this field of study is over.

See also: Abelian Higgs Vortices; Ba¨cklund Transformations; Bethe Ansatz; Bifurcations of Periodic Orbits; Bi-Hamiltonian Methods in Soliton Theory; Billiards in Bounded Convex Domains; Boundary-Value Problems for Integrable Equations; Breaking Water Waves; Calogero–Moser–Sutherland Systems of Nonrelativistic and Relativistic Type; Cauchy Problem for Burgers-type Equations; Cellular Automata; Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups;  @-Approach to Integrable Systems; Einstein Equations: Exact Solutions; Functional Equations and Integrable Systems; Ginzburg–Landau Equation; Hamiltonian Systems: Obstructions to Integrability; Holonomic Quantum Fields; Instantons: Topological Aspects; Integrability and Quantum Field theory; Integrable Discrete Systems; Integrable Systems and Algebraic Geometry; Integrable Systems and Discrete Geometry; Integrable Systems and the Inverse Scattering Method; Integrable Systems in Random Matrix Theory; Inverse Problem in Classical Mechanics; Isochronous Systems; Isomonodromic Deformations; Integrable Systems and Recursion Operators on Symplectic and Jacobi Manifolds; Korteweg–de Vries Equation and Other Modulation Equations; Multi-Hamiltonian Systems; Nonlinear Schro¨dinger Equations; Ordinary Special Functions; Painleve´ Equations; Peakons; q-Special Functions; Quantum Calogero–Moser Systems; Quantum n-Body Problem; Random Matrix Theory in Physics; Recursion Operators in Classical Mechanics; Riemann–Hilbert Methods in Integrable Systems; Riemann–Hilbert Problem; Separation of Variables for Differential Equations; Sine-Gordon Equation; Solitons and Kac–Moody Lie Algebras; Solitons and Other Extended Field Configurations; Twistors; Toda Lattices; Vortex Dynamics; WDVV Equations and Frobenius Manifolds; Yang–Baxter Equations.

Further Reading Ablowitz MJ and Clarkson PA (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press. Ablowitz MJ and Segur H (1981) Solitons and the Inverse Scattering Transform. Philadelphia: SIAM. Babelon O, Bernard D, and Talon M (2003) Introduction to Classical Integrable Systems. Cambridge: Cambridge University Press. Bullough RK and Caudrey PJ (eds.) (1980) Solitons. Heisenberg: Springer. Calogero F (ed.) (1978) Nonlinear Evolution Equations Solvable by the Spectral Transform. London: Pitman. Calogero F (2001) Classical Many-Body Problems Amenable to Exact Treatments. Heidelberg: Springer. Calogero F and Degasperis A (1982) Spectral Transform and Solitons. I. Amsterdam: North Holland. Dodd RK, Eilbeck JC, Gibbon JD, and Morris HC (1982) Solitons and Non-linear Wave Equations. New York: Academic Press. Faddeev LD and Takhtajan LA (1987) Hamiltonian Methods in the Theory of Solitons. Heidelberg: Springer.

Interacting Particle Systems and Hydrodynamic Equations Hoppe J (1992) Lectures on Integrable Systems. Heidelberg: Springer. Konopelchenko GB (1987) Nonlinear Integrable Equations. Heidelberg: Springer. Novikov SP, Manakov SV, Pitaevskii LP, and Zakharov VE (1984) Theory of Solitons: the Inverse Scattering Method. New York: Plenum Press. Moser J (ed.) (1975) Dynamical Systems, Theory and Applications. Heidelberg: Springer.

123

Moser J (1981) Integrable Hamiltonian Systems and Spectral Theory. Pisa: Scuola Normale Superiore. Perelomov AM (1990) Integrable Systems of Classical Mechanics and Lie Algebras. Basel: Birkhauser. Toda M (1981) Theory of Nonlinear Lattices. Heidelberg: Springer. van Diejen JF and Vinet L (eds.) (2000) Calogero-Moser-Sutherland Models. Heidelberg: Springer. Zakharov VE (ed.) (1991), What is Integrability?. Heidelberg: Springer.

Interacting Particle Systems and Hydrodynamic Equations C Landim, IMPA, Rio de Janeiro, Brazil and UMR 6085, Universite´ de Rouen, France ª 2006 Elsevier Ltd. All rights reserved.

Introduction We present the theory of hydrodynamic behavior of interacting particle systems in the context of exclusion processes, in which no more than one particle per site is allowed. Denote by TN = Z=NZ the discrete torus with N points d and let TdN = (TN )d . The state space E N = {0, 1}TN consists of all configurations obtained by distributing particles on the discrete torus TdN respecting the exclusion rule which prevents more than one particle per site. The configurations are denoted by the Greek letter so that (x) is equal to 0 or 1 if site x 2 TdN is vacant or occupied for the configuration . Denote by {x : x 2 Zd } the group of translations in E N : (x )(z) = (x þ z) for each x, z in Zd . Here and below summations are performed modulo N. A d function f : {0, 1}Z ! R with finite support is called a cylinder function. Fix a family of non-negative cylinder functions cj , 1  j  d. Let cx, xþej ( ) = cj (x ) and consider the Markov process { t : t  0} on E N with generator LN given by ðLN f Þð Þ¼

d X X

cx; xþej ð Þ½f ðx; xþej Þf ð Þ

½1

j¼1 x2Td N

Here, {e1 , . . . , ed } stands for the canonical basis of Rd and x,y for the configuration obtained from by exchanging the occupation variables (x) and (y): 8 < ðzÞ if z 6¼ x; y ð x; y ÞðzÞ ¼ ðyÞ if z ¼ x ½2 :

ðxÞ if z ¼ y In this dynamics at each bond {x, x þ ej } the occupation variables (x), (x þ ej ) are exchanged at rate cx, xþej ( ). This happens simultaneously and independently at each bond.

Notice that the total number of particles is conserved by the dynamics since only exchanges are allowed. Denote by N, K (0  K  jTdN j) the hyperplane of all configurations of E N with K particles. Assume that the rates cj are nondegenerate for t to be an irreducible Markov process on each N, K . For 0    1, denote by N the Bernoulli product measure of parameter  on E N . Under N , the variables { (x), x 2 TdN } are independent, with marginals given by N f ðxÞ ¼ 1g ¼  ¼ 1  N f ðxÞ ¼ 0g Assume that the measures N , 0    1 are stationary for the Markov process t . An elementary computation shows that this is the case if each function cj does not depend on (0), (ej ), in which case the process is in fact reversible with respect to N . Let Mþ (Td ) be the space of finite positive measures on the torus Td endowed with the weak topology. For each configuration , let N = N ( , du) be the positive measure on Td obtained by assigning mass N d to each particle: X

ðxÞx=NðduÞ ½3 N :¼ N d x2TdN

where u stands for the Dirac measure on u. The measure N is called the empirical measure associated to the configuration . The integral of a continuous function G : Td ! R with respect to N is denoted by h N ; Gi ¼ N d

X

Gðx=NÞ ðxÞ

x2TdN

Fix a density profile 0 : Td ! [0, 1]. A sequence of probability measures N on E N is said to be associated to 0 if N converges in probability to 0 (u)du under N : ) (  Z   N  N  GðuÞ0 ðuÞ du >  ¼ 0 lim h ; Gi  d N!1 T

124 Interacting Particle Systems and Hydrodynamic Equations

for all continuous functions G : Td ! R and all  > 0. For a continuous profile 0 consider, for instance, the product measure  N0 () on E N whose marginals are given by  N0 ðÞ fðxÞ ¼ 1g ¼ 0 ðx=NÞ It is easy to check that the sequence of probability measures  N0 () is associated to 0 . Denote by Wx, xþej the instantaneous current of particles from x to x þ ej . This is the rate at which a particle jumps from x to x þ ej minus the rate at which a particle jumps from x þ ej to x: Wx; xþej ¼ fðxÞ  ðx þ ej Þgcx; xþej ðÞ Suppose that the mean value of the current vanishes under all stationary states N . This denotes that the average displacement of each particle vanishes in the mean. In particular, in view of the central limit theorem, to observe an evolution of the density in the macroscopic scale, a diffusive rescaling of time is needed. On the other hand, if there is a net flux of particles, the evolution has to be examined in the Euler scale tN. Denote by (N) the time rescaling: N 2 if the mean displacement of particles vanishes and N otherwise. For each probability measure N on E N , let P N be the probability measure on the path space D(R þ , E N ) induced by N and the Markov process t speeded up by (N). Expectation with respect to P N is denoted by E N . N Denote by N t (du) =  (t(N) , du) the empirical measure at time t. Fix a density profile 0 : Td ! [0, 1] and a sequence of probability measures N on E N associated to 0 . The goal of the theory of hydrodynamic limit of interacting particle systems is to show that for each t > 0, N t converges, as N " 1, to a deterministic path (t, du) = (t, u)du whose density is the solution of some partial differential equation, called the hydrodynamic equation. The main tools available are entropy production and Dirichlet forms. Denote by HN ( N j N ) the entropy of a probability measure N on E N with respect to a reference probability measure  N : (Z ) Z HN ð N j N Þ ¼ sup f

f d N  log

EN

ef d N

EN

where the supremum is carried over all functions f : E N ! R. It follows from the general theory of Markov processes that the entropy of the state of the process with respect to an invariant state decreases in time. The rate at which the entropy production decreases can be estimated by the Dirichlet form: let SN t be the

semigroup associated to the generator LN defined in [1] speeded up by (N). An elementary computation gives that Z t  N N N   HN St j þ 2ðNÞ ds IN N SN s 0    HN N jN Here, IN ( N ) is the convex and lower semicontinuous functional given by IN ð N Þ ¼ hf 1=2 ; LN f 1=2 iN where f stands for the Radon–Nikodym derivative d N =dN and h , iN for the scalar product in L2 (N ). Therefore, if the initial state N has entropy with respect to a reference measure N bounded by C0 N d , by convexity of IN ,   N N d HN N SN t j  Z t  þ 2tðNÞN d IN t1 ds N SN  C0 ½4 s 0

for all t  0. This elementary estimate plays a fundamental role in the following sections.

The Entropy Method Consider an exclusion process with generator given by [1]. Fix T > 0, a density profile 0 : Td ! [0, 1] and a sequence of probability measures N associated to 0 . Let Q N be the measure on the path space D([0, T], Mþ (Td )) induced by the process N t and the initial state N . To prove that N converges to (t, u)du in t probability, we first show that the sequence Q N converges to the probability measure Q  concentrated on the deterministic trajectory (t, u)du, whose density is the solution of some partial differential equation with initial condition 0 . It follows from this result and general arguments that N t converges to (t, u)du for each 0  t  T. To prove that Q N converges to Q  , assume that we are able to prove tightness of the sequence Q N . Since there is at most one particle per site, all limit points Q  of the sequence Q N are concentrated on trajectories (t, du) = (t, u)du, which are absolutely continuous with respect to Lebesgue. To characterize the limit points Q  , fix a smooth function G : Td ! R and consider the martingale     N MG; ¼ tN ; G  0N ; G t Z t    ðNÞLN N ½5 s ; G ds 0

Interacting Particle Systems and Hydrodynamic Equations

An elementary computation of its quadratic variation N shows that MG, vanishes in L2 (P N ) as N " 1. t Denote by C0 the space of cylinder functions which have zero mean with respect to all invariant states N . Assume that the currents W0, ej , 1  j  d, belong to C0 so that a diffusive scaling (N) = N 2 is in force. Notice that LN ðxÞ ¼

d X

Wxej ; x  Wx; xþej

j¼1

In particular, after a summation by parts, the integral term on the right-hand side of [5] can be written as Z t d X X N 1d ðrN ½6 uj HÞðx=NÞWx; xþej ðsÞ ds 0

j¼1 x2Td N

where (rN uj H)(x=N) = N{H(x þ ej =N)  H(x=N)}. Notice that this sum is in principle of order N. To illustrate the entropy method, consider the symmetric simple exclusion process obtained by taking cj = 1=2 in [1] and observe that the current W0, ej = (1=2){(0)  (ej )}. A second summation by parts permits to rewrite the martingale [5] as Z  N   N  1 t N  t ; G  0 ; G  s ; N G ds 2 0 where N is the discrete Laplacian. Since the martingale MtG, N vanishes in L2 (P N ), as N " 1, all limit points Q  are concentrated on weak solutions of the linear heat equation. It remains to recall that there is a unique weak solution of the Cauchy problem for the heat equation to conclude that the sequence Q N converges to Q  , the measure concentrated on the deterministic path t (du) = (t, u)du whose density is the solution of the heat equation with initial condition 0 . The symmetric simple exclusion process has the very special property that the martingale MtG, N can be written as a function of the empirical measure. This is not the case for all the other models, for which a further argument is needed to close eqn [5] in terms of the empirical measure. To present the additional arguments needed, assume that cj () = 1 þ [(ej ) þ (2ej )]. In this case, the current W0, ej is equal to fð0Þ  ðej Þg þ fð0Þðej Þ  ðej Þð2ej Þg þ fð0Þð2ej Þ  ðej Þðej Þg A second summation by parts in [6] permits to rewrite it as Z t d X X N d ð@u2j HÞðx=NÞx hðsN2 Þ ds þ oN ð1Þ ½7 0

j¼1 x2Td N

125

where h() = (0) þ 2(0)(ej )  (0)(2ej ). The remainder oN (1) appears because we replaced discrete space derivatives by continuous ones. In contrast with the symmetric simple exclusion process, the martingale MtG, N defined in [5] is not a function of the empirical measure and an argument is needed to close the equation. For each positive integer ‘ and d-dimensional integer x, denote by ‘ (x) the empirical density of particles in a box of length 2‘ þ 1 centered at x: X 1 ðyÞ ‘ ðxÞ ¼ d ð2‘ þ 1Þ jyxj‘ ~ be the For a cylinder function h : E N ! R, let h() expected value of h with respect to the invariant ~ = E N [h()]. For ‘  1 and a cylinder state N : h()  function h, let   X   1 ‘ ~  V‘ ðÞ ¼  ðy hÞðÞ  hð ð0ÞÞ d ð2‘ þ 1Þ jyj‘ Theorem 1 Consider a sequence of probability measures mN on E N such that IN (mN )  C0 N d2 for some 0 <  < 1 and some finite constant C0 . Then, 2 3 X lim sup lim sup E N 4N d x V"N ðÞ5 ¼ 0 "!0

N!1

x2TdN

This statement, due to Guo et al. (1988), permits the replacement of a local function h by a function of the density of particles over a macroscopic cube. It is the main step in the proof of the hydrodynamic behavior of gradient systems, defined below, and its proof can be found in Kipnis and Landim (1999, chapter 5). Assume that the sequence N has entropy with respect to a reference invariant state N bounded by C0 N d for some finite constant C0 . It follows R T from [4] that the sequence of measures T 1 0 ds N SN s satisfies the assumptions of Theorem 1. Therefore, due to the presence of the time integral, we may ~ "N (x)). replace the cylinder function h in [7] by h( Since "N (0) can be written as hN , " i, where " = (2")d 1{[", "]d }, we now have expressed the martingale [5] in terms of the empirical measure. Repeating the arguments presented for the symmetric simple exclusion process, we may conclude that all limit points Q  of the sequence Q N are concentrated on paths t (du) = (t, u)du, whose density is a weak solution of the parabolic equation ( @t ¼ ð þ 2 Þ ð0; Þ ¼ 0 ðÞ

126 Interacting Particle Systems and Hydrodynamic Equations

~ =  þ 2 for h() = (0) þ 2(0)(ej ) because h() (0)(2ej ). It remains to show the uniqueness of weak solutions of this differential equation to conclude. The second integration by parts in [6] was possible because the currents could be written as the difference of local functions and their translations, a very special property not shared by most interacting particle systems. Processes with this attribute are called gradient systems.

the cube {‘, . . . , ‘}d and by L‘ the restriction of the generator LN to the cube ‘ , obtained by suppressing all jumps from ‘ (resp. c‘ ) to c‘ (resp. ‘ ). For 0  K  j‘ j, let ‘ , K be the uniform measure on the configurations of {0, 1}‘ with K particles. The following estimate is needed in the proof of Theorem 2: Theorem 3 that

There exists a finite constant C0 such

hf ; f i ; K  C0 ‘2 hf ; ðL‘ Þf i ; K ‘

Nongradient Models Consider an exclusion process with rates cj () = 1 þ (ej ), in which case the current is given by W0; ej ¼ fð0Þ  ðej Þg þ fð0Þ  ðej Þgðej Þ a cylinder function in C0 . Fix T > 0, a density profile 0 : Td ! [0, 1] and a sequence of probability measures N associated to 0 and having entropy with respect to a reference invariant state N bounded by C0 N d for some finite constant C0 . Recall the definition of the sequence of measures Q N , assumed to be tight. To characterize the limit points of Q N , fix a smooth function G : Td ! R and examine the N martingale MG, introduced in [5]. After an t integration by parts, the integral term of the martingale becomes [6]. While a second integration by parts is possible for the first part of the current (0)  (ej ), the second piece remains Z t d X  X  N 1d ½8 rN uj H ðx=NÞx wj ðsN2 Þ ds 0

j¼1 x2Td N

where wj = {(0)  (ej )}(ej ). Notice the extra factor N multiplying the sum and that wj belongs to C0 . The next result and Theorem 4 are due to Varadhan (1994). Theorem 2 Consider a sequence of probability measures mN on E N such that HN (mN jN )  C0 N d for some 0 <  < 1 and some finite constant C0 . Fix a smooth function G : Td ! R and a cylinder function  in C0 . There exists a seminorm kk such that (

 Z  lim sup EmN  N!1

T

dsN

1d

0

 C0 TkGk22 sup kk2

X x2TdN

 )2  Gðx=NÞx ðsN2 Þ ½9

01

The explicit form of the seminorm kk can be found in Kipnis and Landim (1999, chapter 7). The proof of Theorem 2 requires a sharp estimate on the spectral gap of the generator LN . Denote by ‘

‘

for all ‘  1, 0  K  j‘ j and zero-mean function f in L2 (‘ , K ). This result is due to Quastel (1992) for symmetric simple exclusion processes. Yau developed a general method to prove sharp estimates for the spectral gap of the generator for conservative dynamics (see Lu and Yau (1993) and Yau (1997)). Since the parallelogram identity is easy to check, by polarization we can define a semi-inner product   ,   from the seminorm kk . Denote by H the Hilbert space induced by C0 and the semi-inner product   ,   . Denote by L the generator [1] extended to Zd . Notice that Lf belongs to C0 for any cylinder function f, and that the gradients (ej )  (0), and the currents wj , 1  j  d, also belong to C0 . The next result states that all functions in H can be written as a linear combination of gradients and cylinder functions in the image of the generator. Theorem 4 Denote by LC0 the space {Lg : g 2 C0 }. For each 0    1, H ¼ LC0 fðej Þ  ð0Þ : 1  j  dg In particular, there exists a matrix {Di, j () : 1  i, j  d} and a sequence of functions {fi, k (,) 2 C0 : k  1}, 1  i  d, for which wi þ

d X

Di; j ðÞfðej Þ  ð0Þg  Lfi; k ð; Þ

j¼1

vanishes in H as k " 1. For reversible systems (and more generally for generators satisfying a sector condition), it can be shown that the sequence of local functions fi, k (, ) can be taken independent of  : fi, k (, ) = fi, k (). Moreover, with a little extra effort, one obtains a bound uniform in :   d   X   inf sup wi þ Di; j ðÞfðej Þ  ð0Þg  Lf  ¼ 0  f 2C0 01 j¼1

½10



This estimate together with some algebraic relations in H give a variational formula for the matrix Di, j :

Interacting Particle Systems and Hydrodynamic Equations

for every vector v in Rd ,

Z

2   X d 1   inf  v  DðÞv ¼ vi wi  Lf   ð1  Þ f 2C0  i¼1

It can also be shown that the matrix D is continuous and strictly elliptic. We may now complete the proof of the hydrodynamic behavior. Recall that the main difficulty was to express formula [8] in terms of the empirical measure. Fix 1  i  d and consider a sequence of cylinder functions {fi, k : k  1} satisfying [10] asymptotically as P k " 1. Adding and subtracting the "N "N "N expression 1kd Dj, k ( (0)){ (ej )   (0)}  Lfj, k , [8] becomes the sum of three terms. The first one is just the expression which appears inside the expectation in [9] with G = (rN uj H) and  given by d X

dsN d

0

½11



wj þ

t

Dj; k ð"N ð0ÞÞf"N ðej Þ  "N ð0Þg  Lfj; k

d X X

127

  "N  @u2j ; uk H ðx=NÞ dj; k sN 2 ðxÞ

j; k¼1 x2Td

N

þ oN ð1Þ where dj,0 k = Dj, k . We have already seen in the derivation of the hydrodynamic equation for gradient systems that this sum can be expressed as a function of the empirical measure. Since all limit points are concentrated on paths t (du) which are absolutely continuous, this integral converges to Z d Z t X   ds du @u2j ; uk H ðuÞ dj; k ð ðs; uÞÞ j; k¼1

0

Td

Since the martingale [5] vanishes, all limit points are concentrated on trajectories t (du) = (t, u)du which are weak solutions of @t ¼

d X

@ uj





j; k þ Dj; k ð Þ @uk

j; k¼1

k¼1 N

Since the sequence of measure satisfies the assumptions of Theorem 2, a modification of the proof of this theorem, to take into account the dependence of  on N and ", shows that the limit of the expectation of the absolute value of the first term in the decomposition, as N " 1 and then " # 0, is bounded by C0 Tk@uj Hk22 sup kj;  k2

where D is the strictly elliptic and continuous matrix given by the variational formula [11]. Here, the identity matrix j, k comes from the first piece of the current which permitted a second integration by parts. A uniqueness result of weak solutions of the Cauchy problem with initial condition 0 concludes the proof of the hydrodynamic behavior of this nongradient system.

01

where

Hyperbolic Equations

j;  ¼ wj þ

d X

Dj; k ðÞfðej Þ  ð0Þg  Lfj; k

k¼1

By [10], the penultimate expression vanishes as k " 1. The second term in the decomposition is Z t d X X  rN dsN 1d uj H ðx=NÞx Lfj; k ðsN 2 Þ 0

j; k¼1 x2Td

N

The presence of the generator L and the diffusive rescaling of time permit to show that the expectation of the absolute value of this expression is of order N1 for each fixed k. Finally, the third term is equal to Z t d X X  rN  dsN1d uj H ðx=NÞDj; k 0



j; k¼1 x2Td

N

 "N

"N "N  sN sN2 ðx þ ek Þ  sN 2 ðxÞ 2 ðxÞ A second integration by parts is now possible and one obtains that the previous expression is equal to

Consider the asymmetric simple exclusion process obtained by setting cj () = (0)[1  (ej )] in formula [1]. Notice that the current W0, ej = (0)[1  (ej )] has mean (1  ) with respect to the invariant state N , suggesting the Euler rescaling of time (N) = N. Let  be the partial order on E N defined by   if (x)  (x) for every x in TdN . The asymmetric exclusion process is attractive: there exists a stochastic evolution on E N  E N with the following two properties: (1) it preserves the order, in the sense that t  t for all t  0 if 0  0 and (2) each coordinate evolves according to the original asymmetric exclusion dynamics. This coupling, which may be constructed by letting particles jump together as much as possible, is the main tool in the derivation of the hydrodynamic equation of asymmetric processes. Fix a smooth function G : Td ! R and recall N definition [5] of the martingale MG, . An element tary computation shows that the quadratic variation of this martingale vanishes as N " 1. On the other

128 Interacting Particle Systems and Hydrodynamic Equations

hand, after an integration by parts, the integral term of the martingale becomes Z t d X  X  rN N d uj H ðx=NÞsN ðxÞ 0

j¼1 x2Td N

informally described at the beginning of this section. Rezakhanlou (1991) proved the following theorem: Theorem 5 For every smooth positive function H with compact support in (0, 1)  Td and every " > 0,

 ½1  sN ðx þ ej Þ ds Assume that the state of the process at any macroscopic time s is close to a product measure associated to some profile (s,). Since the martingale vanishes asymptotically, taking expectations in [5], we obtain that the density profile should be a weak solution of the quasilinear hyperbolic equation @t þ

d X

@uj Fð Þ ¼ 0

½12

j¼1

where F(a) = a(1  a). It is well known that solutions of this equation may develop shocks even if the initial profile 0 (  ) is smooth and that there is no uniqueness of weak solutions. Several criteria have been introduced to select the relevant solution among the weak solutions. Kruzˇkov (1970), for instance, in the case where density profile 0 : Td ! R is bounded, proved that there exists a unique measurable function which satisfies the entropy condition @ t j  cj þ

d X

@ui jFð Þ  FðcÞj  0

½13

lim lim

‘!1 N!1

þ

d X

PN

N

"Z 0

1

dt Nd

X

  @t Hðt; x=NÞt‘ ðxÞ  t‘ ðxÞ

x2TdN

) #   ‘   ‘    ð@ui HÞðt;x=NÞ F t ðxÞ  F t ðxÞ  " ¼ 1

i¼1

If we now assume that the second coordinate t is initially distributed according to the stationary state N , it is not difficult to replace t‘ in the above formula by , obtaining a microscopic version of the entropy inequality. In the one-dimensional nearest-neighbor case, by coupling arguments, we may replace the average ‘ (0) over a large microscopic box by an average "N (0) over a small macroscopic box, deriving the entropy inequality [13]. To conclude the proof it remains to show, by means of coupling argument again, that the density profile at time t converges in L1 (Td ) to the initial condition as t # 0. In higher dimensions or in the one-dimensional non-nearest-neighbor case, it has not been proved that replacement of ‘ (0) by "N (0) is allowed. One is thus forced to consider measure-valued solutions of eqn [12]. Details can be found in Kipnis and Landim (1999, chapter 8).

i¼1

in the sense of distributions on (0, 1)  Td , for every c 2 R, and which converges to the initial condition in L1 (Td ) as t # 0: limt ! 0 k t  0 k1 = 0. Fix T > 0 and a density profile 0 : Td ! [0, 1]. To couple the original process with another one starting from a different initial sate, we need to impose the initial distribution to be of product form. Consider, therefore, a sequence of ‘‘product’’ probability measures N associated to 0 and recall the definition of the sequence of measures Q N given in the section ‘‘The entropy method,’’ assumed to be tight. We have to prove that all limit points are concentrated on entropy solutions of [12]. Coupling the original process t with another one, denoted by

t , starting from the Bernoulli product measure with density , and examining the time evolution of P we derive an entropy x2TdN jtN (x)  tN (x)j, inequality at the microscopic level: let N be a sequence of probability measures on the product space E N  E N whose first coordinate is N . Denote by PN

N the measure on the path space D([0, T], E N  E N ) induced by N and the coupling

Relative Entropy Method The relative entropy method, due to Yau (1991), is based on the analysis of the time evolution of the entropy of the state of the process with respect to the product measure associated to the solution of the hydrodynamic equation. While the entropy method requires uniqueness of weak solutions and proves the existence of weak solutions, the relative entropy method requires the existence of a smooth solution and proves the uniqueness of such smooth solutions. Consider the exclusion process with rates cj () = 1 þ [( ej ) þ (2ej )]. We have seen that the hydrodynamic equation of this model is given by the nonlinear parabolic equation @t ¼ f þ 2 g

½14

Fix a profile 0 : Td ! [0, 1] bounded away from 0 and 1: 0 <   0 (u)  1  . Let (t, u) be the solution of the hydrodynamic equation [14] with N initial condition 0 and denote by  (t, ) the product

Interacting Particle Systems and Hydrodynamic Equations

measure with slowly varying parameter associated to the profile (t, ): N f; ðxÞ ¼ 1g ¼ ðt; x=NÞ;  ðt;Þ

for x 2 TdN

Theorem 6 Let { N : N  1} be a sequence of probability measures on E N whose entropy with respect to  N0 () is of order o(N d ):   HN N j N0 ðÞ ¼ oðN d Þ Then, the relative entropy of the state of the process N at the macroscopic time t with respect to  (t, ) is d also of order o(N ):   N d H N N SN j t ðt;Þ ¼ oðN Þ for every t  0 It is not difficult to deduce from this result a strong version of the hydrodynamic limit behavior of the interacting particle system: Corollary 1 Under the assumptions of the theorem, for every cylinder function  and every continuous function H : Td ! R,  X  Hðx=NÞx ðÞ lim E N SN N d N!1



x2TdN

Z Td



 ~ HðuÞð ðt; uÞÞ du ¼ 0

The relative entropy method can be extended to nongradient systems and to asymmetric processes, whose macroscopic evolution is described by quasilinear hyperbolic equations, up to the first shock. The hydrodynamic behavior of an interacting particle system corresponds to a law of large numbers for the empirical measure. The central limit theorem is well understood in equilibrium, but remains to this date an important open question in nonequilibrium. The large deviations for diffusive systems have also been investigated, as well as the hydrodynamic behavior of systems in contact with reservoirs. The Navier–Stokes equations have been derived as a correction of the hydrodynamic equation of asymmetric particle systems. We refer to Kipnis and Landim (1999) for further details.

129

See also: Boltzmann Equation (Classical and Quantum); Bose–Einstein Condensates; Breaking Water Waves; Fourier Law; Interacting Stochastic Particle Systems; Macroscopic Fluctuations and Thermodynamic Functionals; Multi-Scale Approaches.

Further Reading De Masi A and Presutti E (1991) Mathematical Methods for Hydrodynamic Limits, Lecture Notes in Mathematics, vol. 1501. New York: Springer. Fritz J (2001) An Introduction to the Theory of Hydrodynamic Limits, Lectures in Mathematical Sciences, vol. 18. Tokyo: The University of Tokyo, ISSN 09198140. Guo MZ, Papanicolaou GC, and Varadhan SRS (1988) Nonlinear diffusion limit for a system with nearest neighbor interactions. Communications in Mathematical Physics 118: 31–59. Jensen L and Yau HT (1999) Hydrodynamical Scaling Limits of Simple Exclusion Models, IAS/Park City Mathematical Series, vol. 6, pp. 167–225. Providence, RI: American Mathematical Society. Kipnis C and Landim C (1999) Scaling Limits of Interacting Particle Systems. Grundlheren der mathematischen Wissenschaften, vol. 320. New York: Springer. Kruzˇkov SN (1970) First order quasilinear equations in several independent variables. Matematicheskii Sbornik 10: 217–243. Landim C (2004) Hydrodynamic Limits of Interacting Particle Systems, ICTP Lecture Notes, vol. 17, pp. 57–100. Trieste: Abdus Salam International Centre for Theoretical Physics. Lu SL and Yau HT (1993) Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Communications in Mathematical Physics 156: 399–433. Quastel J (1992) Diffusion of color in the simple exclusion process. Communications on Pure and Applied Mathematics XLV: 623–679. Rezakhanlou F (1991) Hydrodynamic limit for attractive particle systems on Zd . Communications in Mathematical Physics 140: 417–448. Spohn H (1991) Large Scale Dynamics of Interacting Particles. Berlin: Springer. Varadhan SRS (1994) Nonlinear diffusion limit for a system with nearest neighbor interactions II. In: Elworthy KD and Ikeda N (eds.) Asymptotic Problems in Probability Theory: Stochastic Models and Diffusion on Fractals, Pitman Research Notes in Mathematics, vol. 283, pp. 75–128. New York: Wiley. Varadhan SRS (2000) Lectures on hydrodynamic scaling. Fields Institute Communications 27: 3–40. Yau HT (1991) Relative entropy and hydrodynamics of Ginzburg–Landau models. Letters in Mathematical Physics 22: 63–80. Yau HT (1997) Logarithmic Sobolev inequality for generalized simple exclusion processes. Probability Theory and Related Fields 109: 507–538.

130 Interacting Stochastic Particle Systems

Interacting Stochastic Particle Systems H Spohn, Technische Universita¨t Mu¨nchen, Garching, Germany ª 2006 Elsevier Ltd. All rights reserved.

Introduction According to the basic principles of mechanics, the motion of atoms and molecules is governed, in the semiclassical approximation, by the deterministic Hamiltonian equations of motion. While all evidence points in this direction, for many problems this Hamiltonian approach is so complicated that it hardly yields any useful results. A simple example are many (109 ) polystyrene balls (size 1 mm) immersed in water. The Hamiltonian description would have to deal with the degrees of freedom of all the fluid molecules and all the polystyrene balls. Clearly, a more useful approach is to collect the incessant bombardment of a polystyrene ball by water molecules into a stochastic force acting on the ball with postulated statistical properties. For example, following Einstein, one could regard successive collisions as independent and occurring after an exponentially distributed waiting time. In addition to such stochastic forces, the polystyrene balls are charged and interact with each other through the screened Coulomb force. On the one-particle level, stochastic models have a long tradition within statistical physics. Considerable part of the classical theory of Markov processes is the mathematical response to such type of description. The aspect of interaction is more recent. Its origin can be traced back to the Metropolis algorithm in early computer simulations (ffi 1953). It was recognized that the Hamiltonian dynamics is a rather slow tool to statistically sample the Gibbs equilibrium distribution Z1 exp [H=kB T]. A more efficient route is to devise a stochastic algorithm which has as its unique stationary measure the Gibbs distribution. Such schemes are now known as Markov Chain Monte Carlo and of extremely wide use, not only in statistical physics but also in quantum chromodynamics (QCD) and other quantum field theories. The time appearing in the stochastic algorithm has no physical significance; it merely counts how often a certain operation is performed. The second clearly identifiable push toward the use of interacting stochastic particle systems came from the study of critical dynamics. Close to a point of second-order phase transition, the equilibrium properties are very effectively handled by means of statistical field theories. Thus, it was natural to

search for an extension into the time domain, which then led to time-dependent Ginzburg–Landau theories, where now time refers to physical time. These are interacting stochastic models, where one keeps only a few basic fields, together with their behavior under time reversal, their vector character, and whether they are dynamically conserved or not. In probability theory, interacting stochastic particle systems date back to the seminal papers by M Kac in 1956 and independently by R L Dobrushin and by F Spitzer in 1970. Spitzer was motivated by spin-flip and spin-exchange dynamics, while Dobrushin had the vision of many locally interacting components. In the early days, one of the prime goals was the construction of the stochastic process in infinite volume, an enterprise which had important mathematical spin-off, for example, the theory of Dirichlet forms on function spaces. Physical models offer a rich menu to the probabilist, but there is also considerable input from other areas. To give just one example: in queueing theory one considers queues in series, that is, a customer served at one counter immediately moves on to the next one. If one regards as field the number of customers at each counter, one has an interacting stochastic particle system, the interaction being mediated through the servers. This article is split into two sections. In the first one, we list and explain a few prototypical interacting stochastic particle systems. Of course, the list is hardly exhaustive and we restrict ourselves from the outset to models from statistical physics. In the second part, we summarize prominent lines of recent research. Again the wealth of material is overwhelming and we draw the line according to the rules of mathematical physics.

Model Systems Our list is determined by the intrinsic mathematical properties of the stochastic particle system. Alternatively, a classification is possible according to the physical system, which would, however, be less transparent for our purposes. We restrict ourselves to models with only position-like degrees of freedom, but if needed velocity-like fields may be included. The most basic distinction is the behavior under time reversal. A model is called (statistically) ‘‘time reversible’’ if a particular history and its timereversed image have the same probability. Technically, one imposes this through the condition of detailed balance. Nonreversible systems are much less explored, but currently a very active area of research.

Interacting Stochastic Particle Systems 131 Reversible Models

1. Spin-flip, Glauber dynamics. One considers spins attached to the sites of a regular lattice, which for symplicity we take as the hypercubic lattice Zd . The spin at site x 2 Zd is denoted by x = 1 and the whole spin configuration is denoted by . Thus, the state space of the Markov d processs is {1, 1}Z = . Spin configurations evolve in time through random spin flips, that is, through a change from x to x according to configuration-dependent rates cx (). cx () is local, in the sense that it depends only on the spins close to x, and is translation invariant, that is, if y is the shift by y, then cxþy (y ) = cx (). If the current spin configuration is (t), then after a short time dt  x ðt þ dtÞ ¼

x ðtÞ x ðtÞ

with probability 1  cx ððtÞÞdt with probability cx ððtÞÞdt

The update is performed independently at each lattice site. Technically, it is more concise to specify the generator, L, of the Markov process. It acts on local functions f : ! R and is given by Lf ðÞ ¼

X

cx ðÞðf ðx Þ  f ðÞÞ

½1

rates cxy () between x and y. They are local, translation invariant and symmetric, that is, cxy () = cyx (). The generator now reads Lf ðÞ ¼

1 X cxy ðÞðf ðxy Þ  f ðÞÞ 2 d

½3

x;y2Z

where xy is the configuration  with the occupancies at sites x and y exchanged. The condition of detailed balance refers to the exchange and reads cxy ðÞ ¼ cxy ðxy ÞeðHð

xy

ÞHðÞÞ

½4

In [4] P we can freely add to H the chemical potential  x x . Thus for stochastic lattice gases there is a one-parameter family of invariant measures, labeled by the chemical potential . 3. Interacting Brownian motions. These motions model, for example, suspensions as mentioned in the ‘‘Introduction’’. One considers a box   Rd containing N Brownian particles. The jth Brownian particle has position xj 2 . Thus, the state space of the Markov process is N . We assume that the Brownian particles interact through a (sufficiently local) even pair potential U. Then the total potential energy is

x2Zd

where  x denotes the configuration  with the spin at site x reversed. The transition probability from the configuration  to the configuration 0 in time t  0 is given by the matrix element (eLt ), 0 of the Markov semigroup eLt . To impose time reversibility, one needs an energy function H() constructed according to the rules of equilibrium statistical mechanics. The condition of detailed balance then reads

HðxÞ ¼

N 1X Uðxi  xj Þ; 2 i;j¼1

x

ÞHðÞÞ

½5

The dynamics of the Brownian particles is given through the stochastic differential equations dxj ðtÞ ¼ 

N X

rUðxj ðtÞ  xi ðtÞÞ dt

i¼1;i6¼j

þ cx ðÞ ¼ cx ðx ÞeðHð

x ¼ ðx1 ; . . . ; xN Þ

pffiffiffiffiffiffiffiffiffi 2D0 dWj ðtÞ;

j ¼ 1; . . . ; N

½6

½2

with  = 1=kB T the inverse temperature. Note that on the right only energy differences appear, which are always well defined. In finite volume the unique invariant measure is the Gibbs measure Z1 eH . 2. Spin-exchange, Kawasaki dynamics, stochastic lattice gases. We model particles hopping on the lattice Zd and switch to the occupation variables x , where x = 0 stands for site x empty and x = 1 stands ford site x occupied. The state space is  = {0, 1}Z . Since the number of particles is conserved, the basic dynamical process is a random jump of a particle from x to a nearby site y, provided y = 0. Therefore, we specify the exchange

Wj (t), j = 1, . . . , N, are a collection of independent Brownian motions and D0 is the diffusion coefficient of a single Brownian particle. Equation [6] has to be supplemented with suitable boundary conditions at the surface @. Since the forces in [6] are the gradient of a potential, time reversibility is automatically satisfied with the invariant measure being Z1 N exp(H(x)=D0 ) dx1    dxN . 4. Ginzburg–Landau models. Ginzburg–Landau models should be viewed as discretized versions of stochastic partial differential equations. At every lattice site x 2 Zd , there is a real-valued field x 2 R, a field configuration dbeing denoted by . Formally, the state space is RZ . Since the single-site space is noncompact, some growth condition at

132 Interacting Stochastic Particle Systems

infinity must be imposed. Next we give ourselves an energy, H(), one standard example being X X HðÞ ¼ ðx  y Þ2 þ Vðx Þ ½7 x;y2Zd ;jxyj¼1

x2Zd

The on-site potential increases sufficiently rapidly, so as to make large field values unlikely. The -field evolves according to the set of stochastic differential equations dx ðtÞ ¼ 

pffiffiffiffiffiffiffiffi @H ððtÞÞdt þ 2= dWx ðtÞ; @x

½8

x 2 Zd where {Wx (t), x 2 Zd } is a collection of independent Brownian motions. If V(x ) = 2x , then (t) is a Gaussian field theory. To have an Ising-type phase transition, one would have to choose V(x ) = 2x þ 4x . It is rather simple to modify [8] as to incorporate a conservation law. To each directed bond (x, y), jx  yj = 1, one associates the current jxy = jyx . If e is a unit vector, jej = 1, then X dx ðtÞ þ jxxþe ðtÞdt ¼ 0; x 2 Zd ½9 e;jej¼1

The current has both a deterministic part, given through the gradient of a chemical potential, and a random part:   @H @H jxy ðtÞdt ¼   ððtÞÞdt þ dWxy ðtÞ; @x @y ½10 jx  yj ¼ 1 where Wxy (t) = Wyx (t) is a collection of independent Brownian motions labeled byPnearest-neighbor bonds. The conserved quantity is x x . Again, the dynamics has a one-parameter family of stationary measures labeled by the ‘‘magnetic field’’. Since in [8] and [10] the drift is the gradient of a potential, Ginzburg–Landau models are reversible. 5. Interface dynamics. The scalar field  describes the location of an interface. The energy of an interface does not depend on its absolute displacements. Thus, interface models are special Ginzburg– Landau models, which have an energy H() invariant under the global shift x ! x þ a for all x 2 Zd . An example is HðÞ ¼

X

Vðx  y Þ

½11

x;y2Zd ;jxyj¼1

with even V. Note that in order to have a normalizable equilibrium measure, the interface must be pinned somewhere. 6. Several components. For lattice gases, there may be several components. In a Ginzburg–Landau theory

instead of a scalar, Ising-like field, one could consider a vector-valued, Heisenberg-like, field and require the energy to be invariant under global rotations of the field variables. The construction is as before and we do not have to repeat it. 7. Constrained, glassy dynamics. The constraint is enforced by setting some of the rates equal to zero. For example, in the case of standard Glauber dynamics, one could allow for a spin-flip only if at least two neighboring spins have the opposite sign. The Gibbs measure is still invariant, but the approach to equilibrium will be slowed down due to the constraint. It may even happen that the configuration space splits into several invariant subsets. After this long and still incomplete list, let us turn to the nonreversible models. Nonreversible Models

Mathematically, one merely has to drop the condition of detailed balance. To have a more concrete example, let Li be the generator for the Glauber dynamics satisfying detailed balance with inverse temperature i , i = 1, 2. Then L = L1 þ L2 generates a nonreversible dynamics provided 1 6¼ 2 . Physically, it corresponds to coupling the spins to two bulk thermal reservoirs of different temperatures. Our example leads to a general point which should be noted: While reversible models have a wide range of physical applicability, for nonreversible models nonequilibrium conditions have to be maintained over sufficiently long time spans, which poses considerable difficulties experimentally. Thus on a theoretical level, the efforts go into exploring properties of, say, semirealistic models. Very roughly there are two broad classes of nonreversible models.

Boundary-driven models We consider a finite volume . Inside  the dynamics is reversible as explained before. At the boundary @ the system is coupled to particle, resp. energy, reservoirs. In case the boundary chemical potential, resp. temperature, is not uniform, the dynamics is nonreversible. To be more concrete let us reconsider the lattice gas discussed in item (2) (see the discussion following eqn [2]). Inside  the generator L is given by [3] and satisfies detailed balance [4]. The boundary generator is X L@ f ðÞ ¼ cx ðÞðf ðx Þ  f ðÞÞ ½12 x2@

where the notation is as in [1] with {1, 1} substituted by {0, 1} . cx () satisfies [2] with the same  as in the bulk, but a chemical potential x depending on x 2 @. x controls the injection/

Interacting Stochastic Particle Systems 133

absorption of particles at x. The generator for the nonreversible dynamics is then L ¼ L þ L@

½13

Bulk-driven models A prototype is the twotemperature model mentioned above. More widely studied is a nonconservative force acting globally. Here the standard example are particles moving in  with periodic boundary conditions and subject to an additional uniform force field of strength F, which clearly cannot be written as the gradient of a potential. In the case of Brownian particles, by changing to a comoving frame of reference, one would be back to the reversible case F = 0. For lattice gases the lattice provides a fixed frame and the driven model has properties very different from the undriven one. This leads us to: 8. Driven lattice gases. The generator L is still given by [3]. Formally, we Pinsert in [4] instead of H the Hamiltonian H()  x (F  x)x . The exchange rates then satisfy the condition of ‘‘local’’ detailed balance as cxy ðÞ ¼ cxy ðxy Þ eðHð

xy

ÞHðÞÞ

eðFðxyÞÞðx y Þ

½14

This means, particles preferentially jump in the direction of F. On the infinite lattice the dynamics admits two classes of stationary measures. First, there is the Gibbs measure with particles piling up along F and formally given by P Z1 e

ðHðÞ

ðFxÞx Þ

x

½15

With respect to this measure the dynamics is reversible. Second, there are translation invariant measures with nonzero steady-state current. This cannot happen for reversible models. A very widely studied particular case is the asymmetric simple exclusion process for which d = 1, H() = 0, and jumps are only to nearest-neighbor sites.

Items of Interest As there are thousands of research papers in mathematical physics alone, it is literally impossible to provide any sort of summary. On the other hand, the type of questions investigated are generic. Thus, we just explain what one would like to understand without paying much attention to the fractal boundary between ‘‘proven’’ and ‘‘unproven.’’ For the construction of the stochastic processes listed above, there is a well-developed probabilistic theory available. Thus, the main focus is on ‘‘qualitative properties’’ of the stochastic particle system. As in

the previous section, we distinguish between reversible and nonreversible models. Reversible Models

1. Equilibrium state. The most basic question concerns the classification of invariant measures in infinite volume. By construction, they are the Gibbs measures for the Hamiltonian appearing in the condition of detailed balance. In principle there could be more, which so far has been excluded only in dimension 1 or 2. Properties of the invariant measure belong to the domain of equilibrium statistical mechanics. Thus we can turn directly to: 2. Spectral analysis of the generator L. We fix some extreme Gibbs measure stationary for L. By detailed balance, eLt is a symmetric Markov semigroup in L2 (, ). Hence, L is self-adjoint and L 0. Furthermore, it has a nondegenerate eigenvalue 0. The rate of approach to equilibrium is determined by the spectral gap of L. Related are logSobolev inequalities which serve as a stronger notion. For models with a conservation law, there is no spectral gap. Thus, the more appropriate question is to study how fast the gap vanishes as the volume  increases. In the case of independent components, the spectral subspaces for L are organized as single excitation, double excitation etc. Such a structure persists as the interaction is turned on which, on a mathematical level, is similar to the particle spectrum of a quantum field theory. Physically more directly relevant are: 3. Spacetime correlations. To be concrete, let us consider a Ginzburg–Landau field theory x (t) starting with a translation invariant Gibbs measure . Then x (t) is a spacetime stationary process. The two-point correlation function is the covariance hx ðtÞ0 ð0Þi  h0 ð0Þi2

½16

Its Fourier transform is directly linked to energy– momentum resolved scattering intensity from a probe which is modeled by the respective Ginzburg–Landau theory. For t = 0, the expression [16] is the static correlation, again belonging to the domain of equilibrium statistical mechanics. The time decay depends on whether the field is dynamically conserved or not. Correlation functions do not always capture the physics of the system well. This is certainly true for: 4. Dynamics at low temperatures. Let us consider the Glauber dynamics for the ferromagnetic Ising model in the finite but large volume . Then there is a very high free energy barrier between configurations typical for the þ phase and those typical for the  phase. If one starts the spin system in the þ phase, one

134 Interacting Stochastic Particle Systems

may study through which configurations the system moves to the  phase and how much time such a process will take. If the two phases are symmetric with the external magnetic field h = 0, the spin system tunnels, while for h < 0 and small the þ phase is metastable. Another widely studied situation, also experimentally, is the quenching from high to low temperatures. In our context this means that the initial measure is Bernoulli, while the Glauber dynamics runs at low temperatures. Then spin clusters coarsen as time proceeds developing well-defined interfaces which are governed through motion by mean curvature. Close to a point of second-order phase transition, one has to deal with: 5. Critical dynamics. The usual Glauber dynamics becomes very slow at the critical point and reliable equilibrium is hard to achieve. It is thus a challenge to design faster algorithms. One proposal is the Swendsen–Wang algorithm which is based on the Fortuin–Kasteleyn representation and flips a whole cluster of spins simultaneously. So far we concentrated on statistical properties. Researchers have been fascinated by the observation that for stochastic particle systems, the transition to a deterministic macroscopic evolution can be handled with full rigor. Such a program has been baptized: 6. Hydrodynamic limit, which is meaningful only for particle systems with one or several conservation laws. Let us discuss then a reversible lattice gas with Hamiltonian H. We start the dynamics with a state of local equlibrium which is Gibbs with a slowly varying chemical potential, that is, " !# X 1 Z exp  HðÞ  ð"xÞx ; "  1 ½17 x

Such a measure is almost time invariant. For small ", at least approximately, such a structure should persist in the course of time at the expense of properly regulating the chemical potential. For our example, the correct timescale is "2 t in microscopic units, and the evolution equation for the density, related thermodynamically to the chemical potential, is a nonlinear diffusion equation of the form @ t ¼ r  Dð t Þr t @t

½18

We turn to the nonreversible models. Nonreversible Models

While for reversible models the study of the stationary Gibbs measure is its own field of inquiry, here the first entry must be:

7. Nonequilibrium steady state. This steady state is determined through the dynamics, since the stationary measure  has to satisfy (Lf ) = 0 for a sufficiently large class of functions f. As in equilibrium, phase transitions may occur. In the nonconservative case it would mean that the infinitely extended system has several extreme stationary measures. In the conservative case, say with the density as locally conserved field, it would mean that there is an interval of densities for which there is no extreme stationary measure. Given the nonequilibrium steady state, one may wonder about its typical fluctuations and large deviations. In contrast to thermal equilibrium, weak long-range correlations are the rule. 8. Spacetime correlations in the steady state. Through the bulk drive the power-law decay of time correlations may change. For example for the symmetric and asymmetric exclusion process, the steady states are Bernoulli with density , denoted by hi . For the on-site density–density correlation, one finds, for large t,  1 t1=2 for F ¼ 0 h0 ðtÞ0 ð0Þi1=2  ffi ½19 4 t2=3 for F 6¼ 0 9. Hydrodynamic limit. The concept of slowly varying conserved fields remains valid; only local equilibrium must be replaced by local stationarity. Generically, there are nonzero currents in the steady state. Therefore, the macroscopic fields change on the timescale "1 t (cf. item (5)) and are governed by a hyperbolic conservation law of the form @ t þ div jð t Þ ¼ 0 @t

½20

in the case of a single conservation law. Here, j( ) is the average steady state in the stationary measure at density . Several conservation laws have an intriguing rich variety of solutions. Even on the level of continuum partial differential equations, such systems of hyperbolic conservation laws still pose unresolved basic problems. See also: Ginzburg–Landau Equation; Glassy Disordered Systems: Dynamical Evolution; Interacting Particle Systems and Hydrodynamic Equations; Macroscopic Fluctuations and Thermodynamic Functionals; Stochastic Differential Equations.

Further Reading Binder K and Heermann D (2002) Monte Carlo Simulations in Statistical Physics. Berlin: Springer. Kac M (1959) Probability and Related Topics in the Physical Sciences. London: Interscience. Kipnis C and Landim C (1999) Scaling Limits of Interacting Particle Systems. Grundlehren, vol. 320. Berlin: Springer.

Interfaces and Multicomponent Fluids 135 Liggett TM (1985) Interacting Particle Systems. Berlin: Springer. Liggett TM (1999) Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren, vol. 324. Berlin: Springer. Marro J and Dickmann R (1999) Nonequilibrium Phase Transitions in Lattice Models. Cambridge: Cambridge University Press. Martinelli F (1999) Lecture on Glauber Dynamics for Discrete Spin Models. Lecture Notes in Mathematics, vol. 1717. Berlin: Springer.

Schmittmann B and Zia RKP (1995) In: Domb C and Lebowitz JL (eds.) Statistical Mechanics of Driven Diffusive Systems, Phase Transitions and Critical Phenomena, vol. 17, London: Academic Press. Spitzer F (1970) Interaction of Markov processes. Advances in Mathematics 5: 246–290. Spohn H (1991) Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Heidelberg: Springer.

Interfaces and Multicomponent Fluids J Kim and J Lowengrub, University of California at Irvine, Irvine, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction Many important industrial problems involve flows with multiple constitutive components. Examples include extractors, separators, reactors, sprays, polymer blends, and microfluidic applications such as DNA analysis, and protein crystallization. Due to inherent nonlinearities, topological changes, and the complexity of dealing with unknown, active, and moving surfaces, multiphase flows are challenging. Much effort has been put into studying such flows through analysis, asymptotics, and numerical simulation. Here, we focus on review on studies of multicomponent fluids using continuum numerical methods. There are many ways to characterize moving interfaces. The two main approaches to simulating multiphase and multicomponent flows are interface tracking and interface capturing. In interface-tracking methods (examples include boundary-integral, volume-of-fluid, front-tracking, immersed-boundary, and immersed-interface methods), Lagrangian (or semi-Lagrangian) particles are used to track the interfaces. In (BIMs), the flow equations are mapped from the immiscible fluid domains to the sharp interfaces separating them thus reducing the dimensionality of the problem (the computational mesh discretizes only the interface). In interface-capturing methods such as level-set and phase-field methods, the interface is implicitly captured by a contour of a particular scalar function. The equations governing the motion of an unsteady, viscous, incompressible, immiscible twofluid system are the Navier–Stokes equations (the subscript i denotes the ith flow component):   @ui i þ ui  rui ¼ r  i þ i g; i ¼ 1; 2 ½1 @t i ¼ pi I þ 2i Di

½2

where i is the density, ui is the fluid velocity, pi is the pressure, i is the viscosity, and g is the gravitational acceleration vector. In eqn [2], i is the stress tensor, I is the identity matrix, and Di is the rate of deformation tensor and defined as Di = (1=2)(rui þ ruTi ). The velocity field is subject to the incompressibility constraint, r  ui ¼ 0

½3

We let  denote the fluid interface. The effect of surface tension is to balance the jump of the normal stress along the fluid interface. This gives rise to a Laplace–Young condition for the discontinuity of the normal stress across : ½n ¼  n

½4

where [] denotes the jump 2  1 across , is the curvature of  (positive for a spherical interface),  is the surface tension coefficient which is assumed to be constant, and n is the unit normal vector along  directed toward fluid 2. The fluid velocity is continuous across . In order to circumvent the problems associated with implementing the Laplace–Young calculation at the exact interface boundary, Brackbill and collaborators developed a method referred to as the continuum surface force (CSF) method. See the review by Scardovelli and Zaleski (1999). In this method, the surface tension jump condition is converted into an equivalent singular volume force that is added to the Navier–Stokes equations. Typically, the singular force is smoothed and acts only in a finite transition region across the interface. The system of equations [1]–[2] and the boundary condition, eqn [4] can be combined into the following distribution formulation that holds in both phases: ðut þ u  ruÞ ¼  rp þ r  ð2DÞ þ g þ F sing ; ru¼0 ½5 where the subscript i is dropped (i.e., it is understood that u = ui in fluid i, etc.,) and Fsing is singular

136 Interfaces and Multicomponent Fluids

surface tension force that is given by F sing =  n, where  is the surface delta-function.

Numerical Methods for Multicomponent Fluid Flows Interface-Tracking Methods

Boundary-integral methods (BIMs) BIMs can be highly accurate for modeling free surface flows with relatively regular interface topologies. The BIM was apparently first used by Rosenhead in 1932 to study vortex sheet roll-up. In this approach, the interface is explicitly tracked, but the flow solution in the entire domain is deduced solely from information possessed by discrete points along the interface. BIMs have been used for both inviscid and Stokes flows. For a review of Stokes flow computations, see Pozrikidis (2001), and for a review of computations of inviscid flows, see Hou et al. (2001). For flows with both inertia and viscosity, volume integrals must be incorporated into the formulation. When inertial forces are negligible (left-hand side term of eqn [1] is dropped), the velocity u(x0 ) at a given point x0 on the interface can be obtained by means of the boundary-integral formulation, Z 1 f ðxÞGðx0 ; xÞ ð þ 1Þuðx0 Þ ¼ 2u1 ðx0 Þ  4   nðxÞ dsðxÞ ½6 

1 4

Z

uðxÞ  Tðx0 ; xÞ  nðxÞ dsðxÞ

½7



where  is the viscosity ratio, u1 is an imposed velocity prevailing in the absence of the interfaces, and f (x) is the capillary force function f = . The tensors G and T are the Stokeslet and stresslet, respectively: ^x ^ I x Gðx0 ; xÞ ¼ þ 3 r r ^x ^ 6^ xx Tðx0 ; xÞ ¼  5 r ^ ¼ x  x0 ; where x

r ¼ j^ xj

½8

½9

The boundary conditions at the interface, that is, the stress balance equation [4] and continuity of the velocity across the interface, are automatically satisfied by the boundary-integral formulation. The normal velocity of the interface (x, t) is given by dx  nðxÞ ¼ uðx; tÞ  nðxÞ dt

½10

The shape of the interface does not depend on the tangential velocity and there are many possible choices that can be taken, see Hou et al. (2001). The principal advantages gained by using BIMs are the reduction of the flow problem by one dimension since the formulation involves quantities defined on the interface only and the potential for highly accurate solutions if the flow has topologically regular interfaces. In addition, highly efficient adaptive surface mesh refinement algorithms have recently been developed to improve the performance and accuracy of the methods (Cristini et al. 2001). The main disadvantages are the development of accurate quadratures of integrals with singular kernels (particularly in 3D) and the need for local surgery of the interface in the event of topological changes. BIMs have been successfully used for simulations of complex multiphase flows: drop deformation and breakup; jets; capillary waves; mixing; drop-to-drop interaction; suspension of liquid drops in viscous flow (e.g., see Cristini et al. (2001), Hou et al. (2001), and Pozrikidis (2001) and the references therein). Volume-of-fluid (VOF) method In the VOF method (see Scardovelli and Zaleski (1999) for a recent review), the location of the interface is determined by the volume fraction cij of fluid 1 in the computational cell, ij . In cells containing the interface 0 < cij < 1, cij = 1 in cells containing fluid 1, and cij = 0 in cells containing fluid 2 as shown in Figure 1b. A VOF algorithm is divided into two parts: a reconstruction step and a propagation step. A typical interface reconstruction is shown in Figure 1c. In the piecewise linear interface construction (PLIC) method, the true interface, as shown in Figure 1a, is approximated by a straight line perpendicular to an interface normal vector nij in each cell ij . The normal vector nij is determined from the volume fraction gradient using data from neighboring cells. With given a volume fraction cij

Fluid 2 nij cij Fluid 1

(a)

0

0

0

0.1

0.5

0.4

0.9

1

1

(b)

nij cij

(c)

Figure 1 VOF representation of an interface: (a) actual interface, (b) volume fraction, and (c) an approximation to the interface is produced using an interface reconstruction method such as piecewise linear approximation as shown.

Interfaces and Multicomponent Fluids 137

and a normal vector nij , the interface is given by the straight line with normal nij such that area beneath the line in cell ij is equal to cij . More recently, parabolic reconstructions of the interface have been used to gain higher-order accuracy for the surface tension force (e.g., the ‘‘parabolic reconstruction of surface tension’’ or PROST algorithm). Once the interface has been reconstructed, its motion by the underlying flow field must be modeled by a suitable advection algorithm. The key here is that the explicit interface reconstruction enables fluxes to be developed that exactly conserve mass and do not diffuse the interface. Capillary effects may be represented by the continuous surface stress (Scardovelli and Zaleski 1999), T ¼ ðI  n  nÞjr~cj;

F sing ¼ r  T

½11

where ~c is a smoothed version of the volume fraction. For the flows in which the capillary force is the dominant physical mechanism, the PROST algorithm discussed above can be used to significantly reduce spurious currents due to inaccurate representation of surface tension terms and associated pressure jump in normal stress. The distribution form of the fluid equations [5] is typically solved using a variant of the projection method for incompressible single phase flows. VOF methods are popular and have been used in commercial multiphase flow codes, in models of inkjet printers, flows with surfactants and in many other applications (e.g., see Scardovelli and Zaleski (1999) and James and Lowengrub (2004) and the references therein). The principal advantage of VOF methods is their inherent volume-conserving property. Nevertheless, spurious bubbles and drops may be created. The reconstruction of the interface from the volume fractions and the computation of geometric quantities such as curvature are typically less accurate than other methods discussed here

since the curvature and normal vectors are obtained by differentiating a nearly discontinuous function (volume fraction). Front-tracking methods The basic idea behind the original front-tracking method is the use of two grids as illustrated in Figure 2. One is a standard, Eulerian finite difference mesh that is used to solve the fluid equations. The other is a discretized interface mesh that is used to explicitly track the interface and compute surface tension force which is then transferred to the finite difference mesh via a discrete delta-function. Front tracking was first proposed by Richtmyer and Morton and further developed by Glimm and co-workers. A similar approach was taken by Unverdi and Tryggvason (see Tryggvason et al. (2001) and Peskin (2002) for recent reviews), who combined a moving grid description of the interface with flow computations on a fixed grid. In this immersed-boundary approach, all the fluid phases are treated together by solving a single set of governing equations. This method has its roots in the original marker-and-cell (MAC) method, where marker particles are used to identify each fluid and the immersed-boundary method of Peskin and McQueen, that was designed to track moving elastic boundaries in homogeneous fluids. The interface is represented discretely by Lagrangian markers that are connected to form a front which lies within and moves through a stationary Eulerian mesh. In Tryggvason’s original implementation, the basic structural unit is a line segment. Since the interface moves and deforms during the computation, interface elements must occasionally be added or deleted to maintain regularity and stability. In the event of merging/breakup, elements must be relinked to effect a change in topology. The interface is represented using an ordered list of marker particles xk = ((x1 )k , (x2 )k ), 1  k  N.

ui – 1/2, j + 1 vi , j + 1/2

Fluid 2 tA

nf B

A

tB

B

A pij ui – 1/2, j

ui – 1/2, j

Xf,k

vi, j – 1/2

Fluid 1

(a)

ui + 1/2, j + 1

(b)

(c)

Figure 2 (a) The basic idea in the front-tracking method is to use two grids – a stationary finite difference mesh and a moving Lagrangian mesh, which is used to track the interface. (b). Blow-up of the subgrid control volume in (a). (c) Control volume for the Eulerian mesh, i, jþ(1=2) .

138 Interfaces and Multicomponent Fluids

The first step in this algorithm is the advection of the marker particles. A simple bilinear interpolation is used to find the velocity inside each grid cell (indicated in Figure 2c). The marker particles are then advected in a Lagrangian manner. Once the points have been advected, a list of connected polynomials (pxi (s), pyi (s)) is constructed using the marker particles. This gives a parametric representation of the interface, with s typically an approximation of the arclength. Both lists are ordered and thus identify the topology of the interface. In later works, higher-order polynomials have been used (e.g., cubic splines) and semi-Lagrangian evolutions have been implemented where other tangential velocities have been used. As the interface evolves, the markers drift along the interface following tangential velocities and more markers may be needed if the interface is stretched by the flow. Typically, the markers are redistributed along the interface to maintain an accurate interface representation. Next, we compute the surface tension force, Z F sing ðx; tÞ ¼ f ðx  xf ðsÞÞnf ds ½12 ðtÞ

where the subscript f means values evaluated at the interface (t) and s is arclength. The discrete numerical implementation of this distribution onto the fixed grid is in the form of a sum over interface elements, xf , k : X f k ðx  xf ;k Þsk ½13 F ij ðxÞ ¼ k

where sk is the average of the straight line distances from the point xf , k to the two neighboring points xf , kþ1 and xf , k1 as indicated by the subgrid control volume shown in Figures 2a and 2b. The delta-function is typically taken to be Peskin’s discrete Dirac delta-function: ðx  xf ;k Þ   8 2 < Q 1 1 þ cos ½xi  ðxf ;i Þk  if jx  x j  2h f ;k ¼ i¼1 4h 2h : 0 otherwise

[14]

Other higher-order alternative forms of the regularized delta-function using the product formula have recently been proposed. Using the Frenet relation, the surface tension force on a short segment of the front is given by Z B Z B @t f fk ¼ ds ¼ ðt B  t A Þ ½15 f nf ds ¼  @s A A where A and B are the segment endpoints that lie on the boundary of the subgrid control volume

(Figures 2a and 2b), and t f is a tangent vector computed by fitting a polynomial to the endpoints of each element. In the case of flows with varying density and/or viscosity between the fluid components, there is a need to calculate the phase indicator function I(x, t) (defined by interface geometry and position), which has the value 0 in fluid 1 and 1 in fluid 2. The indication function can be determined via the solution of the equation Z Iðx; tÞ ¼ r  nf ðx  xf ðs; tÞÞds ½16 ðtÞ

This equation is discretized on the Eulerian mesh and a discrete delta-function (e.g., eqn [14]) is used. The fluid properties such as density and viscosity are determined via the indicator function, that is, (x, t) = 1 þ (2  1 )I(x, t), etc. As in the volume of fluid algorithm, the distribution form of the Navier–Stokes equations [5] are typically solved using a version of Chorin’s projection method. An alternative flow solver that can be used to integrate the flow equations in the presence of an interface is the immersed-interface method (IIM). The IIM was developed by Leveque and Li (see the review Li 2003), and can be used together with front-tracking as well as level-set methods. The IIM directly incorporates jump conditions for the normal stress into the finite difference stencil. The key idea of this method is to use the jump conditions in Taylor series expansions of pressure and velocity near interfaces to derive difference equations that achieve pointwise second-order accuracy. The principal advantage of front-tracking algorithms is their inherent accuracy, due in part to the ability to use a large number of grid points on the interface. Front-tracking methods can be complicated to implement, particularly in 3D, but give the precise location and geometry of the interface. In addition, explicit front tracking permits more than one interface to be present in a single computational cell without coalescence, which can be important in dense bubbly flows, emulsions, etc. One of major handicaps of front-tracking methods is the difficulty in modeling topological changes of the interface such as breakup and coalescence without ad hoc cutand-connect and reconnecting parameterized interface (particularly, difficulties in 3D). Interface-Capturing Methods

Level-set method Level-set methods, introduced by Osher and Sethian (see the recent review papers (Osher and Fedkiw 2001, Sethian and Smereka

Interfaces and Multicomponent Fluids 139

2 1.5

φ0

0.5

–0.5

0 –0.5 –1 –1.5

Fluid 1

–1

Γ

–1.5

Fluid 2

–2 –2 –1.5 –1 –0.5 0 0.5 1 1.5

(a)

where  is usually is one or two grid lengths. After solving eqn [18] to steady state (x, t) is then replaced by d(x, steady ). Note that d(x, steady ) is typically a good approximation of the signed distance function. The density and viscosity are defined as

1 0.5

ð Þ ¼ 2 þ ð1  2 ÞH ð Þ

–2 2

2

1

0

–1

–2 –1

0

1

2

and

(b)

Figure 3 (a) Zero contour of representing the interface . (b) Surface of with zero contour.

2003) and the recent texts (Osher and Fedkiw 2002, Sethian 1999)), are popular computational techniques for tracking moving interfaces. These methods rely on an implicit representation of the interface as the zero set of an auxiliary function (level-set function). The application of these methods to incompressible, multiphase flows started with the work of Osher, Merriman, Sussman, Smereka, Hou, and their collaborators. In the level-set method, the level-set function

(x, t) is defined as follows (see Figure 3): ( > 0 if x 2 fluid 1

ðx; tÞ ¼ 0 if x 2  ðthe interface between fluidsÞ < 0 if x 2 fluid 2 and the evolution of is given by

t þ u  r ¼ 0

½17

which means that the interface moves with fluid. To keep the interface geometry well resolved, the level-set function should be a distance function near the interface. However, under the evolution [17] it will not necessarily remain as such. We note that special velocity extensions v off the interface (i.e., v = u at the interface, v 6¼ u away from interface) have been recently developed to better maintain as a distance function (e.g., Sethian and Smereka (2003) and Macklin and Lowengrub (2005)). Typically, a reinitialization step (solving a Hamilton–Jacobi type equation, eqn [18]) below, is performed to keep as a distance function near the interface while keeping original zero-level set unchanged. More specifically, given a level-set function, , at time t, the contours are redistributed according to the steady-state solution of the equation @d ¼ S ð Þð1  jrdjÞ; @

dðx; 0Þ ¼ ðxÞ

½18

where S is the smoothed sign function defined as

S ð Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

þ 2

½19

ð Þ ¼ 2 þ ð1  2 ÞH ð Þ

½20

where H ( ) is the smoothed Heaviside function given by 8 0 if <  > > <  

H ð Þ ¼ 12 1 þ  þ 1 sinð =Þ if j j   > > : if >  1 The mollified delta-function is  ( ) = dH =d . The surface tension force is given as   r

r

½21 F sing ¼ r   ð Þ jr j jr j The fluid equations [5] are solved using projection methods, the IIM or the ghost-fluid (GF) method (e.g., Osher and Fedkiw (2001, 2002) and Fedkiw et al. (2003)). The GF method is similar to the IIM in that jump discontinuities are incorporated in the finite difference stencil. In the GF algorithm, subcell resolution is used to mark the interface position and the values of discontinuous quantities are artificially extended to grid points neighboring the interface via extrapolation. A fully second order accurate GF method for moving interfaces has recently been developed (Macklin and Lowengrub 2005). Applications of the level-set method include multiphase flows, viscoelastic fluid flows and fluid– structure interactions (e.g., see the reviews Osher and Fedkiw (2001, 2002), Sethian (1999), and Sethian and Smereka (2003)). Advantages of the level-set algorithm include the simplicity with which it can be implemented, the ability to capture merging and breakup of interfaces automatically, and the ease with which the interface geometry can be described using the level-set function. A disadvantage of the level-set method is that mass is not conserved. Accurate numerical simulations of multiphase flow and topology transitions require the computational mesh to resolve both the macroscales (e.g., droplet size, flow geometry) and the microscales to accurately capture local interface geometries near contact region, van der Waals forces, surfactant distribution, and Marangoni stresses. Adaptive mesh

140 Interfaces and Multicomponent Fluids

5 4 3 2 1 0 –1 –2 –3 –4 –5

–3.2

–2

–3.275

–2.25 –2.5

–3.25 –3.28

–2.75 –3

–3.3

–3.25

–3.285

–3.5

–3.35

–3.75 –4

–2

0

2

4

–4 –1

–0.5

0

0.5

1

–3.4 – 0.75

– 0.7

– 0.65

– 0.6

– 0.55

–3.29 –0.655

– 0.65

– 0.645

– 0.64

Figure 4 Each of the first three figures has a boxed region that is magnified in the next figure. The rates of magnification are 5, 10, 40/3, respectively. The meshes in the figure are used to simulate the drop-impacting interface problem. Source: Zheng X, Anderson A, Lowengrub JS, and Cristini V (unpublished).

algorithms have recently been used greatly to increase accuracy and computational efficiency in level-set methods. Typically, the methods involve Cartesian adaptive mesh refinement. Problems tackled using this approach include droplet formation in inkjet printers and wake development behind a ship. Another approach, recently developed, is to use adaptive unstructured mesh refinement (Zheng et al. 2005), as shown in Figure 4, in which the impact of a drop onto a fluid interface is captured. Hybrid Methods

More recently, a number of hybrid methods, which combine good features of each algorithm, have been developed. These include coupled level-set volumeof-fluid (CLSVOF) algorithms, particle level-set methods, marker-VOF methods and level-contour front-tracking methods. Level-set and VOF methods have recently been combined. The volume fraction is used to maintain volume conservation, while the level-set function is used to describe the interface geometry. After every time step, the volume-fraction function and level-set function are made compatible. The coupling between the level-set function and the VOF function c occurs through the normal of the reconstructed interface and through the fact that the level-set function is reset to the exact signed normal distance to the reconstructed interface (where the area below the reconstructed interface is given by the volume-fraction function). In the particle level-set method, Lagrangian disconnected marker particles are randomly positioned near the interface and are passively advected by the flow in order to rebuild the level-set function in under-resolved zones, such as high-curvature regions and near filaments. In these regions, the standard nonadaptive level-set method regularizes excessively the interface structure and mass is lost. The use of marker particles significantly ameliorates these difficulties.

Recently, a hybrid method has been developed, which uses both marker particles, to reconstruct and move the interface, and the volume-fraction function to conserve volume. In this approach, a smooth motion of the interface, typical of marker methods is obtained together with volume conservation, as in standard VOF methods. This work improves both the accuracy of interface tracking, when compared to standard VOF methods, and the conservation of mass, with respect to the original marker method. Finally, a hybrid method that combines a level contour reconstruction technique with front-tracking methods has recently been developed to automatically model the merging and breakup of interfaces in three-dimensional flows. Phase-Field Method

Phase-field, or diffuse-interface, models are an increasingly popular choice for modeling the motion of multiphase fluids (see Anderson et al. (1998) for a recent review). In the phase-field model, sharp fluid interfaces are replaced by thin but nonzero thickness transition regions where the interfacial forces are smoothly distributed. The basic idea is to introduce a conserved order parameter (e.g., mass concentration) that varies continuously over thin interfacial layers and is mostly uniform in the bulk phases (see Figure 5). For density-matched binary liquids (let  = 1 for simplicity), the coupling of the convective Cahn–Hilliard equation for the mass concentration with a modified momentum equation that includes a phase-field-dependent surface force is known as Model H (Hohenberg and Halperin 1977). In the case of fluids with different densities a phase-field model has been proposed by Lowengrub and Truskinovsky. Complex flow morphologies and topological transitions such as coalescence and interface breakup can be captured naturally and in a mass-conservative and energy-dissipative fashion since there is an associated free energy functional.

Interfaces and Multicomponent Fluids 141

1 0.8 0.6 ξ

0.4 0.2 0 –0.2 –2

–1.5

–1

–0.5

0

0.5

1

1.5

2

Figure 5 A concentration prome across an interface with interface thickness, .

The phase field is governed by the following advective Cahn–Hilliard equation: @c þ u  rc ¼ r  ðMðcÞr Þ @t

½22

¼ F0 ðcÞ  2 c

½23

where M(c) = c(1  c) is the mobility, F(c) = (1=4)c2 (1  c)2 is a Helmholtz free energy that describe the coexistence of immiscible phases, and  is a measure of interface thickness and   (see Figure 5). It can be shown that in the sharp interface limit  ! 0, the classical Navier–Stokes system equations and jump conditions are recovered. The pffiffiffi singular surface tension force is F sing = 6 2r  (rc  rc), where  is the surface tension coefficient. An alternative surface tension pffiffiffiforce formulation based on the CSF is Fsing = 6 2r (rc=jrcj)jrcjrc. Recently, very efficient nonlinear multigrid methods have been developed to solve implicit discretizations of the Cahn–Hilliard equation (e.g., Kim et al. (2004)). These schemes have been combined with projection methods to solve the Navier–Stokes equations to perform simulations of multiphase flows. An example of simulation of liquid thread breakup using a phase-field method is shown in Figure 6. A long cylindrical thread of a viscous fluid 1 is in an infinite mass of another viscous fluid 2. If the thread becomes varicose with wavelength , the equilibrium of the column is unstable, provided  exceeds the circumference of the cylinder. This is the Rayleigh capillary instability that results in surface-tensiondriven breakup of the thread. An advantage of the phase-field approach is that it is straightforward to include more complex physical effects. For example, the binary model can be

Figure 6 Time evolution leading to multiple pinch-offs. The evolution is from top to bottom and left to right. The domain is axisymmetric, the initial velocities are zero everywhere, and the concentration field pffiffiffi is given by c(r , z) = 0.5(1  tanh ((r  0.5 0.05 cos (z))=(2 2))) on  = (0, )  (0, 2 ). Densities are matched and viscosity ratio is 0.5.

straightforwardly extended to describe threecomponent flows as follows. Consider a ternary mixture and denote the composition of components 1, 2, and 3, expressed as mass fractions, by c1 , c2 , and c3 , respectively. Therefore, 3 X

ci ¼ 1;

0  ci  1

½24

i¼1

The composition of a ternary mixture (A, B, and C) can be mapped onto an equilateral triangle (the Gibbs triangle (Porter and Easterling 1993)) whose corners represent 100% concentration of A, B, or C as shown in Figure 7a. Mixtures with components lying on lines parallel to BC contain the same percentage of A, those with lines parallel to AC have the same percentage of B concentration, and analogously for the C concentration. In Figure 7a, the mixture at the position marked ‘ ’ contains 60% A, 10% B, and 30% C. Because the concentrations sum to unity, only two of them need to be determined, say c1 , c2 . The evolution of c1 and c2 is governed by the following advective ternary Cahn–Hilliard equation: @c1 þ u  rc1 ¼ r  ðMðc1 ; c2 Þr 1 Þ @t C2

C

A (a)

½25

B C1

C3

(b)

Figure 7 (a) Gibbs triangle. (b) Contour plot of the free energy F (c1 , c2 ) on the Gibbs triangle.

142 Interfaces and Multicomponent Fluids

@c2 þ u  rc2 ¼ r  ðMðc1 ; c2 Þr 2 Þ @t

½26

@Fðc1 ; c2 Þ  2 c1  0:52 c2 @c1

½27

1 ¼

@Fðc1 ; c2 Þ  0:52 c1  2 c2 ½28 @c2 P where M(c1 , c2 ) = 3i 0, there is a  > 0, such that for all initial datum !0 in L1 () satisfying

Z

ð!  !0 Þ2 dx  ; we have :



Z

ð!  !ðtÞÞ2 dx  "; for all t



where !(t) denotes the solution of the Cauchy problem associated with the initial datum !0 by Youdovitch’s theorem.

Now we can state the following result. Theorem (Arnol’d) Let ! be a stationary solution given by [10]. We assume that one of the following assumptions holds: (C1) There are positive constants c1 , c2 , such that c1  f 0  c2 (C2) There are positive constants c1 , c2 , with c2 < 1 (first eigenvalue of the Dirichlet problem on the domain ) such that: c1  f 0  c2 Then ! is stable in the L2 -norm.

Remarks (i) This result was the first nonlinear stability result for stationary flows. (ii) The proof makes use of the conservation of the functionals of the vorticity field. Another significant contribution of Arnol’d to hydrodynamics was to reveal the geometrical aspect of the instability of the perfect-fluid motion. We give a brief insight into this issue. Let us come back to the Lagrangian description of motion. We want to determine the function ’(t, x). Each mapping ’t (x) = ’(t, x) is, for t fixed, a diffeomorphism of  preserving the Lebesgue measure and the orientation (equivalently stated, it  is an element of SDiff()).

In other words, a fluid motion is a curve t ! ’t  (the configdrawn on the ‘‘manifold’’ M = SDiff() uration space of the system). At time t, the relationship @’ ðt; xÞ ¼ uðt; ’ðt; xÞÞ @t states that the velocity field u(t, ’t (x)) belongs to the space tangent to M at ’t . The tangent space at ’ to M is the space of vector fields v(’(x)), where v(x)  satisfying is an incompressible vector field on  v  n = 0 on @. This space is naturally endowed with a norm given by the kinetic energy Z 1 vðxÞ2 dx 2  and thus M is endowed with a Riemannian structure. It is easy to check that the perfect-fluid motions correspond to the curves ’t drawn on M which are the critical points of the action integral: 2 Z Z  @’  1 t2 dt  ðt; xÞ dx; for all t1 < t2 2 t1  @t ðwith the constraints ’ðt1 ; :Þ ¼ ’1 ; ’ðt2 ; :Þ ¼ ’2 Þ That is to say, the perfect-fluid motions are the geodesics of the Riemannian manifold M. The main interest of this geometric framework is to bring back, at least formally, the perfect-fluid motions to well-known objects. Indeed, we know that the Riemannian curvature of a manifold has a profound impact on the behavior of geodesics on it. If the Riemannian curvature is positive, then nearby geodesics oscillate about one another, and if the curvature is negative, geodesics rapidly diverge from one another. More precisely, the stability of geodesics is expressed in terms of the curvature by means of Jacobi’s equation [1]. If ’t is a geodesic curve starting from ’0 , with velocity field v(t) (whose norm is supposed equal to 1), if the sectional curvature of the manifold in all the 2-planes containing v(t) is less than c(< 0), a perturbation of the initial datum will increase at least as exp(ct): ~t Þ dð’0 ; ’ ~0 Þ expðctÞ dð’t ; ’ ~0 denotes the perturbed initial datum and d where ’ the geodesic distance on the manifold. Moreover, if the curvature at every point and for all the sections is less than c, and if M is compact, then the geodesic flow, that is, the one-parameter group of transformations (’0 , v(0)) ! (’t , v(t)), is mixing (in the usual meaning of ergodic theory). Arnol’d succeeded in calculating the sectional curvature for flows on the two-dimensional torus; he showed that the

Inviscid Flows

curvature is negative for ‘‘most’’ of the sections. This gives an enlightening geometrical picture of the instability of Lagrangian flows. It was tempting to connect the above considerations on the instability of two-dimensional flows with the problem of weather forecast. In 1963 EN Lorenz stated that a two-week forecast would be a theoretical bound for predicting the atmospheric motion. Lorenz’s assertion was based on numerical simulations. He took as model for the large-scale atmospheric motion the two-dimensional Euler equations on the torus, which he truncated to a small number of Fourier modes (about 20). This model is highly unstable and displays exponential sensitivity with respect to the initial datum. However, the parallel between the behavior of this system and the instability of the Lagrangian flow is misleading. On the one hand, if we again do the Lorenz computations on Euler equations, taking into account a large number of Fourier modes, we note a striking phenomenon: the flow has a tendency to self-organize into large vortices, called coherent structures, and simultaneously the exponential sensitivity, as measured in terms of the energy norm of the velocity field, disappears. On the other hand, the problem of predicting the Lagrangian flow is very different, the Lagrangian flow can be exponentially unstable, while the corresponding velocity field quietly converges, in the energy norm, towards some equilibrium. We must keep in mind that the meteorologist aims to predict the values of the velocity field at some future time and not the trajectories of the fluid particles. In fact, it appears that Lorenz has ignored phenomena of a statistical nature which occur when a large number of degrees of freedom are considered; thus, his theoretical bound for the prediction of the atmospheric motion has no definite basis. More detailed reflections on this issue can be found in Robert and Rosier (2001).

The Cauchy Problem for the Euler Equations for Compressible Inviscid Fluids As remarked in the introduction, compressible flows yield pressure waves. The equations of motion being nonlinear, these waves interact in an intricate manner giving rise to shocks. This is the main feature of compressible fluid flows. Compressible flows are situated in the more general domain of nonlinear hyperbolic systems, which were intensively studied during the last decades. We only give here an example of the kind of result which can be obtained.

165

The following theorem, which states that for a set of regular initial data, shocks do not occur till some finite time, is a consequence of a more general result on hyperbolic systems due to Majda (1984). We consider  = 0. Then there is a finite time T > 0, depending on the Hs and L1 norms of the initial data, such that the Cauchy problem for [1], [2], [4] has a unique bounded smooth solution p, u 2 C1 ([0, T]  0 for all t, x.

Inviscid Flows and Turbulence Loosely speaking, turbulence is the intricate motion of a slightly viscous flow. Going back to the first half of the last century, there are two main approaches to turbulence. The first is due to Leray. The dissipation of energy is a characteristic feature of three-dimensional turbulence, and Leray thought that, even if very small, the viscosity of the fluid plays an important role, so that to understand turbulence the first step is to study the Navier– Stokes equations. A radically different approach is due to Onsager. Onsager (1949) started with the fundamental remark that the 4/5 law of turbulence, which relates the dissipation of energy to the increments of the velocity field, does not involve viscosity. Furthermore, he observed that the proof of the conservation of energy for the solutions of Euler equations uses an integration by parts which supposes some regularity of the velocity field. He then imagined that an inviscid dissipation mechanism, due to a lack of regularity of the solutions, was at work in Euler equations. In modern terminology, he suggested to model turbulent flows by nonregular (weak) solutions satisfying the Euler equations in the sense of distributions. He also conjectured that if a solution satisfies a Ho¨lder regularity condition of order >1=3, then the energy would be conserved. Onsager’s views were revolutionary and forgotten for a long time. Recent works, such as the proof of Onsager’s conjecture, the construction of weak solutions with energy dissipation, and the discovery of the explicit local form of the energy dissipation for weak solutions, show a renewed interest in these views (see, e.g., Constantin and Titi (1994), Eyink (1994), Robert (2003), and Shnirelman (2003)). See also: Compressible flows: Mathematical Theory; Dissipative Dynamical Systems of Infinite Dimension; Hyperbolic Dynamical Systems; Incompressible Euler Equations: Mathematical Theory; Non-Newtonian Fluids; Partial Differential Equations: Some Examples; Chaos and Attractors; Turbulence Theories.

166 Isochronous Systems

Further Reading Arnold VI and Khesin BA (1998) Topological Methods in Hydrodynamics. Berlin: Springer. Batchelor GK (1967) An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press. Chemin JY (1995) Fluides parfaits incompressibles. Asterisque no 230, SMF. Chen GQ and Wang D (2002) The Cauchy problem for the Euler equations for compressible fluids. In: Friedlander S and Serre D (eds.) Handbook of Mathematical Fluid Dynamics, vol. 1. Amsterdam: Elsevier. Constantin P, Weinan E, and Titi ES (1994) Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Communications in Mathematical Physics 165: 207–209. Eyink G (1994) Energy dissipation vithout viscosity in ideal hydrodynamics. Physica D 78(3–4): 222–240. Frisch U (1995) Turbulence. Cambridge: Cambridge University Press.

Kato T (1975) Quasi-Linear Equations of Evolution with Applications to Partial Differential Equations, Lecture Notes in Math. vol. 448, pp. 25–70. New York: Springer. Majda A (1984) Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. New York: Springer. Marchioro C and Pulvirenti M (1994) Mathematical Theory of Incompressible Non-Viscous Fluids. Berlin: Springer. Onsager L (1949) Statistical hydrodynamics. Nuovo Cimento (suppl. 6): 279–291. Robert R (2003) Statistical hydrodynamics. In: Friedlander S and Serre D (eds.) Handbook of Mathematical Fluid Dynamics, vol. 2. Amsterdam: Elsevier. Robert R and Rosier C (2001) Long-range predictibility of atmospheric flows. Nonlinear Processes in Geophysics 8: 55–67. Serre D (1979) Les invariants du premier ordre de l’equation d’Euler en dimension 3. Comptes Rendus de l’Academie des Sciences Paris, Serie A 289: 267–270. Shnirelman A (2003) Weak solutions of incompressible Euler equations. In: Friedlander S and Serre D (eds.) Handbook of Mathematical Fluid Dynamics, vol. 2. Amsterdam: Elsevier.

Ising Model see Two-Dimensional Ising Model

Isochronous Systems Francesco Calogero, University of Rome, Rome, Italy and Institute, Nazionale de Fisica Nucleare, Rome, Italy ª 2006 Elsevier Ltd. All rights reserved.

Introduction This paper reviews recent developments, following closely (sometimes verbatim) the review paper Calogero F (2004c) (see the Bibliography below); for more traditional investigations of isochronous systems see other entries of this Encyclopedia (and for the mathematical investigation of isochronous centers in the plane, related to the 16th Hilbert problem, see for instance the survey paper referred to at the end of this entry). The isochronous systems treated herein are characterized by the property to possess an open domain having full dimensionality in their phase space such that all the motions evolving from a set of initial data in it are completely periodic with the same fixed period. The natural measure of this open domain might, or it might not, be infinite when the measure of the entire phase space is itself infinite: for instance, if the entire phase space is the twodimensional Euclidian plane, such a domain might

be the exterior, or the interior, of a circle of finite radius. It is justified to call such systems superintegrable, or perhaps partially superintegrable inasmuch as the property of isochronicity of all their motions holds only in a subregion of the entire phase space. This terminology is justified by the observation that, roughly speaking, all confined motions of a superintegrable system – in which all but one of the degrees of freedom are constrained by the existence of the maximal possible number of constants of motion – are completely periodic, although not necessarily all with a fixed period – entailing that isochronicity entails superintegrability, while the converse is not the case (see the entry Integrable systems in this Encyclopedia). A simple trick – amounting essentially to a change of independent, and possible as well of dependent, variables, allows to deform a largely arbitrary dynamical system so that the deformed system obtained from it be isochronous. This ‘‘trick’’, which is now explained, entails therefore that isochronous systems are not rare. Below we provide several examples; others can be found in the further reading suggested at the end of this entry, and/or can be manufactured ad libitum using the trick.

Isochronous Systems 167

The Trick We now show that, given a largely arbitrary dynamical system, it is possible to introduce a deformed version of it featuring a real constant !, that has the following properties: for ! = 0, it coincides with the original, undeformed system; for ! > 0, it possesses an open region having full dimensionality in its phase space such that all solutions evolving from an initial datum in it are ~ which is a finite completely periodic with a period T integer multiple, or perhaps a simple fraction, of the basic period T¼

2 !

½1

right-hand side of [2] is not singular for  = 0 and  = 0 . The relevant result guarantees, not only for the initial datum 0 , but for a (sufficiently small but open) set of initial data in its neighborhood, the existence of a circular disk in the complex -plane, centered at  = 0 (where the initial data are assigned) and having a nonvanishing radius , such that the solutions () corresponding to these initial data are holomorphic in it, namely for j j <  (and note that if () is a multicomponent object, the property to be holomorphic is featured by each and everyone of its components). Let us now introduce the following changes of dependent and independent variables: zðtÞ ¼ expði!tÞ ð Þ

Let us indeed, consider a quite general dynamical system which we write as follows:  0 ¼ Fð;  Þ

½2

Here   () is the dependent variable, which might be a scalar, a vector, a tensor, a matrix, you name it. The independent variable is , and the main limitation on the dynamical system [2] is that it be permissible to treat this variable as complex; this requires that the derivative with respect to this complex variable  that appears in the left-hand side of the evolution equation [2] make sense, namely that this dynamical system be analytic, entailing that the dependent variable  be an analytic function of the complex variable  (but this does not require () to be a holomorphic nor a meromorphic function of ; () might feature all sorts of singularities, including branch points, in the complex -plane, indeed this will generally happen since we generally assume the evolution equation (??) to be nonlinear). The quantity F in the right-hand side of [2] – which has of course the same scalar, vector, matrix. . . character as  – might depend (arbitrarily but analytically) on  as well as on . (Let us also emphasize that this approach is as well applicable to more general dynamical systems that also feature other, ‘‘spacelike’’, independent variables, for instance are a system of PDEs rather than ODEs; the interested reader is referred to the literature cited below). In spite of the generality of this dynamical system, [2], there generally holds a result (‘‘Theorem of existence, uniqueness and analyticity’’) that characterizes the solution () of its initial-value problem determined by the assignment ð0Þ ¼ 0 Here, for notational simplicity, we assign the initial datum 0 at  = 0; and we assume of course that the

   ðt Þ ¼

expði!tÞ  1 i!

½3a

½3b

This transformation is called ‘‘the trick’’. The essential part of it is the change of independent variable [3b]: and let us re-emphasize that, here and hereafter, the new independent variable t is considered as the real, ‘‘physical time’’ variable. Note that [3b] entails  ð0Þ ¼ 0; _ ð0Þ ¼ 1 and, most importantly, that (t) is a periodic function of t with period T, see [1]. More specifically, as the time t increases from zero onwards, the complex variable  travels counterclockwise round and round on the circle C the diameter of which, of length 2/!, lies on the imaginary axis in the complex -plane, with one extreme at the origin,  = 0, and the other at the point  = 2i/!, making a full circle in the time interval T. As for the prefactor exp(i!t) that multiplies () in the right-hand side of [3a], its purpose is to allow, via an appropriate choice of the parameter , the deformed system, see below, to have a neater look; however this choice is hereafter restricted by the condition that  be real and rational, say ¼

p q

with p and q two coprime integers and q > 0. This restriction is essential to guarantee, via [3], that if () is holomorphic in  in the (closed) disk encircled by the circle C, then z(t) is completely periodic (namely, each and everyone of its components is periodic) with the period ~ ¼ qT T

½4

168 Isochronous Systems

The deformed dynamical system is the one that obtains from [2] via the trick [3]. It clearly reads as follows: z_ ¼ i!z þ exp½ið þ 1Þ!t   expði!tÞ  1  F expði!tÞz; i!

½5

which can clearly be satisfied by initial data situated inside an open domain of such data, at least provided ! is sufficiently large (actually, in all the examples reported below no restriction on the value of ! is required, namely such an open domain exists for any arbitrary value of ! > 0).

Examples In this subsection we report tersely several examples of isochronous dynamical systems; in each case we also provide the reference where more information can be found. Except when explicitly otherwise mentioned, these dynamical systems are to be considered in the complex context. The first example we report is a Hamiltonian N-body problem which is a generalization of a wellknown integrable (indeed, superintegrable) system (see Integrable Systems: Overview). It is characterized by the (normal) Hamiltonian N   1X p2n þ !2 z2n 2 n¼1 ðkÞ N K X 1 X fnm þ 4 m;n¼1;m6¼n k¼1 kðzn  zm Þ2k

½6a

and correspondingly by the Newtonian equations of motion €zn þ !2 zn ¼

N X

K X

m¼1;m6¼n k¼1

ðkÞ

fnm

ðzn  zm Þ1þ2k



½6b

(k) Here the 12 N(N  1)K ‘‘coupling constants’’ fnm are arbitrary, except for the symmetry restriction (k) (k) fnm = fmn (see [6a]). The next example we report is a real N-body problem in the horizontal plane, characterized by the Newtonian equations of motions

N   X ^ ; nm þ nm k^ m¼1;m6¼n



And it is plain, on the basis of the arguments we just gave, that this system is isochronous, a sufficient condition for the complete periodicity with period ~ see [4], of its solutions being provided by the T, inequality 2 0g  fG >  > 0g

½7

ð‘; g; Þ 2 T3

Maximal KAM Tori Kolmogorov’s Theorem and the RPC3BP (1954)

Kolmogorov’s invariant tori theorem deals with the persistence, in nearly integrable Hamiltonian systems, of Lagrangian (maximal) tori, which, in general, foliate the integrable limit. Kolmogorov (1954) stated his theorem and gave a precise outline of the proof. Let us briefly recall this milestone of the modern theory of dynamical systems. Let M := Bd  Td (Bd being a d-dimensional ball in Rd centered at the origin) be endowed P with the standard symplectic form dy ^ dx := dyi ^ dxi (y 2 Bd , x 2 Td ). A Hamiltonian function N on M having a Lagrangian invariant d-torus of energy E on which the N-flow is conjugated to the linear dense translation x ! x þ !t, ! 2 Rd nQ d can be put in the form N :¼ E þ !  y þ Qðy; xÞ

½9 jj  1 P (as usual, jj = 1 þ    þ d , !y := di= 1 !i yi , and @y = @y11    @ydd ); in such a case, the Hamiltonian N is said to be in Kolmogorov normal form. The vector ! is called the ‘‘frequency vector’’ of the invariant torus {y = 0}  Td . The Hamiltonian N is said to be nondegenerate if @ y Qð0; xÞ ¼ 0;

8  2 Nd ;

deth@y2 Qð0; Þi 6¼ 0

are conjugate symplectic coordinates and if Del is the corresponding symplectic map, then HKep  Del = (3 M2 )=(2L2 ). Note that the Delaunay variables become singular when C is vertical (the node is no more defined) and in the circular limit (the perihelion is not unique). In these cases different variables have to be used. 5. Let (X(i) , x(i) ) = Del ((Li , Gi , i ), (‘i , gi , i )). Then Hplt expressed in the Delaunay variables {(Li , Gi , i ), (‘i , gi , i ): 1  i  n} becomes

191

½10

where the brackets denote average over Td and @y2 the Hessian with respect to the y-variables. We recall that a vector ! 2 Rd is said to be ‘‘Diophantine’’ if there exist  > 0 and  d  1 such that j!  kj

 ; jkj

8 k 2 Zd nf0g

½11

½8

The set Dd of all Diophantine vectors in R d is a set of full Lebesgue measure. We also recall that Hamiltonian trajectory is called quasiperiodic with (rationally independent) frequency ! 2 Rd if it is conjugate to the linear translation  2 Td !  þ !t 2 Td .

Note that the number of action variables on which the integrable Hamiltonian H(0) Del depends is strictly less than the number of degrees of freedom. This ‘‘proper degeneracy,’’ as we shall see in next sections, brings in an essential difficulty one has to face in the perturbative approach to the many-body problem. In fact, this feature of the many-body problem is common to several other problems of celestial mechanics.

Theorem (Kolmogorov 1954) Consider a oneparameter family of real-analytic Hamiltonian functions H" := N þ "P where N is in Kolmogorov normal form (as in eqn [9]) and " 2 R. Assume that ! is Diophantine and that N is nondegenerate. Then, there exists "0 > 0 and for any j"j  "0 , a real-analytic symplectic transformation " : M ! M putting H" in Kolmogorov normal form, H"  " = N" , with N" := E" þ !  y0 þ Q" (y0 , x0 ). Furthermore, jE"  Ej, k"  idkC2 , and kQ"  QkC2 are small with ".

ð0Þ

ð1Þ

ð0Þ

HDel ¼ HDel þ "HDel ; HDel :¼ 

n X 3 M2 i

i¼1

i

2L2i

192 KAM Theory and Celestial Mechanics

In other words, the Lagrangian unperturbed torus T 0 := {y = 0}  Td persists under small perturbation and is smoothly deformed into the H" -invariant torus T " := " ({y0 = 0}  Td ); the dynamics on such torus, for all j"j  "0 , consists of dense quasiperiodic trajectories. Note that the H" -flow on T " is analytically conjugated by " to the translation x0 ! x0 þ !t with the same frequency vector of N, while the energy of T " , namely E" , is in general different from the energy E of T 0 . Kolmogorov’s proof is based on an iterative (Newton) scheme. The map " is obtained as limk ! 1 (1)      (k) , where the (j) ’s are ("-dependent) symplectic transformations of M successively closer to the identity. It is enough to describe the construction of (1) ; (2) is then obtained by replacing H" with H"  (1) , and so on. The map (1) is "-close to the identity and it is generated by g(y0 , x) := y0 x þ "(bx þ s(x) þ y0 a(x)), where s and a are (resp. scalar- and vector-valued) real-analytic functions on Td with zero average and b 2 R d ; this means that the symplectic map (1) : (y0 , x0 ) ! (y, x) is implicitly given by the relations y = @x g and x0 = @y0 g. It is easy to see that there exists a unique g of the above form such that for a suitable "0 > 0, H"  ð1Þ ¼ E1 þ !  y0 þ Q1 ðy0 ; x0 Þ þ "2 P1 8 j"j  "0

½12

with @y Q1 (0, x0 ) = 0, for any  2 Nd and jj  1; here, E1 , Q1 , and P1 depend on " and, for a suitable c1 > 0 and for j"j  "0 , jE  E1 j  c1 j"j, kQ  Q1 kC2  c1 j"j, and kP1 kC2  c1 . Notice that the symplectic transformation (1) is actually the composition of two ‘‘elementary’’ transfo(1) (1) 0 0 mations: (1) = (1) 1  2 where 2 : (y , x ) ! ( , ) d is the symplectic lift of the T -diffeomorphism given by x = þ "a( ) (i.e., (1) is the symplectic map 2 generated by y0  þ "y0  a( )), while (1) 1 : ( , ) ! (y, x) is the angle-dependent action translation generated by  x þ "(b  x þ s(x)); (1) 2 acts in the ‘‘angle direction’’ and straightens out the flow up to order O("2 ), while (1) 1 acts in the ‘‘action direction’’ and is needed to keep the frequency of the torus fixed. Since H"  1 =: N1 þ "2 P1 is again a perturbation of a nondegenerate Kolmogorov normal form (with same frequency vector !), one can repeat the construction by obtaining a new Hamiltonian of the form N2 þ "4 P2 . Iterating, after k steps, one gets k a Hamiltonian Nk þ "2 Pk . Carrying out the (straightforward but lengthy) estimates, one can k check that kPk kC2  ck  c2 , for a suitable constant c > 1 independent of k (the fast growth of the constant ck is due to the presence of the small

divisors appearing in the explicit construction of the symplectic transformations (j) ). Thus, it is clear that taking "0 small enough the iterative procedure converges (superexponentially fast) yielding the thesis of the above theorem. 6. While the statement of the invariant tori theorem and the outline of the proof are very clearly explained in Kolmogorov (1954), Kolmogorov did not fill out the details nor gave any estimates. Some years later, Arnol’d (1963a) published a detailed proof, which, however, did not follow Kolmogorov’s idea. In the same year, J K Moser published his invariant curve theorem (for areapreserving twist diffeomorphisms of the annulus) in smooth setting. The bulk of techniques and theorems stemmed out from these works is normally referred to as KAM theory; for reviews, see Arnol’d (1988) or Bost (1984–85). A very complete version of the ‘‘KAM theorem’’ both in the real-analytic and in the smooth case (with optimal smoothness assumptions) is given in Salamon (2004); the proof of the real-analytic part is based on Kolmogorov’s scheme. The KAM theory of M Herman, used in his approach to the planetary problem, is based on the abstract functional theoretical approach of R Hamilton (which, in turn, is a development of Nash–Moser implicit function theorem; see Bost (1984–85) for references); it is interesting, however, to note that the heart of Herman’s KAM method is based on the above-mentioned Kolmogorov’s transformation (1) (compare Fe´joz (2002)). 7. In the nearly integrable case, one considers a oneparameter family of Hamiltonians H0 (I) þ "H1 (I, x) with (I, x) 2 M := U  Td standard symplectic action-angle variables, U being an open subset of Rd . When " = 0, the phase space M is foliated by H0 -invariant tori {I0 }  Td , on which the flow is given by x ! x þ @y H0 (I0 )t. If I0 is such that ! := @y H0 (I0 ) is Diophantine and if det @y2 H0 (I0 ) 6¼ 0, then from Kolmogorov’s theorem it follows that the torus {I0 }  Td persists under perturbation. In fact, introduce the symplectic variables (y, x) with y = I  I0 and let N(y):= H0 (I0 þ y), which by Taylor’s formula can be written as H0 (I0 ) þ !  y þ Q(y) with Q(y) quadratic in y and @y2 Q(0) = @y2 H0 (I0 ) invertible. One can then apply Kolmogorov’s theorem with P1 (y, x) := H1 (I0 þ y, x). Notice that Kolmogorov’s nondegeneracy condition det @y2 H0 (I0 ) 6¼ 0 simply means that the frequency map I 2 Bd U ! !ðIÞ :¼ @y H0 ðIÞ

½13

KAM Theory and Celestial Mechanics

is a local diffeomorphism (Bd being a ball around I0 ). 8. The symplectic structure implies that if n denotes the number of degrees of freedom (i.e., half of the dimension of the phase space) and d is the number of independent frequencies of a quasiperiodic motion, then d  n; if d = n, the quasiperiodic motion is called maximal. Kolmogorov’s theorem gives sufficient conditions in order to get maximal quasiperiodic solutions. In fact, Kolmogorov’s nondegeneracy condition is an open condition and the set of Diophantine vectors is a set of full Lebesgue measure. Thus, in general, Kolmogorov’s theorem yields a positive invariant measure set spanned by maximal quasiperiodic trajectories. As mentioned above, the planetary many-body models are properly degenerate and violate Kolmogorov’s nondegeneracy conditions and, hence, Kolmogorov’s theorem – clearly motivated by celestial mechanics – cannot be applied. There is, however, an important case to which a slight variation of Kolmogorov’s theorem can be applied (Kolmogorov did not mention this in 1954). The case referred to here is the simplest nontrivial three-body problem, namely, the restricted, planar, and circular three-body problem (RPC3BP for short). This model, largely investigated by Poincare´, deals with an asteroid of ‘‘zero mass’’ moving on the plane containing the trajectory of two unperturbed major bodies (say, Sun and Jupiter) revolving on a Keplerian circle. The mathematical model for the restricted three-body problem is obtained by taking n = 2 and setting m2 = 0 in eqn [1]: the equations for the two major bodies (i = 0, 1) decouple from the equation for the asteroid (i = 2) and form an integrable twobody system; the problem then consists in studying the evolution of the asteroid u(2) (t) in the given gravitational field of the primaries. In the circular and planar cases, the motion of the two primaries is assumed to be circular and the motion of the asteroid is assumed to take place on the plane containing the motion of the two primaries; in fact (to avoid collisions), one considers either inner or outer (with respect to the circle described by the relative motion of the primaries) asteroid motions. To describe the Hamiltonian Hrcp governing the motion of the RCP3BP problem, introduce planar Delaunay variables ((L, G), (‘, g)) ˆ for the asteroid (better, for the reduced heliocentric Sun–asteroid system). Such variables, which are closely related to the above (spatial) Delaunay variables, have the following physical interpretation: G is proportional to the absolute value of the angular momentum of

193

the asteroid, L is proportional to the square root of the semimajor axis of the instantaneous Sun– asteroid ellipse, ‘ is the mean anomaly of the asteroid, while gˆ the argument of the perihelion. Then, in suitably normalized units, the Hamiltonian governing the RPC3BP is given by 1 G 2L2 þ "H1 ðL; G; ‘; g; "Þ

Hrcp ðL; G; ‘; g; "Þ :¼ 

½14

where g := gˆ  ,  2 T being the longitude of Jupiter; the variables ((L, G), (‘, g)) are symplectic coordinates (with respect to the standard symplectic form); the normalizations have been chosen so that the relative motion of the primary bodies is 2 periodic and their distance is 1; the parameter " is (essentially) the ratio between the masses of the primaries; the perturbation H1 is the function x(2)x(1)  1=jx(2)  x(1) j expressed in the above variables, x(2) being the heliocentric coordinate of the asteroid and x(1) that of the planet (Jupiter): such a function is real-analytic on {0 < G < L}  T2 and for small " (for complete details, see, e.g., Celletti and Chierchia (2003)). The integrable limit 2 Hð0Þ rcp : = Hrcp j" = 0 = 1=ð2L Þ  G

has vanishing Hessian and, hence, violates Kolmogorov’s nondegeneracy condition (as described in item (7) above). However, there is another nondegeneracy condition which leads to a simple variation of Kolmogorov’s theorem, as explained briefly below. Kolmogorov’s nondegeneracy condition det2y H0 (I0 ) 6¼ 0 allows one to fix d-parameters, namely, the d-components of the (Diophantine) frequency vector ! = @y H0 (I0 ). Instead of fixing such parameters, one may fix the energy E = H0 (I0 ) together with the direction {s! : s 2 R} of the frequency vector: for example, in a neighborhood where !d 6¼ 0, one can fix E and !i =!d for 1  i  d  1. Notice also that if ! is Diophantine, then so is s! for any s 6¼ 0 (with same  and rescaled ). Now, it is easy to check that the map I 2 H01 (E) ! (!1 =!d , . . . , !d1 =!d ) is (at fixed energy E) a local diffeomorphism if and only if the (d þ 1)  (d þ 1) matrix @y2 H0

@y H0

@y H0

0

!

evaluated at I0 is invertible (here the vector @y H0 in the upper right corner has to be interpreted as a column while the vector @y H0 in the lower left corner has to be interpreted as a row). Such

194 KAM Theory and Celestial Mechanics

‘‘iso-energetic nondegeneracy’’ condition, rephrased in terms of Kolmogorov’s normal forms, becomes  2  h@y Qð0; Þi ! 6¼ 0 ½15 det ! 0 Kolmogorov’s theorem can be easily adapted to the fixed energy case. Assuming that ! is Diophantine and that N is isoenergetically nondegenerate, the same conclusion as in Kolmogorov’s theorem holds with N" := E þ !"  y0 þ Q" (y0 , x0 ), where !" = " ! and j"  1j is small with ". In the RCP3BP case, the isoenergetic nondegeneracy is met, since ! 2 Hð0Þ @ðL;GÞ Hð0Þ @ðL;GÞ 3 rcp rcp det ¼ 4 ð0Þ L @ðL;GÞ Hrcp 0 Therefore, one can conclude that on each negative energy level, the RCP3BP admits a positive measure set of phase points, whose time evolution lies on twodimensional invariant tori (on which the flow is analytically conjugate to linear translation by a Diophantine vector), provided the mass ratio of the primary bodies is small enough; such persistent tori are a slight deformation of the unperturbed ‘‘Keplerian’’ tori corresponding to the asteroid and the Sun revolving on a Keplerian ellipse on the plane where the Sun and the major planet describe a circular orbit. In fact, one can say more. The phase space for the RCP3BP is four dimensional, the energy levels are three dimensional, and Kolmogorov’s invariant tori are two dimensional. Thus, a Kolmogorov torus separates the energy level, on which it lies, into two invariant components, and two Kolmogorov’s tori form the boundary of a compact invariant region so that any motion starting in such region will never leave it. Thus, the RCP3BP is ‘‘totally stable’’: in a neighborhood of any phase point of negative energy, if the mass ratio of the primary bodies is small enough, the asteroid stays forever on a nearly Keplerian ellipse with nearly fixed orbital elements L and G. Arnol’d’s Theorem

Consider again the planetary (1 þ n)-body problem governed by the Hamiltonian Hplt in eqn [5]. In the integrable approximation, governed by the Hamiltonian H(0) plt , the n planets describe Keplerian ellipses focused on the Sun. Arnol’d (1963b) has stated the following theorem. Theorem (Arnol’d 1963b) Let " > 0 be small enough. Then, there exists a bounded, Hplt -invariant set F (") M of positive Lebesgue measure corresponding to planetary motions with bounded relative distances; F (0) corresponds to Keplerian

ellipses with small eccentricities and small relative inclinations. This theorem represents a major achievement in celestial mechanics solving more than tri-c´entennial mathematical problem. Arnol’d (1963b) gave a complete proof of this result only in the planar three-body case and gave some indications of how to extend his approach to the general situation. However, to give a full proof of Arnol’d’s theorem in the general case turned out to be more than a technical problem and new ideas were needed: the complete proof (due, essentially, to M Herman) has been given only in 2004. In the following subsections, we briefly review the history and the ideas related to the proof of Arnol’d’s theorem. As for credits: the proof of Arnol’d’s theorem in the planar 3BP case is due to Arnol’d himself (Arnol’d 1963b); the spatial 3BP case is due to Laskar and Robutel (1995) and Robutel (1995); the general case is due to Herman (1998) and Fe´joz (2004). The exposition we have given does not always follow the original references. The planar three-body problem Recall the Hamiltonian Hpln of the planar (1 þ n)-body problem given in item (3) of the section ‘‘The planetary (1 þ n)-body problem.’’ A convenient set of symplectic variables for nearly circular motions are the ‘‘planar Poincare´ variables.’’ To describe such variables, consider a single, planar two-body system with Hamiltonian jXj2 M ; X 2 R2 ; 0 6¼ x 2 R 2  jxj 2 ðwith respect to dX ^ dxÞ

½16

and introduce – as done before formula [14] for H(0) rcp – planar Delaunay variables ((L, G), (‘, g)) (here, g = gˆ = argument of the perihelion). To remove the singularity of the Delaunay variables near zero eccentricities, Poincare´ introduced variables ((, ), (, )) defined by the following formulas:  ¼ L;

H ¼LG

 ¼ ‘ þ g; h ¼ g pffiffiffiffiffiffiffi 2H cos h ¼ pffiffiffiffiffiffiffi 2H sin h ¼

½17

As Poincare´ showed, such variables are symplectic and analytic in a neighborhood of (0, 1)  T  {0, 0}; notice that the symplectic map ((, ), (, )) ! (X, x) depends on the parameters , M, and ". In Poincare´ variables, the two-body Hamiltonian in eqn [16]

KAM Theory and Celestial Mechanics

becomes =(22 ), with  := (=m0 )3 =M. Now, re-insert the index i, let i : ((i , i ), (i , i )) ! (X(i) , x(i) ) and (, , , ) = (1 (1 , 1 , 1 , 1 ), . . . , n (n , n , n , n )). Then, the Hamiltonian for the planar (1 þ n)-body problem takes the form Hpln   ¼ H0 ðÞ þ "H1 ð; ; ; Þ  3 n 1X i i 1 H0 :¼  ; i :¼ 2 2 i¼1 i m0 Mi compl

H1 :¼ H1

½18

princ compl

X 1i 2. To overcome these problems Herman proposed a new approach, which is described below. Instead of Kolmogorov’s nondegeneracy assumption – which says that the frequency map [13] I ! !(I) is a local diffeomorphism – one may consider weaker nondegeneracy conditions. In particular, in Fe´joz (2004), one considers nonplanar frequency maps. A smooth curve u 2 A ! !(u) 2 Rd , where A is an open nonempty interval, is called ‘‘nonplanar’’ at u0 2 A if all the u-derivatives up to order (d  1) at u0 , !(u0 ), !0 (u0 ), . . . , !(d1) (u0 ) are linearly independent in R d ; a smooth map u 2 A Rp ! !(u) 2 R d , p  d, is called nonplanar at u0 2 A if there exists a smooth curve ’ : Aˆ ! A such that !  ’ is nonplanar at t0 2 Aˆ with ’(t0 ) = u0 . A S Pyartli has proved (see, e.g., Fe´joz (2004)) that if the map u 2 A Rp ! !(u) 2 Rd is nonplanar at u0 , then there exists a neighborhood

197

B A of u0 and a subset C B of full Lebesgue measure (i.e., meas(C) = meas(B)) such that !(u) is Diophantine for any u 2 C. The nonplanarity condition is weaker than Kolmogorov’s nondegeneracy conditions; for example, the map  4  I1 2 !ðIÞ :¼ @I þ I1 I2 þ I1 I3 þ I4 4  3  ¼ I1 þ 2I1 I2 þ I3 ; I12 ; I1 ; 1 violates both Kolmogorov’s nondegeneracy and the isoenergetic nondegeneracy conditions but is nonplanar at any point of the form (I1 , 0, 0, 0), since !(I1 , 0, 0, 0) = (I13 , I12 , I1 , 1) is a nonplanar curve (at any point). As in the three-body case, the frequency map is that associated with the averaged secular Hamiltonian Hsec :¼ Hð0Þ ðÞ þ "Hð1Þ Z d ð1Þ  Hð1Þ H ð; ; ; p; qÞ :¼ ð2Þn

½28

which has an elliptic equilibrium at = = p = q = 0 (as above,  is regarded as a parameter). It is a remarkably well-known fact that the quadratic part of H(1) does not contain ‘‘mixed terms,’’ namely,  ð1Þ Hð1Þ ¼ H0 þ " Qpln  þ Qpln  þ Qspt p  p  þ Qspt q  q þ O4 ½29 where the function H(1) 0 and the symmetric matrices Qpln and Qspt depend upon  while O4 denotes terms of order 4 in ( , , p, q). The eigenvalues of the matrices Qpln and Qspt are the first Birkhoff invariants of H(1) (with respect to the symplectic variables ( , , p, q)). Let 1 , . . . , n and &1 , . . . , &n denote, respectively, the eigenvalues of Qpln and Qspt ; then the frequency map for the (1 þ n)-body problem will be defined as (recall eqn [18]) ^ "Þ  ! ð!;

½30

with  ^ :¼ !

1 n ;...; 3 3  1 n

 ½31

 :¼ ð ; &Þ :¼ ðð 1 ; . . . ; n Þ; ð&1 ; . . . ; &n ÞÞ Herman pointed out, however, that the frequencies and & satisfy two independent linear relations, namely (up to renumbering the indices), &n ¼ 0;

n X

ð i þ &i Þ ¼ 0

½32

i¼1

which clearly prevents the frequency map to be nonplanar; the second relation in eqn [32] is usually

198 KAM Theory and Celestial Mechanics

called ‘‘Herman resonance’’ (while the first relation is a well-known consequence of rotation invariance). The degeneracy due to rotation invariance may be easily taken care of by considering (as in the three-body case) the (6n  2)-dimensional invariant symplectic manifold Mver , defined by taking the total angular momentum C to be vertical, that is, C  k1 = 0 = C  k2 . But, when n > 2, Jacobi’s reduction of the nodes is no more available and to get rid of the second degeneracy (Herman’s resonance), the authors bring in a nice trick, originally due – once more! – to Poincare´. In place of considering Hnbp restricted on Mver , Fe´joz considers the modified Hamiltonian H nbp :¼ Hnbp þ C23 ;

C3 :¼ C  k3 ¼ jCj

½33

where 2 R is an extra artificial parameter. By an analyticity argument, it is then possible to prove that the (rescaled) frequency map ^ 1 ; . . . ; n ; &1 ; . . . ; &n1 Þ 2 R3n1 ð; Þ ! ð!; is nonplanar on an open dense set of full measure and this is enough to find a positive measure set of Lagrangian maximal (3n  1)-dimensional invariant tori for H nbp ; but, since H nbp and Hnbp commute, a classical Lagrangian intersection argument allows one to conclude that such tori are invariant also for Hnbp yielding the complete proof of Arnol’d’s theorem in the general case. Notice that this argument yields (3n  1)-dimensional tori, which in the three-body case means five dimensional. Instead, the tori found in the section ‘‘The spatial three-body problem’’ are four dimensional. The point is that in the reduced phase space, the motion of the nodeline – denoted as 2 (t) in eqn [25] – does not appear. We conclude this discussion by mentioning that the KAM theory used in Fe´joz (2004) is a modern and elegant function-theoretic reformulation of the classical theory and is based on a C1 local inversion theorem (F Sergeraert and R Hamilton) on ‘‘tame’’ Frechet spaces (which, in turn, is related to the Nash–Moser implicit function theorem; see Bost (1984–85)).

Lower Dimensional Tori The maximal tori for the many-body problems described above are found near the elliptic equilibria given by the decoupled Keplerian motions. It is natural to ask what happens of such elliptic equilibria when the interaction among planets is taken into account. Even though no complete answer has yet been given to such a question, it

appears that, in general, the Keplerian elliptic equilibria ‘‘bifurcate’’ into elliptic n-dimensional tori. This section presents a short and nontechnical account of the existing results on the matter (the general theory of lower-dimensional tori is, mainly, due to J K Moser and S M Graff for the hyperbolic case and V K Melnikov, H Eliasson, and S B Kuksin for the technically more difficult elliptic case; for references, see, e.g., Chierchia et al. (2004)). The normal form of a Hamiltonian admitting an n-dimensional elliptic invariant torus T of energy E, proper frequencies !ˆ 2 Rn , and ‘‘normal frequencies’’  2 Rp in a 2d-dimensional phase space with d = n þ p is given by ^yþ N :¼ E þ !

p X j¼1

j

j2 þ j2 2

½34

Here the symplectic form is given by dy ^ dx þ d ^ d , y 2 Rn , x 2 Tn , ( , ) 2 R 2p ; T is then given by T := {y = 0}  { = = 0}. Under suitable assumptions, a set of such tori persists under the effect of a small enough perturbation P(y, x, , ). Clearly, the union of the persistent tori (if n < d) forms a set of zero measure in phase space; however, in general, n-parameter families persist. In the many-body case considered in this article, the proper frequencies are the Keplerian frequencies given by the map  ! !() (eqn [31]), which is a ˆ local diffeomorphism of Rn . The normal frequencies , instead, are proportional to " and are the first Birkhoff invariants around the elliptic equilibria as discussed above. Under these circumstances, the main nondegeneracy hypothesis needed to establish the persistence of the Keplerian n-dimensional elliptic tori boils down to the so-called Melnilkov condition: j 6¼ 0 6¼ i  j ;

8j 6¼ i

½35

Such condition has been checked for the planar three-body case in Fe´joz (2002), for the spatial three-body case in Biasco et al. (2003) and for the planar n-body case in Biasco et al. (2004). The general spatial case is still open: in fact, while it is possible to establish lower-dimensional elliptic tori for the modified Hamiltonian H nbp in [33], it is not clear how to conclude the existence of elliptic tori for the actual Hamiltonian Hnbp since the argument used above works only for Lagrangian (maximal) tori; on the other hand, the direct asymptotics techniques used in Biasco et al. (2003) do not extend easily to the general spatial case. Clearly, the lower-dimensional tori described in this section are not the only ones that arise in n-body dynamics. For more lower-dimensional tori in the planar three-body case, see Fe´joz (2002).

KAM Theory and Celestial Mechanics

Physical Applications The above results show that, in principle, there may exist ‘‘stable planetary systems’’ exhibiting quasiperiodic motions around coplanar, circular Keplerian trajectories – in the Newtonian many-body approximation – provided the masses of the planets are much smaller than the mass of the central star. A quite different question is: in the Newtonian many-body approximation, is the solar system or, more in generally, a solar subsystem stable? Clearly, even a precise mathematical reformulation of such a question might be difficult. However, it might be desirable to develop a mathematical theory for important physical models, taking into account observed parameter values. As a very preliminary step in this direction, consider one of the results of Celletti and Chierchia (see Celletti and Chierchia (2003), and references therein). In Celletti and Chierchia (2003), the (isolated) subsystem formed by the Sun, Jupiter, and asteroid Victoria (one of the main objects in the Asteroidal belt) is considered. Such a system is modeled by an order-10 Fourier truncation of the RPC3BP, whose Hamiltonian has been described in the section ‘‘Kolmogorov’s theorem and the RPC3BP (1954).’’ The Sun–Jupiter motion is therefore approximated by a circular one, the asteroid Victoria is considered massless, and the motions of the three bodies are assumed to be coplanar; the remaining orbital parameters (Jupiter/Sun mass ratio, which is approximately 1/1000; eccentricity and semimajor axis of the osculating Sun–Victoria ellipse; and ‘‘energy’’ of the system) are taken to be the actually observed values. For such a system, it is proved that there exists an invariant region, on the observed fixed energy level, bounded by two maximal two-dimensional Kolmogorov tori, trapping the observed orbital parameters of the osculating Sun–Victoria ellipse. As mentioned above, the proof of this result is computer assisted: a long series of algebraic computations and estimates is performed on computers, keeping a rigorous track of the numerical errors introduced by the machines.

Acknowledgments The author is indebted to J Fe´joz for explaining his work on Herman’s proof of Arnol’d’s theorem prior to publication. The author is also grateful for the collaborations with his colleagues and friends L Biasco, E Valdinoci, and especially A Celletti.

199

See also: Averaging Methods; Diagrammatic Techniques in Perturbation Theory; Gravitational N-Body Problem (Classical); Hamiltonian Systems: Stability and Instability Theory; Hamilton–Jacobi Equations and Dynamical Systems: Variational Aspects; Korteweg–de Vries Equation and Other Modulation Equations; Stability Problems in Celestial Mechanics; Stability Theory and KAM.

Further Reading Arnol’d VI (1963a) Proof of a Theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russian Mathematical Survey 18: 13–40. Arnol’d VI (1963b) Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi MatematiI¨ceskih Nauk 18(6(114)): 91–192. Arnol’d VI (ed.) (1988) Dynamical Systems III, Encyclopedia of Mathematical Sciences, vol. 3. Berlin: Springer. Biasco L, Chierchia L, and Valdinoci E (2003) Elliptic twodimensional invariant tori for the planetary three-body problem. Archives for Rational and Mechanical Analysis 170: 91–135. Biasco L, Chierchia L, and Valdinoci E (2004) N-dimensional invariant tori for the planar (N þ 1)-body problem. SIAM Journal on Mathematical Analysis, to appear, pp. 27 (http://www.mat. uniroma3.it/users/chierchia/WWW/english_version.html). Bost JB (1984/85) Tores invariants des syste`mes dynamiques hamiltoniens. Se´minaire Bourbaki expose 639: 113–157. Celletti A and Chierchia L (2003) KAM stability and celestial mechanics. Memoirs of the AMS, to appear, pp. 116. (http:// www.mat.uniroma3.it/users/chierchia/WWW/english_version. html). Chierchia L and Qian D (2004) Moser’s theorem for lower dimensional tori. Journal of Differential Equations 206: 55–93. Fe´joz J (2002) Quasiperiodic motions in the planar three-body problem. Journal of Differential Equations 183(2): 303–341. Fe´joz J (2004) De´monstration du ‘‘the´ore`me d’Arnol’d’’ sur la stabilite´ du syste`me plane´taire (d’apre`s Michael Herman). Ergodic Theory & Dynamical Systems 24: 1–62. Herman MR (1998) De´monstration d’un the´ore`me de V.I. Arnol’d. Se´minaire de Syste`mes Dynamiques and manuscripts. Kolmogorov AN (1954) On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Doklady Akademii Nauk SSR 98: 527–530. Laskar J and Robutel P (1995) Stability of the planetary threebody problem. I: Expansion of the planetary Hamiltonian. Celestial Mechanics & Dynamical Astronomy 62(3): 193–217. Robutel P (1995) Stability of the planetary three-body problem. II: KAM theory and existence of quasi-periodic motions. Celestial Mechanics & Dynamical Astronomy 62(3): 219–261. Salamon D (2004) The Kolmogorov–Arnol’d–Moser theorem. Mathematical Physics Electronic Journal 10: 1–37. Siegel CL and Moser JK (1971) Lectures on Celestial Mechanics. Berlin: Springer.

200 Kinetic Equations

Kinetic Equations C Bardos, Universite´ de Paris 7, Paris, France ª 2006 Elsevier Ltd. All rights reserved.

Introduction In most physical cases, the evolution of a system of N indistinguishable interacting particles XN = (x1 , x2 , . . . , xN ) with velocities VN = (v1 , v2 , . . . , vN ) is described by a Hamiltonian system dXN @HðXN ; VN Þ ¼ @VN dt dVN @HðXN ; VN Þ ¼ @XN dt

½1

dN in the phase space RdN X  RV . When N becomes large, it is natural to consider replacing the above discrete phase space by a continuous phase space of dimension 1  d  3, Rdx  Rdv and to introduce a measure f (x, v, t) that describes the density of particles which, at the point x 2 Rd and at time t, have velocity v. This measure may also be interpreted as a generalization of the empirical measure

N ðtÞ ¼

1 X x ðtÞ;v ðtÞ N 1iN i i

defined in the phase space Rdx  Rdv by the above system of N particles. In this way, one constructs a link between the microscopic and the macroscopic descriptions. The macroscopic physical quantities are, for instance, the first moments of this density: Z ðx; tÞ ¼ f ðx; v; tÞdv ðdensityÞ Rd Z v vf ðx; v; tÞdv ðmomentumÞ ðx; tÞuðx; tÞ ¼ Rdv

ðx; tÞEðx; tÞ ¼

Z

jvj2 f ðx; v; tÞdv ðenergyÞ Rdv 2

Kinetic theory studies the intermediate stage shown in Figure 1. Its first successes were related to classical thermodynamics and in particular to the molecular hypothesis. The contributions of Maxwell (1860, 1872) and of Boltzmann (1867) led to the ‘‘Boltzmann’’

Hamiltonian Systems

1 2 → Kinetic equations → Macroscopic equations

Figure 1 Illustration of the role of kinetic equations in linking microscopic and macroscopic properties.

equation, described in the companion article of Mario Pulvirenti (see Boltzmann Equation (Classical and Quantum)). In 1905, Lorentz used the same point of view to describe the motion of electrons in a metal. However, the different physical context leads to some basic differences between the Boltzmann equation and the Lorentz equation. The Boltzmann equation is derived under the assumption that the driving forces result from collisions between pairs of molecules. Therefore, the problem is nonlinear with a quadratic nonlinearity. In the Lorentz model the driving force is the interaction of the electrons with the atoms of the metal, which remain fixed. Collisions between electrons are ignored, so that the Lorentz equation is linear. The most general form of a kinetic equation is as follows: @t f ðx; v; tÞ þ rv Hf  rx f ðx; v; tÞ  rx Hf  rv f ðx; v; tÞ ¼ Cðf Þ

½2

The term C(f ) represents the effect of interactions either between particles or with the background. Without this term, the eqn [2] is reduced to the classical Liouville equation @t f ðx; v; tÞ þ rv Hf  rx f ðx; v; tÞ  rx Hf  rv f ðx; v; tÞ ¼ 0

½3

which says that the function f is transported by the flow of the Hamiltonian Hf (x, v). This Hamiltonian depends on the model and may involve the unknown function f itself. In the simplest case H(x, v) = jvj2 =2, eqn [3] and its solutions are given by @t f ðx; v; tÞ þ v  rv f ðx; v; tÞ ¼ 0 f ðx; v; tÞ ¼ f ðx  vt; v; 0Þ

½4

Nowadays kinetic equations appear in a variety of sciences and applications, such as astrophysics, aerospace engineering, nuclear engineering, particle– fluid interactions, semiconductor technology, social sciences, and biology, for example in chemotaxis and immunology. They are used first to model phenomena and then to obtain a qualitative and quantitative description of situations involving sufficiently many particles so as to prohibit any computation at the level of particles, and yet the medium is still too rarefied to allow the use of macroscopic equations. As detailed in the next section, a macroscopic description requires that the function f (x, v, t) be close to local thermodynamical equilibrium. For classical and quantum Boltzmann equations (see Boltzmann

Kinetic Equations

Equation (Classical and Quantum)) these equilibria are either Maxwellian, Bose–Einstein, or Fermi– Dirac distributions. Several effects, especially the influence of the boundary, may prevent the system from reaching local thermodynamical equilibrium and, therefore, even in macroscopic descriptions, kinetic equations may still be used to take into account the effect of the boundary. In this case, the term ‘‘Knudsen boundary layer’’ is currently used. Finally, one should keep in mind that there exist some macroscopic phenomena which cannot be deduced from the corresponding microscopic physics by the mediation of a kinetic equation. Once again, returning to the companion article (see Boltzmann Equation (Classical and Quantum)) one observes that, since the only equilibria are Maxwellian, the macroscopic equations are those describing perfect gases. A real gas with a nontrivial van der Waals law is ‘‘too dense’’ to be explained by this theory. The alternative seems to go directly from the microscopic direction to the macroscopic description. This is a subject which is still under investigation and for which the reader may consult Olla et al. (1993).

Kinetic Equations Entropy and Irreversibility At the level of particles, the basic laws of physics are reversible. Yet these same laws are not reversible when seen at the level of a macroscopic description. This lack of reversibility is measured by the decay of entropy (mathematicians prefer convex functions; therefore, the mathematical entropy considered in this contribution is the negative of the physical entropy, and with irreversibility it decays). The kinetic equations lie in between, as shown in Figure 1; the decay of entropy should appear along one of the two arrows of this diagram. Since the appearance of irreversibility is related to loss of information and averaging, it should be driven by a ‘‘mixing’’ process. In general two mechanisms are responsible for such effects: 1. an ergodic or a relaxation mechanism by which a process averages itself; and 2. the introduction of some external random parameter. Observable quantities are then defined as averages over that parameter. It seems important to compare these two ‘‘processes.’’ This will be illustrated below with the most classical examples of the theory.

201

The Diffusion Limit for the Neutron Transport Equation

Equations very similar to the one introduced by Lorentz are used to describe the interaction of neutrons with atoms in a nuclear reactor: this is the reason why these types of equations are often called neutron transport equations. An important issue is the derivation of a macroscopic diffusion equation. Assuming that neutrons are not subject to acceleration effects, considering the problem with constant modulus of velocity (jvj = 1), introducing a ‘‘small’’ parameter  which here corresponds to the absorption of the medium, one can study the following simplified model: @t f þ v  rx f Z  ðxÞ  þ kðv; v0 Þf ðv0 Þdv0 ¼ 0 f   jv0 j¼1

½5

In [5] one assumes, for the kernel k(v, v0 ), the following properties: 8v; v0; Z

kðv; v0 Þ ¼ kðv0 ; vÞ; 0 < kðv; v0 Þ kðv; v0 Þdv ¼ 1

½6

jv0 j¼1

and denotes by K the operator Z f 7! Kf ¼ kðv; v0 Þf ðv0 Þdv0 jv0 j¼1

In the simplest case (say without boundary) eqn [5] is well-posed both for positive and negative time but hypothesis [6] has the following important consequences: 1. For positive time, it defines, for each  > 0, a contraction semigroup in any Lp space and, therefore, the sequence of solutions or a subsequence thereof converges, say weakly, to a limit f (x, v, t). 2. One also observes that v 7! 1 is (up to a multiplicative constant) the only solution of the equation Z f  Kf ¼ f ðvÞ  kðv; v0 Þf ðv0 Þdv ¼ 0 ½7 jv0 j¼1

Therefore, the 1 in front of the collision term forces the limit f (x, v, t) to be independent of v. In this simple problem, this is the thermodynamical equilibrium. Dividing by  and integrating over jvj = 1 gives the relation Z @t f ðx; v; tÞdv jvj¼1 Z 1 þ rx vf ðx; v; tÞdv ¼ 0 ½8  jvj¼1

202 Kinetic Equations

Now using the Fredholm alternative implies the existence and uniqueness of a function v 7! (v) such that Z ðvÞ  kðv; v0 Þðv0 Þdv0 0 jv j¼1 Z ¼ v; ðv0 Þdv0 ¼ 0 ½9 jv0 j¼1

Multiply eqn [5] by (v) and integrate over jvj = 1 to obtain Z ðxÞ lim ððI  KÞÞðvÞf ðx; v; tÞdv !0  jvj¼1 Z ðxÞ ¼ lim ðvÞðI  KÞf ðx; v; tÞdv !0  jvj¼1 Z ¼  lim rx ðvÞ  vf ðx; v; tÞdv ½10 !0

jvj¼1

Since the operator (I  K) is self-adjoint nonnegative, with 0 as the leading eigenvalue, the matrix D¼

Z

ðvÞ  vdv

jvj¼1

¼

Z

ðvÞ  ðI  KÞðvÞdv

jvj¼1

is positive definite, and one finally obtains the diffusion equation   D rx f ¼ 0 @t f  rx ½11 ðxÞ The above derivation is an example of what is called the ‘‘moments method.’’ It is implicit even in the papers of Maxwell. It has been systematically used in several domains:

 To understand the relation between the Boltzmann equation and the Euler and Navier–Stokes equations (Golse 2005);  To compute the critical size of a nuclear assembly. One shows that this size is well approximated by the size of the domain for which the Laplacian, with appropriate boundary conditions, has leading eigenvalue 0. It is for the spectral analysis of this problem that the averaging lemma (see the section ‘‘Some specific mathematical tools’’) was derived.  To analyze the macroscopic limit for the solution of the radiative transfer equations, which describe the propagation of the intensity of photons in a large class of phenomena ranging from stellar atmospheres to the cooling of glass, including

optical tomography in biomedical imaging. In a simplified form, the so-called ‘‘grey model,’’ these equations can be reduced to @t I ðx; v; tÞ þ v  rx I ðx; v; tÞ Z  1  1 I ðx; v0 ; tÞÞdv0 I ðx; v; tÞ þ   4 jv0 j¼1 Z  1  I ðx; v0 ; tÞÞ dv0 ¼ 0 4 jv0 j¼1

½12

In contrast to the previous example, the problem is, in many cases, nonlinear. The opacity  is a positive function that depends on the intensity I through Z ~Iðx; tÞ ¼ 1 I ðx; v0 ; tÞ dv0 4 jv0 j¼1 and which goes to 1 with ~I going to zero. The moments method can be applied with the averaging lemma, and one shows that the limit of I is a function that is independent of v and satisfies the following degenerate parabolic equation:   1 rx I ¼ 0 ½13 @t I  rx 3ðIÞ This equation is similar to the one obtained in the description of porous media and contains the following information: for initial data I(x, 0) with compact support, in contrast to the behavior of solutions of the standard diffusion equation, the solution I(x, t) remains compactly supported in x. The boundary of this support is the thermal front and for a finite time, up to saturation (by water in porous media, by reacted deuterium in laserconfined fusion), this front remains fixed. What made the analysis of the above macroscopic limit simple was the existence of an  > 0 dependent process which, for vanishing , forces the solution to converge to a ‘‘thermodynamical’’ equilibrium. The irreversibility was already present in the first arrow of Figure 1. This is what made the analysis of the second arrow simple. The subtleties of the appearance of the irreversibility in the first arrow may be well explained by the next examples. The Linear Billiard Model

In the absence of an external electric field, the model proposed by Lorentz could be viewed as a limit of a system of particles evolving freely between spherical obstacles and reflecting on these obstacles according to the law of geometric optics. Along these lines, two types of results have been proved in two space variables.

Kinetic Equations

In 1973, Gallavotti considered the case where the obstacles are randomly spaced under a Poisson configuration and proved the following theorem: Theorem 1 Consider obstacles(balls) of radius  and center ci . Assume that the probability of finding exactly N such obstacles in a bounded measurable set  R2 is given by the ‘‘Poisson law’’ PðdcN Þ ¼ e jj

N  dc1 dc2    dcN N!

½14

 

½15

with cN ¼ c1 ; c2 ; . . . ; cN

and

 ¼

Denote by E the expectation with respect to the above Poisson distribution. For given  and cN introduce OcN ; ¼ R2 n [1iN fjx  ci j  g

½16

and fcN ,  , the solution of the problem @t fcN ; ðx; v; tÞ þ v  rx fcN ; ðx; v; tÞ ¼ 0 in OcN ;  S1

½17

with specular reflection on the boundary and v-independent initial data: fcN ; ðx; v; 0Þ ¼ ðxÞ in OcN ;  S1

½18

h ðx; t; Þ ¼ E ½fcN ; 

½19

Then

converges weakly for t 0 to the solution of the transport equation  @t f ðx; v; tÞ þ v  rf ðx; v; tÞ þ  2f ðx; vÞ  Z 1 0 0 0 f ðx; v Þjv  v jdv ¼ 0  ½20 4 S f ðx; v; 0Þ ¼ ðxÞ

in R2  S1

½21

The situation is completely different when the obstacles are periodically spaced, a situation which seems closer to Lorentz’s original idea. Golse (2003) (and previous contributions quoted in this article) obtained the following result: Theorem 2 Assume that the obstacles are periodically spaced and conveniently scaled, defining the domain O ¼ R2 n [ 2fx; jx  jj  2 g j2Z

½22

203

Then there exists a family of continuous uniformly bounded initial data such that no subsequence extracted from the family of solutions of @t f þ v  rx f ¼ 0

in O  S1

½23

with specular reflections on the boundary, converges to solutions of equations of the type [20]. This pathology is related to the existence of particles that can travel freely for a very long time before meeting the obstacles, and the proof with some arithmetic (Diophantine approximations and continued fractions) relies on the analysis of such trajectories. A comparison between the Theorems 1 and 2 shows that the ergodic property of the free flow on the periodic lattice is not strong enough to lead to a collisional kinetic equation unless some complementary randomness is introduced. The examples of this section should be compared with the rigorous derivation of the Boltzmann equation by Lanford (see Boltzmann Equation (Classical and Quantum)). The reader should observe that this derivation corresponds to the same type of scaling (finite mean free path). However, no extra randomness is needed in this case. The proof uses the fact that configurations leading only to a finite number of binary collisions are of full measure. This corresponds to an ergodicity property which is enforced by the fact that the problem is genuinely nonlinear.

Mean-Field Scaling and Vlasov Equations The neutron transport equation is devoted to the interaction with obstacles and the Boltzmann equation to binary collisions. A simpler situation from the mathematical point of view corresponds to the case where each particle is under the action of the average of all other particles. Then the name ‘‘mean field limit’’ is used. The simplest example is the derivation of a Vlasov-type equation from a system of N classical particles interacting with a C2 potential V(jxj). The following Hamiltonian is used: Hðx1 ; . . . ; xN ; v1 ; . . . ; vN Þ X jvk j2 1 X þ ¼ Vðjxk  xl jÞ 2N 1l6¼kN 2 1kN

½24

and the name mean-field scaling is related to the factor N 1 before the potential. Assuming that the particles are undistinguishable, one introduces the joint probability density FN  FN (x1 , . . . , xN ,

204 Kinetic Equations

v1 , . . . , vN ) in the N-particle phase space, which satisfies the Liouville equation X  @t FN þ fHN ; FN g :¼ @t FN þ vk rxk FN 1kN

1 X  rx ðVðjxk  xl ÞÞ 2N 1l6¼kN k   rvk FN ¼ 0 ½25 From [25], with the notations Xn ¼ ðx1 ; . . . ; xn Þ; Vn ¼ ðv1 ; . . . ; vn Þ n XnN ¼ ðxnþ1 ; . . . ; xN Þ; VN ¼ ðxnþ1 ; . . . ; xN Þ one deduces an infinite hierarchy of equations for the marginals Z n n FN ðXn ; Vn ; tÞ ¼ fN ðXN ; VN ; tÞdXnN dVN n for 1  n  N; FN  0 for N < n :

@t Fn ðXn ; Vn ; tÞ þ

X

n vn rxi FN ðXn ; Vn ; tÞ

1in

 1 X n rvi rxi Vðjxi  xj jÞFN ðXn ; Vn ; tÞ N 1i 0, EW! (t) converges weakly with  going to zero to a solution F(t) of the kinetic equation

Oscillatory solutions of the Schro¨dinger equation, with wavelength of the order of the Planck constant, tend to behave like particles. This is described in detail by different tools of high-frequency approximation. In particular, the limit of the Wigner transform of the density (x, t)  (y, t):   Z 1 hy  i y e x þ ;t Wðx; ; tÞ ¼ 2 ð2Þ3d R3d   hy   ½32 x  ; t dy 2

@t Fðt; x; Þ þ  rx Fðt; x; Þ Z ¼ jTð ; 0 Þj2 ðj j2  j 0 j2 ÞðFðt; x; 0 Þ

is a solution of a Liouville equation. Therefore, one should expect that in the presence of ‘‘many’’ obstacles (‘‘many potentials’’) the limit should be given by a kinetic equation. As shown by the previous section the introduction of randomness seems compulsory in reaching this goal. Consider a big cube  = L of size L in R3 . Let ! = (x ),  = 1, 2, . . . , N denote the configuration of random obstacles distributed uniformly in . The density of obstacles is  = N=L3 and the expectation with respect to this uniform measure is denoted by  Z Y  E :¼ dx Ld 1N

With V(jxj) a smooth, short-range potential, the random potential created by the obstacles is X V! ðxÞ ¼ Vðjx  x jÞ 1N

then one of the typical results (low-density limit, which corresponds to the quantum version of Gallavotti classical result) obtained, reads as follows: Theorem 3 (Erdo¨s and Yau 1988) Assume that the density of obstacles is  = 0  with a fixed 0 .

 Fðt; x; ÞÞd 0

½36

where T is the amplitude of the scattering operator associated to the Schro¨dinger equation with the short range potential V. The proof uses several ingredients including scattering theory with expansion in term of Dyson series; see Erdo¨s and Yau (1998).

Semiconductor Modeling In modern computers, the electronic devices are so small that the electric current may have no space/time to reach a thermodynamical equilibrium. Therefore, this turns out to be a field where the kinetic equations are the most naturally used. Details of what can be deduced from a mathematical analysis can be found in Poupaud (1994). The equations involve the distribution of electrons fe (x, k, t) and holes fh (x, k, t) and have the following form: @t fe ðt; x; kÞ þ ve ðkÞrx fe ðt; x; kÞ q þ rx Uðt; xÞ  rk fe ðt; x; kÞ h 1 ¼ ðQe ðfe Þðt; x; kÞ þ Re ðfe ; fh Þðt; x; kÞÞ  @t fh ðt; x; kÞ þ vh ðkÞrx fh ðt; x; kÞ q  rx Uðt; xÞ  rk fh ðt; x; kÞ h 1 ¼ Qh ðfh Þðt; x; kÞ þ Rh ðfh ; fe Þðt; x; kÞ 

½37

½38

206 Kinetic Equations

The variable k ranges over a torus B of R3 which, in physics books, carries the name of Brillouin zone. The velocities of propagation of electrons and holes are determined in terms of the energy band by the formula ve;h ¼

1 rk Ee;h ðkÞ h 

½39

The potential U is determined in terms of the doping profile C(x), the conductivity r , and the density of electrons and holes according to the formula  Z q 1 x Uðt; xÞ ¼ CðxÞ  fe ðt; x; kÞdk r jBj B  Z 1 þ f ðt; x; kÞdk ½40 jBj B h Finally Qe,h and Re,h are binary integral operators in the variable k 2 B which model collisions and generation–recombination processes. Concerning the ‘‘mathematical approach’’ the situation is as follows. The relations [39] can be deduced from the highfrequency analysis of the solution of the Schro¨dinger equation i h@t

x h2 ¼  þV h  2

½41

with V a periodic potential constructed on the dual lattice of B. The method uses the Bloch decomposition of the solution and the Wigner series (Poupaud 1994). No mathematical derivation of the collisions operator is currently available. The situation should be compared to what is said in the section ‘‘Derivation of kinetic equations from the Schro¨dinger equation,’’ but in a much more complicated setting. On the other hand, the collision operators Qe,h and Re,h , as given by phenomenological arguments, have enough good relaxation properties to allow a rigorous limit of the system [37]–[38] for  going to zero (Poupaud 1994). This leads to the justification of the so-called drift–diffusion models and to the possibility of constructing correctors (with respect to ) and to treating the effect of heterojunctions by boundary layer analysis.

Some Specific Mathematical Tools Few proofs were given in the above exposition and details would not be suitable for a review article. However, the mathematical approach to kinetic equations has generated some new tools, and it may be useful to give the most prominent ones.

The Averaging Lemma

Compactness results appear in spectral theory and in the construction of solutions of nonlinear equations (whenever strong convergence is needed for the limit). Being hyperbolic, the transport operator v  rx propagates singularities along characteristics. Therefore, at first sight it seems hopeless that one might obtain any regularizing effect from the free streaming part of a kinetic model. The key to obtaining regularizing effects from the transport operator v  rx is to seek those effects not on the number density itself, but on velocity averages thereof; in other words, on the macroscopic densities. Here is the prototype of all velocity averaging results. Theorem 4 Let F be a bounded family in L2 (Rd  Rd ). Assume that the family v  rx F is also bounded in L2 (Rd  Rd ). Then, for each 2 L2 (Rd ), the family of moments  (x) defined by Z  ðxÞ ¼ F ðx; vÞ ðvÞdv Rd

is relatively compact in L2 (Rd ). For the proof one starts with the expression G = F þ v  rx F takes the Fourier transform with respect to x of this relation and writes for ^ ( ) the expression Z ^ G ð ; vÞ ðvÞdv ½42 ^ ¼ 1 þ iv:

Rd Then use the Cauchy–Schwarz inequality to obtain !Z Z j ðvÞj2 dv 2 j^  j  jG ð ; vÞj2 dv ½43 2 Rd 1 þ jv  j Rd and complete the proof by standard arguments. The averaging lemma was first observed by Agoshkov (1984) for abstract results concerning the regularity of solutions of kinetic equations in domains with boundary. Independently, it was rediscovered in the improved form given above by Golse, Perthame, and Sentis (1985) and used for the spectral theory in the diffusion approximation. The extension to Lp , p > 1, spaces and to L1 (with use of entropy estimate) were instrumental in proving the validity of the Rosseland approximation for the radiative transfer equations and for the proof of existence by Lions and Di Perna of renormalized solutions of the Boltzmann equation. A more refined result needs to be used to establish the incompressible limit of the solutions of the Boltzmann equations; see Golse (2005) for details and a complete list of references.

Kinetic Equations The Dispersive Property

Rdv

of the

f ðx; v; 0Þ ¼ f 0 ðx; vÞ

½44

Consider for the solutions in elementary kinetic equations @t f þ v  rx f ¼ 0;

Rdx



the local density ðx; tÞ ¼

Z Rdv

f ðx; v; tÞdv

½45

From the relation jðx; tÞj ¼ ¼ 

Z Z Z

Rdv

f 0 ðx  vt; v; tÞdv sup jf 0 ðx  vt; wÞjdv

Rd w2Rd

½46

deduce with an elementary change of variable the following estimate, which carries the name of dispersion lemma, 1 jtjd

kf 0 kL1 ðRd ;L1 ðRd ÞÞ x

v

½47

From interpolation and duality arguments follows: Proposition 1 The macroscopic density  defined by [45] satisfies the inequality kkLq ðRt ;Lp ðRd ÞÞ  CðdÞkf 0 kLa ðR2d Þ x

½48

for any choice of real numbers a, p, and q such that 1p< 1a¼

d ; d1

However, the estimates for kinetic equations are not easily translated into estimations for the Schro¨dinger equation because the properties of the initial data in terms of norms cannot be simply estimated in terms of the inverse Wigner transform. Spaces with Fourier transform in Lp , p 6¼ 2, are not easy to characterize and not natural for the Schro¨dinger equation. The above estimates have been very useful in analyzing the largetime behavior of solutions and also in proving the regularity of the three-dimensional Vlasov equation. The Entropy and Entropy Dissipation

f ðx; v; tÞdv

Rv

jðx; tÞj 

207

2 d ¼ q 1  1p

2p 2d < p þ 1 2d  1

½49

The values a = 1, p = 1, and q = 1 are obvious. The other limiting values are the interesting ones. They are given by p = d=(d  1), that is, p = d0 then q = 2 and a = 2d=(2d  1). These inequalities carry the name of Strichartz inequalities because they are very similar to classical inequalities obtained by Strichartz for the solution of the free Schro¨dinger equation. This should not be surprising since the Wigner transform of the densities Z 1 f ðx; v; tÞ ¼ eiyv ðx þ 12 y; tÞ ð2Þd  ðx  12 y; tÞdy ½50 then turns out to be a solution of the transport equation @t f þ v  rx f ¼ 0

½51

For solutions of the Boltzmann equation the Boltzmann H function Z Hðf Þ ¼ f ðx; vÞ log f ðx; vÞdx dv R3 R3

decreases in time and the same is true for the relative entropy to an absolute Maxwellian M(v) = 2 (2)3=2 ejvj =2 :     Z f f ln HðFjMÞ ¼  f þ M dx dv M R3 R3 This leads to the systematic introduction in the theory of the notion of relative entropy. It turned out to be instrumental in proving relaxation toward equilibrium of solutions of kinetic (or similar) equations and for the analysis of hydrodynamical limits. A striking example considered by Desvillettes and Villani is the linearized Fokker–Planck equation in any space dimension: @t F þ v  rx F  rx VðxÞ  rv F ¼ rv ðrv F þ FvÞ

½52

When x 7! V(x) is a smooth potential strictly convex at infinity, this system has a unique steady state given by the relation 2

F1 ðx; vÞ ¼ eVðxÞ MðvÞ ¼ eVðxÞ

ejvj

=2

ð2Þd=2

For any solution of [52] one has Z F Fjrv log j2 dx dv ¼ 0 @t HðFjMÞ þ d d M R R

½53

½54

which says that the entropy dissipation is the relative Fisher information (with respect to v) of F. Now, to study the relaxation to equilibrium, one uses the logarithmic Sobolev inequality: Z 1 F HðFjMÞ  Fjrv log j2 dx dv ½55 d d 2 R R M Details, references, and extensions can be found in Arnold et al. (2004).

208 Knot Homologies

Conclusions Kinetic equations have been studied since the end of the nineteenth century, both from the physical and mathematical points of view, but it seems that since the middle of the last century the interest in this approach has considerably increased. The fact that these equations are well adapted to the description of media which have not ‘‘thermalized’’ (because they are too rarefied or because the domain where they evolve is too small) has been a basic reason for their use in many applied fields; to the ones already quoted one may add the analysis of the air between the reading head and a compact disk, the computations of the characteristics of an ionic motor, and many others. As a consequence, mathematical progress has been very important. Without going into the details, this contribution is focused on this, and in particular on what can be obtained by the deterministic approach and where the introduction of randomness seems compulsory. The kinetic formulation turned out to be well adapted to large-scale computers, in particular with Monte Carlo simulations. One should observe that the point of view of modern functional analysis contributes stability estimates to the understanding and improvement of numerical methods. For an introduction to such numerical methods, the reader should first concentrate on the Boltzmann equation itself, which has been one of the basic motivations; consult the book of Sone (2002) the references therein and in particular the book of Bird (1994). See also: Boltzmann Equation (Classical and Quantum); Breaking Water Waves; Einstein’s Equations with Matter; Fourier Law; Interacting Stochastic Particle Systems;

Nonequilibrium Statistical Mechanics: Dynamical Systems Approach; Partial Differential Equations: Some Examples; Quantum Dynamical Semigroups.

Further Reading Arnold A, Carrillo J, Dolbeault J, et al. (2004) Entropies and equilibria of many-particle systems: an essay on recent research. Monatschefte fu¨r Mathematik 142(1): 35–43. Bird GA (1994) Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford: Oxford University Press. Chavanis PH and Sommeria J (1998) Monthly Notices of the Royal Astronomical Society 296: 569. Erdo¨s L and Yau HT (1998) Linear Boltzmann equation as a scaling weak limit of quantum Lorentz gas. Advances in Differential Equations and Mathematical Physics, Contemporary Mathematics 217: 137–155. Glassey RT (1996) The Cauchy Problem in Kinetic Theory. Philadelphia, PA: SIAM. Golse F (2003a) On the statistics of free-path lengths for the periodic Lorentz gas. In: Zambrini J (ed.) Proceedings of the XIVth International Congress of Mathematical Physics, Lisbon 2003. Singapore: World Scientific. Golse F (2003b) The mean-field limit for the dynamics of large particle systems. Journe´es Equations aux de´rive´es partielles Forges-les-Eaux, 2 6 juin 2003 GDR 2434 (CNRS). Golse F (2005) The Boltzmann equation and its hydrodynamic limits. In: Dafermos C and Feireisl E (eds.) Handbook of Differential Equations, vol. 2: Evolutionary Equations. Boston: Elsevier. Olla S, Varadhan R, and Yau HT (1993) Hydrodynamical limit for Hamilton system with weak noise. Communications in Mathematical Physics 155: 523–560. Poupaud F (1994) Mathematical theory of kinetic equations for transport modelling in semiconductors. In: Perthame B (ed.) Advances in Kinetic Theory and Computing Series in Applied Mathematics, vol. 22, pp. 141–168. Singapore: World Scientific. Sone Y (2002) Kinetic Theory and Fluid Dynamics. Boston: Birkha¨user.

Knot Homologies and normalized so that P of the unknot is equal to 1. Let PN (K) be the specialization of PK given by

J Rasmussen, Princeton University, Princeton, NJ, USA ª 2006 Elsevier Ltd. All rights reserved.

PN ðKÞ ¼ PK ðqN ; qÞ

Introduction A knot homology is a theory which assigns to a knot K (or link L) in S3 a graded homology group whose graded Euler characteristic is a knot polynomial associated to K. In all known examples, the knot polynomials in question are specializations of the HOMFLY polynomial PK (a, q), which we take to be determined by the skein relation 1

2

-Þ ¼ ðq  q1 ÞPð aPð% -Þ  a1 Pð%

Þ

½1

½2

Then for each N 0, there is a bigraded knot i, j homology HN (K), which satisfies X i;j ð1Þi qj dim HN ðKÞ ½3 PN ðKÞ ¼ i;j

We refer to the first grading i as the homological grading, and the second grading j as the polynomial or q-grading. The idea of a knot homology was introduced by Khovanov (2000) in a seminal paper, in which he defined the homology theory corresponding to the

Knot Homologies

Jones polynomial (N = 2). In subsequent work, he defined such a theory for N = 3, and then, in collaboration with Rozansky, for any N > 0. Recently, the two authors have introduced a triply graded homology theory Hi, j, k (K) whose graded Euler characteristic gives the entire HOMFLY polynomial: X ð1Þi qj ak dim Hi;j;k ðKÞ ½4 PK ða; qÞ ¼ i;j;k

All of these theories are combinatorial in nature. In contrast, the knot homology for N = 0 arises from a very different source – the Heegaard Floer homology of Ozsva´th and Szabo´. This theory traces its roots back to invariants of 3- and 4-manifolds defined using Seiberg–Witten and Donaldson theory. The definition of H0 (K) is not combinatorial, but because of its connections with these invariants, the theory is known to carry a good deal of geometric information about the knot K. The interplay between the two apparently different sorts of knot homologies (N > 0 and N = 0) has enhanced our understanding of both sides. This article will mostly focus on the cases N = 0 and N = 2, which are the oldest and best-studied examples of knot homologies and are related to the two best-known specializations of the HOMFLY polynomial – the Alexander and Jones polynomials. We have chosen to use a uniform notation to emphasize the similarities between theories, but the reader should be aware that other notation is more common in the literature. H0 is often referred to as the knot Floer homology (written HFK), and is usually normalized with a polynomial grading of j0 = j=2, corresponding to the substitution t = q2 , which gives the standard normalization of the Alexander polynomial. H2 is generally called the reduced Khovanov homology, and often denoted by Khr or Khred .

Construction Seen from a distance, all knot homologies are defined in much the same way. Given a knot K, we must first choose some additional data D which give a concrete geometric presentation of the knot. Using this data, we write down a bigraded chain i, j complex (CN (D), dN ). This complex depends on our initial choice of D, but when we take i, j homology, we are left with groups HN (K) which are invariants of the knot K (cf. the simplicial homology of a topological space X, where the chain groups depend on the choice of some initial geometric data – a triangulation of X – but the homology groups are invariants of X).

209

In all cases, the generators of CN (D) correspond naturally to terms which appear in a classical model for computing PN (K). In other words, we can write X ð1ÞiðÞ q jðÞ ½5 PN ðKÞ ¼ 2S

where the sum runs over a set of states S determined by D, and the functions i and j are also determined i, j by D. CN (D) is the free abelian group generated by { 2 Sji() = i, j() = j} and the differential dN is chosen to preserve the j-grading: j(dN x) = j(x). It follows that CN (D) decomposes into an infinite direct sum of complexes, one for each value of j, and [3] is a consequence of [5]. Beyond these global similarities, the definition of CN (D) varies with the value of N. In the second half of the article, we give explicit details of the constructions for N = 0 and N = 2.

Filtered Complexes and Deformations An important characteristic shared by all the CN ’s is the existence of deformations with homology Z. Recall that (CN (D), dN ) is a graded chain complex: j(dN x) = j(x). By a deformation of such a complex, 0 we mean a new chain complex (CN (D), dN þ dN ) in which the underlying group remains the same, but the differential has been perturbed by the addition of 0 a new term dN which strictly raises the j-grading: 0 j(dN (x)) > j(x). Any deformation of a graded complex is naturally a filtered complex, and as such, gives rise to a spectral sequence. The E0 term of this spectral sequence is the original unperturbed complex (CN (D), dN ), so the underlying group of the E1 term is just HN (K). Thus, it is independent of the choice of initial data D. In fact, it can often be shown that all terms in the spectral sequence beyond the first one are invariants of K. This is known to be the case for N = 0 and N = 2, and is most likely true for all other N as well (cf. the Leray–Serre sequence associated to a fibration, where the first two terms depend on a choice of geometric data but the E2 and higher terms are all invariants of the fibration). For each value of N, CN (D) admits a natural deformation whose homology is Z in homological grading 0, and zero in every other grading. When N = 0, 2, the filtration grading of this generator is known to be an invariant of K. (This is probably the case for N > 2 as well.) Equivalently, this is the j-grading of the surviving copy of Z in the spectral sequence. When N = 0, this invariant is conventionally normalized to be half the j-grading of the

210 Knot Homologies

generator, and is called (K). When N = 2, it is called s(K).

Geometric Properties Some elementary properties of the HN ’s generalize those of the HOMFLY polynomial. If K1 #K2 denotes the connected sum of K1 and K2 , then over Q HN ðK1 #K2 Þ ffi HN ðK1 Þ  HN ðK2 Þ

½6

 is the mirror image of K, and if K i;j  HN ðKÞ

ffi

i;j HN ðKÞ

½7

Moreover, H0 satisfies an additional symmetry H0i;j ðKÞ ffi H0ij;j ðKÞ

½8

generalizing the symmetry of the Alexander polynomial: P0 (q) = P0 (q1 ). (With integer coefficients, these equalities all hold at the chain level. The correct statements about the homology can be obtained from the Kunneth formula and universal coefficient theorem.) HN (K) also contains deeper information related to the genus of surfaces bounding K. If K is a knot in S3 , recall that g(K) – the Seifert genus of K – is the minimal genus of an orientable surface smoothly embedded in S3 and bounding K. If we view S3 as the boundary of the 4-ball B4 , we can define a second quantity g (K) – the slice genus – by relaxing the requirement that the surface be embedded in S3 and instead requiring it to be embedded in B4 . Both s(K) and (K) give lower bounds on the slice genus of K: jðKÞj  g ðKÞ

½9

jsðKÞj  2g ðKÞ

½10

These bounds are far from independent. In fact, in all known examples, s(K) = 2(K). It is an open problem to determine whether this is true for all knots. From [6], it follows that s and  are additive under connected sum. Thus, both invariants define homomorphisms from the concordance group of knots in S3 to Z. The inequalities in eqns [9] and [10] are not always sharp, but there is one case where equality is known to hold. This is when K is represented by a diagram with all positive crossings (or, more generally, K is quasipositive.) In this case, the slice genus is also equal to the Seifert genus, and all three are easily computed using Seifert’s algorithm. The proof of [10] depends on the fact that for N > 0, HN is functorial in the following sense.

If S S3 [0, 1] is a smoothly embedded, orientable cobordism between links L1 and L2 , then for each N > 0, there is an induced map SN : HN (L1 ) ! HN (L2 ). SN is a graded map: it preserves the homological grading, and lowers the j-grading by (N  1)(S). Under deformation, it becomes a filtered map which induces a rational isomorphism on the deformed homologies. H0 and Heegaard Floer Homology

The proof of [9] depends on the close connection between the knot Floer homology and the Heegaard Floer homology. Roughly speaking, the Heegaard Floer groups of 3-manifolds obtained by surgery on K are determined by the groups H0i, j (K) together with additional differentials obtained by relaxing the requirement that nz () = nw () = 0. The relation with the slice genus again arises by studying maps induced by cobordisms, but in this case, the relevant cobordism is the surgery cobordism between S3 and the 0-surgery on K. This connection also leads to another important property of H0 : it detects the Seifert genus. If we let M(K) be the largest value of j for which the group H0, j (K) is nontrivial, then MðKÞ ¼ 2gðKÞ

½11

This fact generalizes a well-known inequality involving the degree of the Alexander polynomial: if m(K) is the largest power of q appearing in P0 (K), then m(K)  2g(K).

Computations The difficulty of computing HN (K) varies with the value of N. When N = 1, the theory is essentially 0 trivial: H 0, 1 (K) ffi Z for any knot K, and all other groups vanish. Of the remaining knot homologies, H2 (K) is the easiest to compute. The theory for alternating knots was worked out by E S Lee, and extensive calculations have also been made for nonalternating knots using computer programs written by Bar-Natan and Shumakovitch. Computing H0 is more difficult, on account of the noncombinatorial nature of d0 . Three families of knots for which H0 is well understood are alternating knots, (1,1) knots (described in the next section), and knots which admit lens space surgeries. Beyond this, there is an array of techniques which may or may not work in any given case. The best of these is probably a setup introduced by Ozsva´th and Szabo´, in which the generators of C0 (D) correspond to states in the Kauffman state model of the Alexander polynomial. Combining this method with the known

Knot Homologies

results for alternating knots and (1,1) knots gives a fairly good understanding of H0 (K) for knots with 10 or fewer crossings; for larger knots, relatively little is known. Few computations of HN for N > 2 have been made, although the definition in this case is purely combinatorial.

211

β1

α1 Figure 1 Heegaard splitting of S 3 corresponding to the standard decomposition of S 3 into two solid tori.

Thin and Thick Knots

For simple knots, both H0 and H2 are thin. This means that there exists a constant cN (K)(N = 0, 2) i, j such that H N (K) is trivial unless j  2i = cN (K). In such cases, we necessarily have c0 (K) = 2(K) (resp. c2 (K) = s(K)), and HN (K) is completely determined by cN (K) and PN (K). The relationship is best expressed in terms of the Poincare´ polynomial of HN (K): X i;j P N ðKÞ ¼ ti qj dim HN ðKÞ i;j

¼ ðtÞcN ðKÞ=2 PN ðKÞðqðtÞ1=2 Þ

½12

If K is an alternating knot, both H0 (K) and H2 (K) are thin, and c0 (K) = c2 (K) = (K). (Note that in this case the bound on g (K) coming from  and s coincides with the classical bound coming from the signature.) Many nonalternating knots are thin as well; in all examples in which both groups have been computed, either both H0 (K) and H2 (K) are thin, or neither is. In addition, all such knots appear to have c0 (K) = c2 (K) = (K). Those knots whose homologies are not thin are called thick. There are a dozen such knots with ten or fewer crossings: using the standard numbering in the knot tables (see, e.g., Rolfsen (1976)) these are 819 , 942 , 10124 , 10128 , 10132 , 10136 , 10139 ,10145 , 10152 , 10153 , 10154 , and 10161 . It is a curious and as yet unexplained coincidence that, for all of these knots, the ranks of H0 (K) and H2 (K) are equal. There is an analogous notion of thinness when N > 2, but there exist alternating knots for which HN cannot be thin for N  0 (this can be seen from the HOMFLY polynomials).

Construction of H0 We now turn to a more detailed description of the definition of H0 (K). The geometric data D used to define C0 is a Heegaard diagram for the complement of K. One convenient way to specify such a diagram is by a doubly pointed Heegaard diagram of S3 . The data for such a diagram consist of a surface  of genus g, two g-tuples of attaching circles {1 , . . . , g } and {1 , . . . , g } on , and two points z, w 2  which are disjoint from all the ’s and ’s. Each set

of attaching circles is composed of g disjoint simple closed curves, arranged so that when g is cut along them the result is a sphere with 2g holes. Any such set of attaching circles determines a unique genus-g handlebody H with boundary  and the property that each attaching circle bounds a disk in H. The choice of  and  curves determines the underlying 3-manifold in which the knot is embedded. Starting with  [0, 1], we fill in one component of the boundary with the handlebody determined by the -curves, and the other component with the handlebody determined by the -curves to obtain a closed 3-manifold. By hypothesis, this manifold is required to be S3 . A simple Heegaard diagram of S3 with g = 1 is shown in Figure 1. To go from a doubly pointed Heegaard diagram to a diagram of the knot complement, we remove neighborhoods of z and w and replace them with a tube to get a surface 0 of genus g þ 1. We also add an additional -handle gþ1 , which runs from z to w in  in such a way that it does not intersect the other ’s, and then comes back over the tube. This process is illustrated in Figure 2. A Heegaard diagram of S3 K determines a presentation of 1 (S3  K) with one generator xi for each -circle and one relator wj for each -circle. To find the relator wj , one travels along j , recording each intersection with some i by appending x 1 i to the relator. The sign is determined by the sign of the intersection. As an example, consider the two doubly pointed diagrams of Figure 3, both of which correspond to the same Heegaard diagram of S3 . (It is isotopic to the one shown in Figure 1.) The fundamental groups of the associated knot complements can be read off from the corresponding genus2 Heegaard splittings. Starting from the point where

α1

z

α1

α2

w Figure 2 Going from a doubly pointed diagram to a Heegaard diagram of the knot complement.

212 Knot Homologies

which is the Alexander polynomial of the unknot. If we abelianize the relator in the second presentation, we see that jx1 j = jx2 j = q2 , so

z

w p1

β1

φ2 p3

p1

β1

φ1 p2

w p3

α1

  1 1 dx1 x2 x1 x1 2 x1 x2 x1     1  1 1  ¼ jx2 j  x2 x1 x1 þ x2 x1 x1 2 x1 2 x1 x2

z p2

α1

(a)

¼ q2  1 þ q2

(b)

Figure 3 Doubly pointed Heegaard diagrams for the unknot and the trefoil. Opposite sides of the square are identified to form a torus. The dotted line represents 2 :

1 intersects the left-hand side of the square and moving to the right, we get 1 ðS3  K1 Þ ¼ hx1 ; x2 jx1 x1 1 x1 ¼ 1i 1 1 1 ðS3  K2 Þ ¼ hx1 ; x2 jx2 x1 x1 2 x1 x2 x1 ¼ 1i

The first group is isomorphic to Z, and the knot in Figure 3a is the unknot. The second is isomorphic to 1 of the complement of the trefoil knot, and in fact the knot in Figure 3b is the left-handed trefoil. The definition of C0 (D) is based on a classical method for computing the Alexander polynomial known as the Fox calculus, which takes as its input a presentation of 1 (S3  K). According to Fox calculus, P0 ðKÞ ¼ qn detðdxi wj Þ1i;jg

½13

Here dxi wj is an element of the group ring Z½H1 ðS3  KÞ ffi Z½q 2  It is determined by the following rules: dxi xj ¼ ij

d0 x ¼

i

i

½15 ½16

i

where j j : 1 ðS3  KÞ ! H1 ðS3  KÞ ffi Z ¼ hq2 i

½17

is the abelianization map. The factor of qn is chosen so that P0 (K)(1) = 1 and P0 (K)(q) = P0 (K) (q1 ). As an example, consider the two presentations above. In the first presentation, j j sends x1 to 1 and x2 to q2 , so       dx x1 x1 x1 ¼ 1  x1 x1  þ x1 x1  1

1

1

1

¼11þ1 ¼1

½18

½20

which is the Alexander polynomial of the trefoil. When g = 1, the complex C0 (D) is generated by the points of 1 \ 1 . These intersection points may be naturally identified with the appearances of the generator x1 in w1 , and thus with the monomials appearing in dx1 w1 . For example, the three monomials which appear on the right-hand sides of eqns [18] and [19] correspond, respectively, to the points labeled p1 , p2 , and p3 in Figure 3. The j-grading of each generator is given by the exponent of q which the corresponding monomial contributes to the Alexander polynomial. Thus, all three generators in Figure 3a have j-grading 0, while in Figure 3b, the generators p1 , p2 , and p3 have j-gradings 2, 0, and 2 respectively. For general g, the monomials appearing in the determinant of eqn [13] correspond to intersection points of the two totally real tori  = 1 g and  = 1 g inside the symmetric product Symg . The knot Floer homology is the Lagrangian Floer homology of  and  inside the symplectic manifold Symg (  z  w). The generators of C0 (D) are the points of  \ ; the differential is defined by counting holomorphic disks with boundary on  and . To be precise, for x 2  \ ,

½14

dxi ab ¼ dxi a þ jajdxi b   dx x1 ¼ x1 

½19

X

#MðÞy

½21

22 ðx;yÞ; ðÞ¼1 nz ðÞ¼nw ðÞ¼0

Here 2 (x, y) denotes the set of homotopy classes of maps of the strip D = {a þ ib j b 2 [0, 1]} into Symg  which take the right-hand boundary to  and the left-hand boundary to , and which limit to x as b !1 and to y as b ! 1. () denotes the formal dimension of the space of pseudoholomorphic disks in this homotopy class. There is a natural action by translation on the space of such maps, so when

() = 1 we can divide out by this action and obtain an oriented zero-dimensional moduli space M(). Finally, by nz () and nw () we denote the intersection number of such a strip with the divisors determined by z and w inside of Symg . The requirement that they vanish forces the strip to lie

Knot Homologies

in Symg (  z  w). It can be shown that, for  2 2 (x, y), jðxÞ  jðyÞ ¼ nz ðÞ  nw ðÞ

½22

so j(d0 x) = j(x). When g = 1, computing the differential amounts to counting maps of the strip into the Heegaard torus. This can be done algorithmically using the Riemann mapping theorem, so computation of H0 is purely combinatorial. Knots of this form are called (1,1) knots. They are one of our few windows into the behavior of H0 for large knots. As an example, consider the diagram of Figure 3a. The two shaded regions represent the domains of classes 1 2 2 (p1 , p2 ) and 3 2 2 (p3 , p2 ). The Riemann mapping theorem implies that up to reparametrization, there is a unique holomorphic map of the strip into each region, so #M(1 ) = 1 = #M(2 ). The differential in C0 (D1 ) is given by d0 ðp1 Þ ¼ p2 ¼ d0 ðp3 Þ d0 ðp2 Þ ¼ 0 and H0 (U) ffi Z. This reflects the fact that we could have chosen the more efficient diagram of S3  U shown in Figure 1, simply by moving 1 to remove two of the intersection points. For comparison, consider the diagram for the trefoil shown in Figure 3b. All three generators of C0 (D2 ) have different j-gradings, so we must have d0  0. Thus, H0 (T) ffi Z3 . The two disks 1 and 2 are still present, but now nz (1 ) = nw (2 ) = 1, so neither disk contributes to the differential. This is reflected in the fact that 1 cannot be moved to reduce the number of intersection points without passing through either z or w. Deformations

In this case, finding an appropriate deformation of C0 (D) is simple: we just drop the condition that nz () = 0 in the definition of the differential. If a homotopy class  2 2 (x, y) contributes nontrivially to the sum, it must have a holomorphic representative, which necessarily intersects the divisor in Symg  defined by z non-negatively. Thus, nz ()  0. From [22], it follows that j(x)  j(y) = nz ()  0, so this new differential has the form d0 þ d00 , where d00 strictly lowers the j-grading. The fact that the homology of C0 (D) with respect to the perturbed differential is Z goes back to the knot Floer homology’s roots in Heegaard Floer homology. By dropping the condition that nz () = 0, we have effectively forgotten about the basepoint z, and thus about the knot. The new

213

complex simply computes the Heegaard Floer group c 3 ), which is isomorphic to Z. When g = 1, this HF(S can be seen directly: if we remove the basepoint z, any genus-1 Heegaard diagram of S3 can be isotoped into the standard diagram of Figure 1.

Construction of H2 In this case, the geometric data D needed to define the chain complex C2 (D) is a planar diagram of the knot, and the classical model on which the construction of C2 (D) is based is the Kauffman state model for the Jones polynomial. There is a related ~ 2 (D), known as the unreduced homology theory H Khovanov homology, whose graded Euler characteristic is (q þ q1 )P2 (K). This is the original categorification of the Jones polynomial defined in Khovanov (2000). ~ 2 (D), we consider complete resoluTo construct C tions of the planar diagram D. As shown in Figure 4, there are two different ways to resolve each crossing of D. If D has n crossings, there will be 2n ways to resolve all n, one for each vertex of the cube [0, 1]n . To a vertex v, we associate the crossingless planar diagram Dv obtained from the corresponding resolution of D. Thus, each vertex of the cube is decorated by a 1-manifold Dv . If e is an edge joining vertices v0 and v1 (where v0 has one more 0 coordinate than v1 ), we write e : v0 ! v1 , and decorate e with a two-dimensional cobordism Se from Dv0 to Dv1 . Se is a product cobordism outside a neighborhood of a single crossing, where it is the one-handle cobordism between the 0-resolution and the 1-resolution. The resulting cobordism is necessarily composed of a union of product cobordisms (cylinders) together with a single nontrivial cobordism (a pair of pants). Thus, starting from D, we have constructed an n-dimensional cube whose vertices are decorated by 1-manifolds and whose edges are decorated by cobordisms between them. This is the cube of resolutions of D. ~ 2 (D) is to The next step in the construction of C apply a graded (1 þ 1)-dimensional TQFT A to the cube of resolutions. A is a functor which associates to each 1-manifold X a group A(X), and to each two-dimensional cobordism W : X1 ! X2 a homomorphism A(W) : A(X1 ) ! A(X2 ). If we apply A to all the manifolds and cobordisms of the cube of

0

1

Figure 4 0- and 1-resolutions of a crossing.

214 Knot Homologies

mð1  1Þ ¼ 1

Table 1 Summary of cube of resolutions Vertex v Edge e : v1 ! v2

ð1Þ ¼ 1  X þ X  1

! 1-manifold Dv

!

Group A(Dv )

Cobordism Se : Dv1 ! Dv2

!

Homomorphism A(Se ) : A(Dv1 ) ! A(Dv2 )

!

resolutions, we obtain a new cube, decorated with groups and cobordisms between them. This process is summarized in Table 1. ~ 2 (D). We can now describe the chain complex C As a group, M ~ 2 ðDÞ ¼ C AðDv Þ ½23

mð1  XÞ ¼ mðX  1Þ ¼ X ðXÞ ¼ X  X

½28

mðX  XÞ ¼ 0

½29

Note that the multiplication m makes A into a commutative ring isomorphic to Z[X]=(X2 ). A is a graded TQFT. In other words, there is a grading q on A and its tensor products, determined by qð1Þ ¼ 1 qða  bÞ ¼ qðaÞ þ qðbÞ

v

where the sum runs over all vertices of the cube of resolutions. For x 2 A(Dv ), the differential is given by X d2 x ¼ ð1ÞsðeÞ AðSe ÞðxÞ ½24

AðS1 Þ ¼ A ¼ h1; Xi

½25

General principles then imply that A

n a

 S1 ¼ An

qðmða  bÞÞ ¼ qða  bÞ  1

If we define j(x) = k(D) þ q(x) þ i(x), it follows that j(d2 x) = j(x). Taking the graded Euler characteristic gives X ~ 2 ðDÞÞ ¼ qkðDÞ ðC ðqÞiðvÞ ðq þ q1 Þnv ½33 v

where nv is the number of components of Dv . If we define k(D) to be the writhe of D, this is precisely Kauffman’s formula for the unnormalized Jones polynomial. ~ 2 (D) for a simple twoFigure 6 illustrates C crossing link. The figure shows the original link (in the center), the cube of resolutions, and basis vectors ~ 2 (D), together with their j-gradings. We leave it for C ~ 2 (L) is to the reader to check that the homology H four dimensional, supported in j-gradings 1 and 3 at the vertex labeled 00, and in gradings 5 and 7 at the vertex labeled 11.

½26 01 j=5 j=3

j=3

Δ:A

Figure 5 Maps induced by pairs of pants.

½32

qððaÞÞ ¼ qðaÞ  1

j=5

A

½31

From eqns [27]–[29], it is easy to see that

To specify the maps induced by cobordisms, it is enough to describe the maps associated to the two pairs of pants shown in Figure 5. They are given by

m:A A

½30

qðXÞ ¼ 1

e:v!v0

The signs in this sum are determined by assigning a sign (1)s(e) to each edge e in such a way that every two-dimensional face of the cube has an odd number of  signs on its edges. (This ensures that d2 = 0.) There are many ways to do this, but they all result in isomorphic complexes. ~ 2 (D) is easily The homological grading i on C determined. For x 2 A(Dv ), we set i(x) = i(v)  c(D), where i(v) is the sum of all the coordinates of v, and c(D) is a constant. Clearly, i(d2 x) = i(x) þ 1. In order to have invariance, it turns out that c(D) must be chosen to be equal to the number of negative crossings in D. It remains to specify the TQFT A. At the level of groups, A(S1 ) is a free abelian group of rank 2:

½27

A A

j=1

11 j=7 1 1 1 X X 1 j=5 X X j=3

1 X

1 1 1 X X 1 X X

1 X 00

10

Figure 6 The cube of resolutions for the Hopf link.

j=5 j=3

Knot Invariants and Quantum Gravity

To get the reduced chain complex C2 (D), we must divide the graded Euler characteristic by a factor of (q þ q1 ). This is accomplished by choosing a marked point on K and requiring that for each resolution Dv , the vector associated to the circle containing the marked point lie in the subspace of A spanned by X. If D is a diagram of a knot, the resulting homology H2 (K) is independent of the choice of marked point. For links, H2 (L) depends on the component of the link on which the marked point lies.

Deformations in the N = 2 theory are constructed using a technique introduced by E S Lee. The idea is to replace the graded TQFT A with a filtered TQFT A0 . As a group, we still have A(S1 ) = A, but the multiplication and comultiplication maps are perturbations of those for A: m0 ð1  1Þ ¼ 1 ½34

m0 ð1  XÞ ¼ m0 ðX  1Þ ¼ X 0 ðXÞ ¼ X  X þ s1  1 m0 ðX  XÞ ¼ rX þ s

polynomial X2  rX  s has simple roots, the TQFT A0 decomposes as a direct sum of two onedimensional TQFTs. This implies that for a knot, ~ 0 (K) decomposes as a the deformed homology H 2 direct sum of two copies of H1 (K). This group is ~ 0 (K) ffi Z  Z. If s = 0, always isomorphic to Z, so H 2 the same strategy can be used to define deformations of the reduced chain complex C2 (D). In this case, we find that the deformed homology is isomorphic to a single copy of Z. See also: Floer Homology; Gauge Theory: Mathematical Applications; The Jones Polynomial; Knot Theory and Physics; Topological Quantum Field Theory: Overview.

Deformations

0 ð1Þ ¼ 1  X þ X  1  r1  1

215

½35 ½36

The new terms involving r and s have q gradings strictly greater than the terms which are shared with eqns [27]–[29]. Thus, the differential defined by replacing m and  by m0 and 0 will be a ~ 2 (D). perturbation of the original differential on C The simplicity of the homology with respect to the new differential depends on the fact that when the

Further Reading Bar-Natan D (2002) On Khovanov’s categorification of the Jones polynomial. Algebraic and Geometric Topology 2: 337–370. Crowell R and Fox R (1963) Introduction to Knot Theory. Boston: Ginn and Co. Kauffman L (1987) State models and the Jones polynomial. Topology 26: 395–407. Khovanov M (2000) A categorification of the Jones polynomial. Duke Mathematical Journal 101: 359–426. Ozsva´th P and Szabo´ Z (2004) Heegaard diagrams and holomorphic disks. In: Donaldson S, Eliashberg Y, and Gromov M (eds.) Different Faces of Geometry, pp. 301–348. New York: Kluwer/Plenum. Ozsva´th P and Szabo´ Z (2004) Holomorphic disks and topological invariants for closed three-manifolds. Annals of Mathematics 159: 1027–1158. Ozsva´th P and Szabo´ Z (2004) Holomorphic disks and knot invariants. Advances in Mathematics 186: 58–116. Rolfsen D (1976) Knots and Links, Mathematics Lecture Series, No. 7. Berkeley: Publish or Perish. Viro O (2004) Khovanov homology, its definitions and ramifications. Fundamentals of Mathematics 184: 317–342.

Knot Invariants and Quantum Gravity R Gambini, Universidad de la Repu´blica, Montevideo, Uruguay J Pullin, Louisiana State University, Baton Rouge, LA, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction As in all other physical theories, one expects that gravitational phenomena will ultimately be ruled by quantum mechanics. This requires to consider the quantization of the best available theory of gravity, namely Einstein’s general relativity. This problem has been considered since the 1930s (see Loop Quantum

Gravity). The application of the rules of quantum mechanics to general relativity is immediately problematic. Unlike other physical interactions, general relativity describes gravitational phenomena through a distortion of spacetime rather than through a field living in spacetime. Therefore, its quantization is bound to be very different from that of other physical theories. In particular, the well-established framework of perturbative quantum field theory, used with remarkable success in describing electroweak and strong interactions (in the latter case at least in certain regimes), runs into trouble when applied to general relativity. At present, it is not clear if this is a fundamental problem or if there might exist an implementation of perturbative quantum field theory that works well in the gravitational case. On the

216 Knot Invariants and Quantum Gravity

other hand, there exist examples of field theories where perturbative methods fail but that nevertheless can be quantized. This suggests that the consideration of nonperturbative techniques in the quantization of the gravitational field could be a promising avenue. In particular, canonical quantization methods appear attractive for attempting a nonperturbative quantization of gravity. Canonical methods force the introduction, in a clear way, of a Hilbert space of states and definition of the quantum operators of interest. The application of canonical methods to classical general relativity was pioneered by Dirac and Bergmann in the late 1950s. During the 1960s, the resulting canonical theories were considered in a quantum setting by DeWitt. At the time it appeared that making progress in the canonical quantization of general relativity was going to be quite a challenge. In particular, the canonical theory has constraints, which have to be implemented as operator identities quantum mechanically. The wave functions were functionals of the spatial metric of spacetime. One of the operator identities to be satisfied implies that the wave functions only depend on properties of the spatial metric that are invariant under spatial diffeomorphisms. This is a direct consequence of general relativity being a theory that is independent of coordinate choice since a diffeomorphism changes the assignment of coordinates to points in the manifold. Finding such wave functions already presented a challenge, since there is no well-grounded mathematical theory of functionals of diffeomorphism-invariant classes of metrics. Moreover, the other operator identity to be imposed, known as the Hamiltonian constraint or Wheeler–DeWitt equation, was a nonpolynomial complicated operator equation that does not admit a simple geometrical interpretation and needs to be regularized. Since one does not have a background metric to rely upon, traditional regularization techniques of quantum field theory are not suitable to deal with the Hamiltonian constraint. These difficulties severely hampered development of canonical methods for the quantization of general relativity for approximately two decades. The situation started to change when Ashtekar noticed that one could choose a different set of variables to describe general relativity canonically. Instead of using as variable the spatial metric qab , Ashtekar ~ a. chooses to use a set of (densitized) frame fields E i The relationship between the metric and the ~ aE ~b densitized frames is det (qab )qab = E i i and we are assuming the Einstein summation convention, that is, the index i is summed from 1 to 3 (such an index labels which vector in the triad one is referring to). The resulting theory has an additional symmetry

with respect to usual general relativity, in the sense that it is invariant under the choice of frame. This symmetry operates on the index i as if it were an SO(3) symmetry. As canonical momenta the usual choice is to pick the extrinsic curvature of the 3-geometry. Ashtekar chooses a variable related to it that behaves under frame transformations as an SO(3) connection, Aia . The resulting theory is there~ a , Ai ), with i fore cast in terms of a canonical pair (E a i an SO(3) index. One can therefore consider the canonical pair as that of a Yang–Mills theory associated with the SO(3) group. In fact, associated with the extra symmetry under triad rotations the theory has a new set of constraints that take ~ a = 0 with Da the the form of a Gauss law, Da E i covariant derivative formed with the connection Aia . This allows us to view the phase space of a Yang– Mills theory as the kinematical arena on which to discuss quantum gravity. The theory is of course different from the Yang–Mills theory. In particular, it still has constraints that imply that it is invariant under spacetime diffeomorphisms. In the canonical picture, these constraints appear asymmetrically as one constraint is associated with time evolution (‘‘Hamiltonian constraint’’) and a set of three constraints is associated with spatial diffeomorphisms (‘‘diffeomorphism constraint’’). If one quantizes the theory starting from the Ashtekar formulation, given the resemblance with Yang–Mills theory, the natural choice for a representation of the quantum wave functions is to consider wave functions of the connection [A] that are invariant under SO(3) transformations. Such a representation is known as ‘‘connection representation.’’ There is significant experience in Yang–Mills theory in constructing such wave functions. In particular, it is known that if one considers the parallel transport operator defined by a connection around a closed curve (holonomy) and one takes its trace (‘‘Wilson loop’’), the resulting object is invariant under SO(3) transformations. What is more important, the set of traces of holonomies along all possible closed loops is an overcomplete basis for all gauge-invariant functions. More recently, it has been shown that one can construct a less redundant complete basis using techniques from spin networks. We will discuss later on how to do this. Since any gauge-invariant functional can be expanded in the basis of Wilson loops, one can choose to represent it through the coefficients of such an expansion. These coefficients are functions of the curve upon which the corresponding element of the basis of Wilson loops is based. The representation of wave functions in terms of such coefficients is called ‘‘loop representation.’’ Wave

Knot Invariants and Quantum Gravity

functions in the loop representation are functions of a closed curve (more precisely of families of closed curves, or spin networks, as we will discuss below). We still have to deal with the diffeomorphism and Hamiltonian constraints. The diffeomorphism constraint when written in the loop representation implies that the wave functions are not functions of loops but rather of topologically invariant properties of the loops under general diffeomorphisms of the spatial manifold containing the loops. Such functions are technically known in the mathematical literature as ‘‘knot invariants.’’ This is the first point of connection between knot invariants and quantum gravity; they constitute the kinematical arena of the theory. One still has to deal with the Hamiltonian constraint, which has to be imposed as an operator equation. We shall see that knot theory also seems to have a lot to say about solutions of the Hamiltonian constraint. This is quite remarkable, since the Hamiltonian constraint embodies in detail the specific dynamics of Einstein’s theory of gravitation, and to our knowledge this is an input that has never gone into the ideas of knot theory. In terms of the Ashtekar variables, the Hamiltonian constraint takes the form H ¼ Ea  Eb  ðBc þ Ec Þabc

regularization was chosen. Classically, the condition Eai Bai is satisfied for the de Sitter geometry, so one could envision the state as a quantum state associated with such geometry. The exact solution of the above equation is given by a state that is the exponential of the integral on the spatial slice of the Chern–Simons form built from the connection  Z CS ½A ¼ exp k d3 x trðA ^ dA  2 þ A^A^A ½3 3 and the constant k needs to be chosen as k = 6= for the state to be a solution. One can ask, ‘‘what is the expression of this state in the loop representation?’’ To answer this, one needs to compute the coefficients of its expansion in the basis of Wilson loops W [A], where as we stated earlier,  should be a collection of (intersecting) loops (later we will discuss the generalization to spin networks). The expression for the coefficients will be a function only of the loops  and is given by Z CS ½ ¼ DAW ½ACS ½A ½4

½1

where we have used a conventional vector notation for the frame indices and kept explicit the spatial indices. abc is the Levi-Civita totally antisymmetric tensor. We have included a possible cosmological constant . The Ashtekar formulation can be constructed in different ways. In the original formulation, the connection Aia was a complex variable and the Hamiltonian took the form we listed above. However, the resulting theory was only equivalent to real general relativity if the variables satisfied certain reality conditions. One can choose to use a real connection instead, but then the Hamiltonian constraint has additional terms. At the moment, we will concentrate on the constraint as listed above. The constraint has to be implemented as a quantum operator acting on wave functions. Since it involves the product of operators, it needs to be regularized. Most regularization methods are problematic in this context, since they use a metric, and here the metric is a quantum operator, not an external fixed quantity. If we ignore these difficulties, one observes that, if one were to choose a quantum state, for instance in the connection representation, for which, ^ a ½A ¼ B ^ a ½A E i i

217

½2

the state would be annihilated by the Hamiltonian constraint, and this would be true no matter what

This expression is invariant under diffeomorphisms of the manifold or, equivalently, under smooth deformations of the curve . That is, it is what in the mathematical literature is called ‘‘knot invariant.’’ In fact, this integral has been studied by Witten in the context of Chern–Simons theory and has been shown to be related to the Kauffman bracket knot polynomial, which in turn is related to the celebrated Jones polynomial. Therefore, the implication of these results is that the Kauffman bracket knot polynomial appears to be the representation in the loop representation of a state of quantum gravity that solves the quantum Einstein equations (with a cosmological constant). The reader may be intrigued by the word ‘‘polynomial’’ in this context. It should be noted that the Chern–Simons state CS [A] depended on a parameter k, which had to take a certain value for it to solve the quantum Einstein equations. The resulting knot invariant is a polynomial in exp (k). If one expands out the result, an infinite power series in k results. There will be infinite coefficients in the series, but they are just combination of the finite number of coefficients of the polynomial. Knot polynomials are a powerful tool for analyzing and distinguishing knots. The coefficients of the polynomials are all knot invariants. Typically, for ‘‘simple’’ knots, the first few coefficients of the knot polynomial are nonzero. As one considers more complicated knottings, higher

218 Knot Invariants and Quantum Gravity

coefficients become nonvanishing. The ultimate goal of knot theory is to be able to consider two arbitrary knots and to unambiguously determine if the two knots are related by a smooth transformation. The knot polynomials appear as promising tools for achieving this task that has remained elusive up to now. Returning to quantum gravity, to have a well-known knot polynomial as a solution of the quantum Einstein equations is a remarkable fact. The first connection we outlined between knot theory and quantum gravity was less unexpected: if one describes a theory that is diffeomorphism invariant in terms of loops, the appearance of knots is inevitable. But we are now finding that knot invariants from the mathematical literature, which were constructed without any knowledge of the details of the dynamics of the Einstein equations, seem to manage to solve such equations. This is either a big coincidence or a pointer to some unexplained deep connection yet to be understood. Notice, for instance, that other theories of gravity would not have the Kauffman bracket as a quantum state. There is a certain technicality about the Kauffman bracket that makes it difficult to argue with precision that it is a state of quantum gravity. To understand this technicality better, it is perhaps best to concentrate on the form of the quantum state written above if the connection is an abelian connection. In that case, the integral in question, Z I CS abelian ½ ¼ DA dya expðiAa Þ  Z   exp d3 xabc Aa @b Ac ½5 by turning it into a Gaussian integral. The result is I I ðx  yÞc ½6 CS abelian ½ ¼ dxa dyb abc jx  yj   This integral has problems, since the integrand is ill-defined when x = y. Notice that the integral would be well defined if the two contour integrals were evaluated on different, nonintersecting curves. The result would be the well-known formula for the Gauss linking number of the curves, yielding zero if they are not linked and and integer multiple of 4 if they were. So the integral we were trying to compute was actually the Gauss linking number of the curve with itself. Such a quantity is not well defined for ordinary curves. To deal with this problem, mathematicians introduced the concept of framed knots. A framed knot is a curve with a prescription to determine a second curve from it. One way to see it is to construct another curve that is ‘‘infinitesimally close’’ in space to the original

one. It is clear that there is no canonical way to compute such a second curve. Then, when one considers quantities like the self-linking number, one makes them well defined by evaluating the two integrals on the two curves, the original one and the one yielded by the prescription. In reality, the notion of framing is a bit more elaborate than what we hint at here, since one could consider invariants constructed with more than two integrals and could still be ill-defined if one only considers two curves. The notion has to be extended as well to handle intersections in the curves. We will ignore these subtleties in this discussion. The Kauffman bracket knot invariant is an invariant of framed knots, just like the self-linking number. It is not well defined for a single curve. It requires a framing of the knot. In quantum gravity, there is no compelling reason to consider framed curves. It is true that framed curves arise naturally in q-deformed field theories and perhaps a q-deformed version of quantum gravity is what needs to be considered to accommodate the Chern–Simons state, but at the moment there are no proposals along these lines that have widespread consensus. So, it appears the Kauffman bracket does not have a natural role to play as a state of quantum gravity. However, it is known that the frame dependence of the Kauffman bracket knot polynomial can be captured in an overall factor that depends on the self-linking number. If one strips the polynomial of this factor, one gets the Jones polynomial, which is a knot invariant of single curves. Could it be that this polynomial has a chance of being a solution of the quantum Einstein equations? To determine this, the analogy with Chern– Simons theory is no longer useful, since there is no straightforward way to transform the relation between the Kauffman and Jones polynomials into relations between states in the connection representation. To analyze if the Jones polynomial could be a solution of the quantum Einstein equations, one needs to write the quantum Einstein equations directly in terms of loops. There have been several attempts to rewrite the quantum Einstein equations directly in the loop representation. In one of these attempts, the curvature that appears in the Hamiltonian constraint was represented by the ‘‘loop derivative.’’ This is a differential operator that can be introduced in the space of loops by considering that two loops that differ by a small element of area are ‘‘close.’’ One can build an attractive differential calculus in loop space that actually encodes many of the kinematical properties that are useful to formulate Yang–Mills theory.

Knot Invariants and Quantum Gravity

The Hamiltonian constraint in terms of the loop derivative is an operator that has an explicit form. The coefficients of the Jones polynomial can also be given an explicit form by computing perturbatively the integral in the Chern–Simons theory. The results are generalizations of the types of integrals that arise in the self-linking number, but involving a larger number of integrals. One can therefore envisage carrying out an explicit computation in which one checks if the coefficients of the Jones polynomial are annihilated or not by the Hamiltonian constraint of quantum gravity in the loop representation. Such a calculation has been carried out for the first few coefficients. It turns out that the second coefficient (the first coefficient is normalized to unity, so it trivially satisfies the constraint) is indeed annihilated by the Hamiltonian constraint of vacuum quantum gravity (with zero cosmological constant). It has been shown that the third coefficient is not, and there are good arguments to indicate that other coefficients will not be states of quantum gravity. So, a remarkable result has been found in that one of the coefficients of the Jones polynomial (related to the Arf and Casson invariants) is annihilated by a version of the quantum Hamiltonian constraint of general relativity. The result is quite nontrivial; it requires a fair amount of calculation to actually show that the coefficient is annihilated. The meaning of this quantum state and the deep reason why it is annihilated remain at present a mystery. The quantum Hamiltonian constraint based on the loop derivative makes certain assumptions about the space of functions one is using to quantize the theory. In quantum field theory, not all classical operators have a well-defined quantum counterpart. The choice being made is to assume that the curvature Fab is a well-defined quantum operator defined by the loop derivative. Differentiability of knot polynomials is not a new idea. It is the core idea of the Vassiliev knot invariants, which are defined by a set of identities, one of them acting as a ‘‘derivative in knot space.’’ It can be shown that the loop derivative is a concrete implementation of the Vassiliev derivative and, therefore, Vassiliev invariants are the ‘‘arena’’ in which this version of quantum gravity takes place. The Hamiltonian based on the loop derivative has problems, in the sense that it is obtained by a regularization procedure that requires extra external geometric structures. This is common practice in Yang–Mills theory, where one has at hand a fixed external background metric. However, in gravity the geometry is a dynamical object and, if one constructs expressions that resort to some fixed external geometry, one gets inconsistencies. In particular, it is expected that the Hamiltonian based on the loop

219

derivative will not reproduce the correct Poisson algebra of canonical general relativity. This sort of problem plagued early attempts to construct a quantum version of the Hamiltonian constraint in the early 1990s. A point that we mentioned earlier but did not elaborate upon, is that the Wilson loops constitute an overcomplete basis of states. Therefore, if one takes a quantum state and expands it on such a basis, one gets that the coefficients of the expansion satisfy certain identities, called the Mandelstam identities. These are nonlinear identities that states in the loop representation have to satisfy. These identities are very inconvenient at the time of constructing quantum states. The identities stem from the fact that if one chooses a matrix representation of the group of interest, the fact that one is in a given representation is indicated by certain identities the matrices satisfy. To break free from these constraints, one possibility is to consider multiple representations when constructing Wilson loops. To do this, one considers piecewise-continuous graphs with intersections (the nonintersecting case is a trivial subcase). Along the lines connecting the intersections one considers holonomies in a given representation for a given line. In the case of the group SU(2), which is the one of interest in quantum gravity, such representations are labeled by a (half-) integer. One then considers invariant tensors in the group to ‘‘tie the holonomies together’’ at intersections. The resulting object is a gauge-invariant object for a given connection based on a ‘‘spin network.’’ The latter is an embedded piecewise-continuous graph with an assignment of integers to each of its lines and an assignment of ‘‘intertwiners’’ at each intersection (if the intersections are trivalent or lower, one can choose canonical intertwiners and forget about them). One can then consider the ‘‘spin network representation’’ in which one expands gauge-invariant states in terms of the basis of Wilson nets. Knot polynomials for these types of graphs have been considered in the mathematical literature (‘‘polynomials of colored graphs’’). The construction with the Chern–Simons state can be repeated, and there exist suitable generalizations of the Kauffman bracket and Jones polynomials. The Hamiltonian based on the loop derivative can also be introduced in this context; again, its action is well defined on suitable generalizations of Vassiliev invariants for these kinds of graphs. This opens the possibility of encoding the quantum dynamics of general relativity as a combinatorial action in the space of Vassiliev invariants. An alternative Hamiltonian based on assuming that the holonomies and the volume operators are well defined quantum mechanically (but not the curvature) has been introduced that has the advantage of not

220 Knot Theory and Physics

requiring external structures for its regularization. In fact, it can be explicitly checked that it satisfies the correct Poisson algebra without anomalies at the quantum level. The exploration of the action of this Hamiltonian constraint on knot polynomials has not been carried out as systematically as for the one based on the loop derivative, but it has been explicitly shown that the first coefficient in the expansion of the Jones polynomial is annihilated by this Hamiltonian constraint. The first coefficient, written in terms of loops, was simply the numeral 1 and was automatically annihilated. In terms of spin network states, the first coefficient is the ‘‘chromatic evaluation’’ of the network (the result of computing the Wilson loop on a connection that is pure gauge). It is somewhat nontrivial to show that this quantity is actually annihilated by the Hamiltonian constraint in question. At the moment, the issue of what the correct Hamiltonian constraint is that describes a realistic and physically correct theory of quantum gravity is still open to debate. There are certain concerns that the action of the operators considered up to now is too simple to encompass the true dynamics of general relativity. Constructing a semiclassical theory that could confirm or deny the viability of the proposals is a complicated task, since one has to make contact with physics that is not diffeomorphism invariant in the context of a theory that is. Moreover, in canonical quantum gravity, there exists the ‘‘problem of time.’’ Since the Hamiltonian vanishes, the dynamics implied by it is trivial, and one has to disentangle the true dynamics by relational constructions among the variables of the theory. One then needs to compare the resulting predictions with classical general relativity. Whether the current proposals are viable and whether knot theory will play a role at a ‘‘kinematical

level’’ or it will actually play a key role in the detailed dynamics of quantum general relativity is yet to be seen. It is reassuring that in partial constructions, celebrated knot polynomials have appeared to have some knowledge of the dynamics of the Einstein equations. Quantum gravity being an unfinished symphony, we cannot entirely conclude how great an impact knot theory will have on it in the end. One can only note that beautiful mathematical results seem to tie in naturally with the partial constructions that have been carried out thus far. See also: BF Theories; Braided and Modular Tensor Categories; Finite-Type Invariants; Finite-type invariants of 3-Manifolds; The Jones Polynomial; Knot Theory and Physics; Loop Quantum Gravity; Mathematical Knot Theory; Quantum Dynamics in Loop Quantum Gravity; Quantum Geometry and its Applications; Yang–Baxter Equations.

Further Reading Ashtekar A (2005) Gravity and the quantum. New Journal of Physics 7: 200. Gambini R and Pullin J (1996) Loops, Knots, Gauge Theories and Quantum Gravity. Cambridge: Cambridge University Press. Rovelli C (2001) Notes for a brief history of quantum gravity. In: Ruffini R (ed.) Rome 2000, Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories, p. 742. Singapore: World Scientific. Rovelli C (2004) Quantum Gravity. Cambridge: Cambridge University Press. Smolin L (2001) Three Roads to Quantum Gravity. New York: Basic Books. Thiemann T, Introduction to Modern Canonical General Relativity. Cambridge University Press (arXiv.org:gr-qc/0110034) (to appear).

Knot Theory and Physics L H Kauffman, University of Illinois at Chicago, Chicago, IL, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction This article is an introduction to some of the relationships between knot theory and theoretical physics. Knots themselves are macroscopic physical phenomena in three-dimensional space, occurring in rope, vines, telephone cords, polymer chains, DNA, certain species of eel, and many other places in the natural and manmade world. The study of topological invariants of

knots leads to relationships with statistical mechanics and quantum physics. This is a remarkable and deep situation where the study of a certain (topological) aspects of the macroscopic world is entwined with theories developed for the subtleties of the microscopic world. The present article is an introduction to the mathematical side of these connections, with some hints and references to the related physics. We begin with a short introduction to knots, links, braids, and the bracket polynomial invariant of knots and links. The article then discusses Vassiliev invariants of knots and links, and how these invariants are naturally related to Lie algebras and to Witten’s gauge-theoretic approach. This part

Knot Theory and Physics

of the article is an introduction to how Vassiliev invariants in knot theory arise naturally in the context of Witten’s functional integral. The article is divided into several sections beyond the introduction. Section two is a quick introduction to the topology of knots and links. The third one discusses Vassiliev invariants and invariants of rigid vertex graphs. The fourth section introduces the basic formalism and shows how Witten’s functional integral is related directly to Vassiliev invariants. The fifth section discusses the loop transform and loop quantum gravity in this context. The final section is an introduction to topological quantum field theory, and to the use of these techniques in producing unitary representations of the braid group, a topic of intense interest in quantum information theory.

Knots, Braids, and Bracket Polynomial The purpose of this section is to give a quick introduction to the diagrammatic theory of knots, links, and braids. A knot is an embedding of a circle in three-dimensional space, taken up to ambient isotopy. That is, two knots are regarded as equivalent if one embedding can be obtained from the other through a continuous family of embeddings of circles in 3-space. A link is an embedding of a disjoint collection of circles, taken up to ambient isotopy. Figure 1 illustrates a diagram for a knot. The diagram is regarded both as a schematic picture of the knot, and as a plane graph with extra structure at the nodes (indicating how the curve of the knot passes over or under itself by standard pictorial conventions). Ambient isotopy is mathematically the same as the equivalence relation generated on diagrams by the Reidemeister moves. These moves are illustrated in Figure 2. Each move is performed on a local part of the diagram that is topologically identical to the part of the diagram illustrated in this figure (these figures are representative examples of the types of Reidemeister moves) without changing the rest of the diagram. The Reidemeister moves are useful in

Figure 1 A knot diagram.

221

I

II

III

Figure 2 The Reidemeister moves.

doing combinatorial topology with knots and links, notably in working out the behavior of knot invariants. A knot invariant is a function defined from knots and links to some other mathematical object (such as groups or polynomials or numbers) such that equivalent diagrams are mapped to equivalent objects (isomorphic groups, identical polynomials, identical numbers). Another significant structure related to knots and links is the Artin braid group. A braid is an embedding of a collection of strands that have their ends in two rows of points that are set one above the other with respect to a choice of vertical. The strands are not individually knotted and they are disjoint from one another. See Figures 3–5 for illustrations of braids and moves on braids. Braids can be multiplied by attaching the bottom row of one braid to the top row of the other braid. Taken up to ambient isotopy, fixing the endpoints, the braids form a group under this notion of multiplication. In Figure 3 we illustrate the form of the basic generators of the braid group, and the form of the relations among these generators. Figure 4 illustrates how to close a braid by attaching the top strands to the bottom strands by a collection of parallel arcs. A key theorem of Alexander states that every knot or link can be represented as a closed braid. Thus, the theory of braids is critical to the

s1

s2

s3

s1–1

Braid generators

=

s1–1s1 = 1

=

s1s2s1 = s2s1s2

=

s1s3 = s3s1

Figure 3 Braid generators.

222 Knot Theory and Physics

Hopf link

Trefoil knot

Figure-8 knot

where the small diagrams represent parts of larger diagrams that are identical except at the site indicated in the bracket. We take the convention that the letter chi, , denotes a crossing where the curved line is crossing over the straight segment. The barred letter denotes the switch of this crossing, where the curved line is undercrossing the straight segment. In computing the bracket, one finds the following behavior under Reidemeister move I:

Figure 4 Closing braids to form knots and links.

theory of knots and links. Figure 5 illustrates the famous Borrowmean rings (a link of three unknotted loops such that any two of the loops are unlinked) as the closure of a braid. We now discuss a significant example of an invariant of knots and links, the bracket polynomial. The bracket polynomial can be normalized to produce an invariant of all the Reidemeister moves. This normalized invariant is known as the Jones (1985) polynomial. The Jones polynomial was originally discovered by a different method than the one given here. The bracket polynomial, hKi = hKi(A), assigns to each unoriented link diagram K a Laurent polynomial in the variable A, such that 1. If K and K0 are regularly isotopic diagrams, then hKi = hK0 i. 2. If K q O denotes the disjoint union of K with an extra unknotted and unlinked component O (also called ‘‘loop’’ or ‘‘simple closed curve’’ or ‘‘Jordan curve’’), then

hi ¼ A3 h^i and hi ¼ A3 h^i where  denotes a curl of positive type as indicated in Figure 6, and  indicates a curl of negative type, as also seen in this figure. The type of a curl is the sign of the crossing when we orient it locally. Our convention of signs is also given in Figure 6. Note that the type of a curl does not depend on the orientation we choose. The small arcs on the righthand side of these formulas indicate the removal of the curl from the corresponding diagram. The bracket is invariant under regular isotopy and can be normalized to an invariant of ambient isotopy by the definition fK ðAÞ ¼ ðA3 ÞwðKÞ hKiðAÞ where we chose an orientation for K, and where w(K) is the sum of the crossing signs of the oriented link K. w(K) is called the writhe of K. The convention for crossing signs is shown in Figure 6. The State Summation

hK q Oi ¼ hKi where  ¼ A2  A2 3. hKi satisfies the following formulas: hi ¼ Ah  i þ A1 hÞði hi ¼ A1 h  i þ AhÞði

In order to obtain a closed formula for the bracket, we now describe it as a state summation. Let K be any unoriented link diagram. Define a state, S, of K to be a choice of smoothing for each crossing of K. There are two choices for smoothing a given crossing, and thus there are 2N states of a diagram with N crossings. In a state we label each smoothing with A or A1 as in the expansion formula for the bracket. The label is called a vertex weight of the

+

– +

+ or

: – b

CL(b)

Figure 5 Borromean rings as a braid closure.

: Figure 6 Crossing signs and curls.

– or

Knot Theory and Physics

state. There are two evaluations related to a state. The first one is the product of the vertex weights, denoted hKjSi. The second evaluation is the number of loops in the state S, denoted kSk. Define the state summation, hKi, by the formula X hKi ¼ hKjSikSk1 S

It follows from this definition that hKi satisfies the equations hi ¼ Ah  i þ A1 hÞði hK q Oi ¼ hKi hOi ¼ 1 The first equation expresses the fact that the entire set of states of a given diagram is the union, with respect to a given crossing, of those states with an A-type smoothing and those with an A1 -type smoothing at that crossing. The second and the third equation are clear from the formula defining the state summation. Hence, this state summation produces the bracket polynomial as we have described it at the beginning of the section. Remark By a change of variables one obtains the original Jones polynomial, VK (t), for oriented knots and links from the normalized bracket: VK ðtÞ ¼ fK ðt1=4 Þ Remark The bracket polynomial provides a connection between knot theory and physics, in that the state summation expression for it exhibits it as a generalized partition function defined on the knot diagram. Partition functions are ubiquitous in statistical mechanics, where they express the summation over all states of the physical system of probability weighting functions for the individual states. Such physical partition functions contain large amounts of information about the corresponding physical system. Some of this information is directly present in the properties of the function, such as the location of critical points and phase transition. Some of the information can be obtained by differentiating the partition function, or performing other mathematical operations on it. In fact, by defining a generalization of the bracket polynomial, defined on knot diagrams but not invariant under the Reidemeister moves, we can capture significant partition functions that are physically meaningful. There is no room in this survey to detail how this generalization can be used to express the Potts model for planar graphical configurations, and how it expresses the relationship between the Potts model and the Temperley–Lieb

223

algebra in diagrammatic form. There is much more in this connection with statistical mechanics in that the local weights in a partition function are often expressed in terms of solutions to a matrix equation called the Yang–Baxter equation, that turns out to fit perfectly invariance under the third Reidemeister move. As a result, there are many ways to define partition functions of knot diagrams that give rise to invariants of knots and links. The subject is intertwined with the algebraic structure of Hopf algebras and quantum groups, useful for producing systematic solutions to the Yang–Baxter equation. In fact, Hopf algebras are deeply connected with the problem of constructing invariants of threedimensional manifolds in relation to invariants of knots. We have chosen, in this survey article, not to discuss the details of these approaches, but rather to proceed to Vassiliev invariants and the relationships with Witten’s functional integral. The reader is referred to Kauffman (1987, 1994, 2002), Jones (1985), and Reshetikhin and Turaev (1991) for more information about relationships of knot theory with statistical mechanics, Hopf algebras, and quantum groups. For topology, the key point is that Lie algebras can be used to construct invariants of knots and links. This is shown nowhere more clearly than in the theory of Vassiliev invariants that we take up in the next section.

Vassiliev Invariants and Invariants of Rigid Vertex Graphs In this section we study the combinatorial topology of Vassiliev invariants. As we shall see, by the end of this section, Vassiliev invariants are directly connected with Lie algebras, and representations of Lie algebras can be used to construct them. This aspect of link invariants is one of the most fundamental for connections with physics. Just as symmetry considerations in physics lead to a fundamental relationship with Lie algebras, topological invariance leads to a fundamental relationship of the theory of knots and links with Lie algebras. If V(K) is a (Laurent polynomial valued or, more generally, commutative ring valued) invariant of knots, then it can be naturally extended to an invariant of rigid vertex graphs by defining the invariant of graphs in terms of the knot invariant via an ‘‘unfolding of the vertex.’’ That is, we can regard the vertex as a ‘‘black box’’ and replace it by any tangle of our choice. Rigid vertex motions of the graph preserve the contents of the black box, and hence implicate ambient isotopies of the link obtained by replacing the black box by its contents.

224 Knot Theory and Physics

Invariants of knots and links that are evaluated on these replacements are then automatically rigid vertex invariants of the corresponding graphs. If we set up a collection of multiple replacements at the vertices with standard conventions for the insertions of the tangles, then a summation over all possible replacements can lead to a graph invariant with new coefficients corresponding to the different replacements. In this way, each invariant of knots and links implicates a large collection of graph invariants. The simplest tangle replacements for a 4-valent vertex are the two crossings, positive and negative, and the oriented smoothing. Let V(K) be any invariant of knots and links. Extend V to the category of rigid vertex embeddings of 4-valent graphs by the formula VðK Þ ¼ aVðKþ Þ þ bVðK Þ þ cVðK0 Þ where Kþ denotes a knot diagram K with a specific choice of positive crossing, K denotes a diagram identical to the first with the positive crossing replaced by a negative crossing and K denotes a diagram identical to the first with the positive crossing replaced by a graphical node. There is a rich class of graph invariants that can be studied in this manner. The Vassiliev invariants (Bar-Natan 1995) constitute the important special case of these graph invariants where a = þ1, b = 1 and c = 0. Thus, V(G) is a Vassiliev invariant if VðK Þ ¼ VðKþ Þ  VðK Þ Call this formula the exchange identity for the Vassiliev invariant V. See Figure 7. V is said to be of finite type k if V(G) = 0 whenever jGj > k, where jGj denotes the number of (4-valent) nodes in the graph G. The notion of finite type is of extraordinary significance in studying these invariants. One reason for this is the following basic lemma.

2

1 2 1

1

2

Figure 8 Chord diagrams.

The upshot of this lemma is that Vassiliev invariants of type k are intimately involved with certain abstract evaluations of graphs with k nodes. In fact, there are restrictions (the four-term relations) on these evaluations demanded by the topology and it follows from results of Kontsevich (see Bar-Natan (1995) that such abstract evaluations actually determine the invariants. The knot invariants derived from classical Lie algebras are all built from Vassiliev invariants of finite type. All of this is directly related to Witten’s functional integral (Witten 1989). In the next few figures we illustrate some of these main points. In Figure 8 we show how one associates a so-called chord diagram to represent the abstract graph associated with an embedded graph. The chord diagram is a circle with arcs connecting those points on the circle that are welded to form the corresponding graph. In Figure 9 we illustrate how the four-term relation is a consequence of topological invariance. In Figure 10 we show how the four-term relation is a consequence of the abstract pattern of the commutator

Lemma If a graph G has exactly k nodes, then the value of a Vassiliev invariant vk of type k on G, vk (G), is independent of the embedding of G. Proof

Omitted.

(K⏐*)

&

(K⏐+)

(K⏐–)

V(K⏐*) = V(K⏐+) – V(K⏐–) Figure 7 Exchange identity for Vassiliev invariants.

Figure 9 The four-term relation from topology.

Knot Theory and Physics

a

Ta

–

=

T a T b – T b T a = fcab T c –

=

=

–

=

=

–

Figure 10 The four-term relation from categorical Lie algebra.

identity for a matrix Lie algebra. That is, we show how a diagrammatic version of the formula T a T b  T b T a ¼ fcab T c fits directly with the four-term relation. The formula we have quoted here states that the commutator of the matrices T a and T b is equal to a sum of the matrices T c with coefficients (the structure coefficients of the Lie algebra) fcab . Such a relation is the most concrete way to define a matrix Lie algebra. There are other levels of abstraction that can be employed here. The same diagrammatic can be interpreted directly in terms of the Jacobi identity that defines a Lie algebra. We shall content ourselves with this matrix point of view here, and add that it is assumed here that the structure coefficients are invariant under cyclic permutation, an assumption that is not needed in the general case. The four-term relation is directly related to a categorical generalization of Lie algebras. Figure 11 illustrates how the weights are assigned to the chord diagrams in the Lie algebra case – by inserting Lie algebra matrices into the circle and taking a trace of a sum of matrix products. The relationship between Vassiliev invariants and Lie

b

a

tr( Σ T aT bT aT b) ab

Figure 11 Calculating Lie algebra weights.

225

algebras has been known since Bar-Natan’s thesis (see also Kauffman (1995). In Bar-Natan (1995) the reader will find a good account of Kontsevich’s theorem, showing how Lie algebra weight systems, and in fact any weight system satisfying the fourterm relation, can be used to construct knot invariants. Conceptually, the ideas behind the Kontsevich theorem are directly related to Witten’s approach to knot invariants via quantum field theory. We give an exposition of this approach in the next section of this article. Example Let PK (t) = fK (et ) (A = et ) where fK (A) is the normalized bracket polynomial invariant discussed in the last section. Then PK (t) is expressed as a power series in t with coefficients vn (K), n = 0, 1, 2, . . . , that are invariants of the knot or link K. It is not hard to show that these coefficient invariants (extended to graphs so that the Vassiliev exchange identity is satisfied) are Vassiliev invariants of finite type. In fact, most of the so-called polynomial invariants of knots and links (relatives of the bracket and Jones polynomials) give rise to Vassiliev invariants in just this way. Thus, Vassiliev invariants of finite type are ubiquitous in this area of knot theory. One can think of Vassiliev invariants as building blocks for the other invariants, or that these invariants are sources of Vassiliev invariants.

Vassiliev Invariants and Witten’s Functional Integral Edward Witten (1989) proposed a formulation of a class of 3-manifold invariants as generalized Feynman integrals taking the form Z(M), where Z ZðMÞ ¼ DAeðik=4ÞSðM;AÞ Here M denotes a 3-manifold without boundary and A is a gauge field (also called a gauge potential or gauge connection) defined on M. The gauge field is a 1-form on a trivial G-bundle over M with values in a representation of the Lie algebra of G. The group G corresponding to this Lie algebra is said to be the gauge group. In this integral, the action S(M, A) is taken to be the integral over M of the trace of the Chern–Simons 3-form A ^ dA þ (2=3)A ^ A ^ A. (The product is the wedge product of differential forms.) Z(M) integrates over all gauge fields modulo gauge equivalence. The formalism and internal logic of Witten’s integral supports the existence of a large class of topological invariants of 3-manifolds and associated invariants of knots and links in these manifolds.

226 Knot Theory and Physics

The invariants associated with this integral have been given rigorous combinatorial descriptions but questions and conjectures arising from the integral formulation are still outstanding. Specific conjectures about this integral take the form of just how it implicates invariants of links and 3-manifolds, and how these invariants behave in certain limits of the coupling constant k in the integral. Many conjectures of this sort can be verified through the combinatorial models. On the other hand, the really outstanding conjecture about the integral is that it exists! At the present time there is no measure theory or generalization of measure theory that supports it in full generality. Here is a formal structure of great beauty. It is also a structure whose consequences can be verified by a remarkable variety of alternative means. The formalism of the Witten integral implicates invariants of knots and links corresponding to each classical Lie algebra. In order to see this, we need to introduce the Wilson loop. The Wilson loop is an exponentiated version of integrating the gauge field along a loop K in three space that we take to be an embedding (knot) or a curve with transversal selfintersections. For this discussion, the Wilson loop will be denoted by the notation WK ðAÞ to denote the dependence on the loop K and the field HA. It is usually indicated by the symbolism A tr(Pe K ). Thus,  H  A WK ðAÞ ¼ tr Pe K Here the P denotes path ordered integration – we are integrating and exponentiating matrix valued functions, and so must keep track of the order of the operations. The symbol tr denotes the trace of the resulting matrix. This Wilson loop integration exists by normal means and does not require functional integration. With the help of the Wilson loop functional on knots and links, Witten writes down a functional integral for link invariants in a 3-manifold M: ZðM; KÞ ¼ ¼

Z Z

 H  A DAeðik=4ÞSðM;AÞ tr Pe K DAe

ðik=4ÞS

WK ðAÞ

Here S(M, A) is the Chern–Simons Lagrangian, as in the previous discussion. We abbreviate S(M, A) as S and write WK (A) for the Wilson loop. Unless

otherwise mentioned, the manifold M will be the three-dimensional sphere S3 . An analysis of the formalism of this functional integral reveals quite a bit about its role in knot theory. One can determine how the Witten integral behaves under a small deformation of the loop K. Theorem (i) Let Z(K) = Z(S3 , K) and let Z(K) denote the change of Z(K) under an infinitesimal change in the loop K. Then ZðKÞ ¼ ð4i=kÞ

Z

dAeðik=4ÞS ½VolTa Ta WK ðAÞ

where Vol = rst dxr dxs dxt . The sum is taken over repeated indices, and the insertion is taken of the matrices Ta Ta at the chosen point x on the loop K that is regarded as the center of the deformation. The volume element Vol = rst dxr dxs dxt is taken with regard to the infinitesimal directions of the loop deformation from this point on the original loop. (ii) The same formula applies, with a different interpretation, to the case where x is a double point of transversal self-intersection of a loop K, and the deformation consists in shifting one of the crossing segments perpendicularly to the plane of intersection so that the self-intersection point disappears. In this case, one Ta is inserted into each of the transversal crossing segments so that Ta Ta WK (A) denotes a Wilson loop with a self-intersection at x and insertions of Ta at x þ 1 and x þ 2 , where 1 and 2 denote small displacements along the two arcs of K that intersect at x. In this case, the volume form is nonzero, with two directions coming from the plane of movement of one arc, and the perpendicular direction is the direction of the other arc. Remark One shows that the result of a topological variation has an analytic expression that is zero if the topological variation does not create a local volume. Thus, we have shown that the integral of e(ik=4)S(A) WK (A) is topologically invariant as long as the curve K is moved by the local equivalent of regular isotopy. In the case of switching a crossing, the key point is to write the crossing switch as a composition of first moving a segment to obtain a transversal intersection of the diagram with itself, and then to continue the motion to complete the switch. Up to the choice of our conventions for constants, the switching formula is, as shown below (see Figure 12).

Knot Theory and Physics

z

=z

= (c/k)z

–z

227

with summation over the repeated indices. Each of these operators has the property that its action on the Wilson loop has a geometric or topological interpretation. One has

+ O (1/k 2 )

dðKÞ ¼  bðKÞ G Figure 12 The difference formula.

ZðKþ Þ  ZðK Þ Z ¼ ð4i=kÞ DAeðik=4ÞS Ta Ta hK jAi ¼ ð4i=kÞZðT a T a K Þ where K denotes the result of replacing the crossing by a self-touching crossing. We distinguish this from adding a graphical node at this crossing by using the double-star notation. A key point is to notice that the Lie algebra insertion for this difference is exactly what is done (in chord diagrams) to make the weight systems for Vassiliev invariants (without the framing compensation). Thus, the formalism of the Witten functional integral takes one directly to these weight systems in the case of the classical Lie algebras. In this way, the functional integral is central to the structure of the Vassiliev invariants.

The Loop Transform and Quantum Gravity Suppose that (A) is a (complex-valued) function defined on gauge fields. Then we define formally the loop transform b(K), a function on embedded loops in three-dimensional space, by the formula Z bðKÞ ¼ DA ðAÞWK ðAÞ If  is a differential operator defined on (A), then we can use this integral transform to shift the effect of  to an operator on loops via integration by parts: Z d  ðKÞ ¼ DA ðAÞWK ðAÞ Z ¼  DA ðAÞWK ðAÞ When  is applied to the Wilson loop, the result can be an understandable geometric or topological operation. One can illustrate this situation with operators G and H: G ¼ Fija dxi =Aaj ðxÞ H ¼ ars Fija =Asi =Arj

where this variation refers to the effect of varying K. As we saw in the previous section, this means that if b(K) is a topological invariant of knots and links, d(K) = 0 for all embedded loops K. This then G condition is a transform analog of the equation G (A) = 0. This equation is the differential analog of an invariant of knots and links. It may happen that  b(K) is not strictly zero, as in the case of our framed knot invariants. For example, with   Z ðAÞ ¼ exp ðik=4Þ trðA ^ dA þ ð2=3ÞA ^ A ^ AÞ d(K) is zero for flat deformations we conclude that G (in the sense of the previous section) of the loop K, but can be nonzero in the presence of a twist or curl. In this sense, the loop transform provides a subtle variation on the strict condition G (A) = 0. In Ashtekar et al. (1992) and other publications by Ashtekar, Rovelli, Smolin, and their colleagues, the loop transform is used to study a reformulation and quantization of Einstein gravity. The differentialgeometric gravity theory of Einstein is reformulated in terms of a background gauge connection and in the quantization, the Hilbert space consists in functions (A) that are required to satisfy the constraints b G = 0 and H = 0. Thus, we see that G(K) can be partially zero in the sense of producing a framed knot b invariant, and that H(K) is zero for non-selfintersecting loops. This means that the loop transforms of G and H can be used to investigate a subtle variation of the original scheme for the quantization of gravity. This program is being actively pursued by a number of researchers. The Vassiliev invariants arising from a topologically invariant loop transform are of significance to this theory.

Braiding, Topological Quantum Field Theory, and Quantum Computing The purpose of this section is to discuss in a very general way how braiding is related to topological quantum field theory and to the enterprise (Freedman et al. 2002) of using this sort of theory as a model for anyonic quantum computation. The ideas in the subject of topological quantum field theory are well expressed by Michael Atiyah (1990)

228 Knot Theory and Physics

and Edward Witten (1989). The simplest case of this idea is C N Yang’s original interpretation of the Yang–Baxter equation. Yang articulated a quantum field theory in one dimension of space and one dimension of time, in which the R-matrix giving the scattering amplitudes for an interaction of two particles whose (let us say) spins corresponded to the matrix indices so that Rcd ab is the amplitude for particles of spin a and spin b to interact and produce particles of spin c and d. Since these interactions are between particles in a line, one takes the convention that the particle with spin a is to the left of the particle with spin b, and the particle with spin c is to the left of the particle with spin d. If one follows the concatenation of such interactions, then there is an underlying permutation that is obtained by following strands from the bottom to the top of the diagram (thinking of time as moving up the page). Yang designed the Yang–Baxter equation for R so that the amplitudes for a composite process depend only on the underlying permutation corresponding to the process and not on the individual sequences of interactions. In taking over the Yang–Baxter equation for topological purposes, we can use the same interpretation, but think of the diagrams with their under- and over-crossings as modeling events in a spacetime with two dimensions of space and one dimension of time. The extra spatial dimension is taken in displacing the woven strands perpendicular to the page, and allows the use of braiding operators R and R1 as scattering matrices. Taking this picture to heart, one can add other particle properties to the idealized theory. In particular, one can add fusion and creation vertices where, in fusion, two particles interact to become a single particle and, in creation, one particle changes (decays) into two particles. Matrix elements corresponding to trivalent vertices can represent these interactions (see Figure 13). Once one introduces trivalent vertices for fusion and creation, there is the question how these interactions will behave in respect to the braiding operators. There will be a matrix expression for the compositions of braiding and fusion or creation as indicated in Figure 14. Here we will restrict ourselves to showing the diagrammatics with the intent of giving the reader a flavor of these

=R

Figure 14 Braiding.

structures. It is natural to assume that braiding intertwines with creation as shown in Figure 15 (similarly with fusion). This intertwining identity is clearly the sort of thing that a topologist will love, since it indicates that the diagrams can be interpreted as embeddings of graphs in three-dimensional space. Figure 16 illustrates the Yang–Baxter equation. The intertwining identity is an assumption like the Yang–Baxter equation itself, which simplifies the mathematical structure of the model. It is to be expected that there will be an operator that expresses the recoupling of vertex interactions as shown in Figure 17 and labeled by Q. The actual formalism of such an operator will parallel the mathematics of recoupling for angular momentum (see, e.g., Kauffman (1994)). If one just considers the abstract structure of recoupling then one sees that for trees with four branches (each with a single root) there is a cycle of length 5, as shown in

=

Figure 15 Intertwining.

R R

I

I R

R I

I

I

I =

Figure 16 Yang–Baxter equation.

Q

Figure 13 Creation and fusion.

Figure 17 Recoupling.

R R

R

I

I

R

Knot Theory and Physics

Q

Q Q

Q

Q

Figure 18 Pentagon identity.

Figure 20 Decomposition of a surface into trinions.

Figure 17. One can start with any pattern of three vertex interactions and go through a sequence of five recouplings that bring one back to the same tree from which one started. It is a natural simplifying axiom to assume that this composition is the identity mapping. This axiom is called the pentagon identity (Figure 18). Finally, there is a hexagonal cycle of interactions between braiding, recoupling and the intertwining identity as shown in Figure 19. One says that the interactions satisfy the hexagon identity if this composition is the identity. A three-dimensional topological quantum field theory is an algebra of interactions that satisfies the Yang–Baxter equation, the intertwining identity, the pentagon identity and the hexagon identity. There is no room in this summary to detail the way that these properties fit into the topology of knots and three-dimensional manifolds, but a sketch is in order. For the case of topological quantum field theory related to the group SU(2) there is a construction based entirely on the combinatorial topology of the bracket polynomial (see the section

R

Q =

Q

R R Q

Figure 19 Hexagon identity.

229

‘‘Knots, braids, and bracket polynomial’’). For more information on this approach, the reader is referred to Kauffman (1994, 2002). It turns out that the algebraic properties of a topological quantum field theory give it enough power to rigourously model three manifold invariants described by the Witten integral. This is done by regarding the 3-manifold as a union of two handlebodies with boundary an orientable surface Sg of genus g. The surface is divided up into trinions as illustrated in Figure 20. A trinion is a surface with boundary that is topologically equivalent to a sphere with three punctures. In Figure 20 we illustrate two trinions, the second shown as a neighborhood of a trivalent vertex, and a surface of genus 3 that is decomposed into three trinions. It turns out that there is a way to associate a vector space V(Sg ) to a surface with a trinion decomposition, defined in terms of the associated topological quantum field theory, such that the isomorphism class of the vector space V(Sg ) does not depend upon the choice of decomposition. This independence is guaranteed by the braiding, hexagon, and pentagon identities in such a way that one can associate a well-defined vector jM i in V(Sg ) whenever M is a 3-manifold whose boundary is Sg . Furthermore, if a closed 3-manifold M3 is decomposed along a surface Sg into the union of M and Mþ , where these parts are otherwise disjoint 3-manifolds with boundary Sg , then the inner product I(M) = hM jMþ i is, up to normalization, an invariant of the 3-manifold M3 . With the definition of topological quantum field theory given above, knots and links can be incorporated as well, so that one obtains a source of invariants I(M3 , K) of knots and links in orientable 3-manifolds. The invariant I(M3 , K) can be formally compared with the Witten integral Z 3 ZðM ; KÞ ¼ DAeðik=4ÞSðM;AÞ WK ðAÞ

230 Knot Theory and Physics

It can be shown that up to limits of the heuristics, Z(M, K) and I(M3 , K) are essentially equivalent for appropriate choice of gauge groups. This point of view leads to more abstract formulations of topological quantum field theories as ways to associate vector spaces and linear transformations to manifolds and cobordisms of manifolds. (A cobordism of surfaces is a 3-manifold whose boundary consists of these surfaces.) As the reader can see, a three-dimensional TQFT is, at base, a highly simplified theory of point-particle interactions in (2 þ 1)-dimensional spacetime. It can be used to articulate invariants of knots and links and invariants of 3-manifolds. The reader interested in the SU(2) case of this structure and its implications for invariants of knots and 3-manifolds can consult Kauffman (1994, 2002) and Crane (1991). One expects that physical situations involving 2 þ 1 spacetime will be approximated by such an idealized theory. It is thought, for example, that aspects of the quantum Hall effect will be related to topological quantum field theory (Wilczek 1990). One can imagine a physics where the geometrical space is two dimensional and the braiding of particles corresponds to their interactions through circulating around one another in the plane. Anyons are particles that do not just change their wave functions by a sign under interchange, but rather by a complex phase or even a linear combination of states. It is hoped that TQFT models will describe applicable physics. One can think about the possible applications of anyons to quantum computing. The TQFTs then provide a class of anyonic models where the braiding is essential to the physics and to the quantum computation. The key point in the application and relationship of TQFT and quantum information theory is, in our opinion, contained in the structure illustrated in Figure 21. There we show a more complex braiding operator, based on the composition of recoupling with the elementary braiding at a vertex. (This structure is implicit in the hexagon identity of

Q

R

Q –1

B = Q –1 RQ

Figure 21 A more complex braiding operator.

Figure 19.) The new braiding operator is a source of unitary representations of braid group in situations (which exist mathematically) where the recoupling transformations are themselves unitary. This kind of pattern is utilized in the work of Freedman et al. (2002) and in the case of classical angular momentum formalism has been dubbed a ‘‘spin-network quantum simulator’’ by Rasetti and collaborators (see, e.g., Marzuoli and Rasetti (2002). Kauffman and Lomonaco (2006) show how certain natural deformations (Kauffman 1994) of Penrose (1969) spin networks can be used to produce such the Freedman–Kitaev model for anyonic topological quantum computation. It is legitimate to speculate that networks of this kind are present in physical reality. Quantum computing can be regarded as a study of the structure of the preparation, evolution, and measurement of quantum systems. In the quantum computation model, an evolution is a composition of unitary transformations (usually finite-dimensional over the complex numbers). The unitary transformations are applied to an initial state vector that has been prepared prior to this process. Measurements are projections to elements of an orthonormal basis of the space upon which the evolution is applied. The result of measuring a state j i, written in the given basis, is probabilistic. The probability of obtaining a given basis element from the measurement is equal to the absolute square of the coefficient of that basis element in the state being measured. It is remarkable that the above lines constitute an essential summary of quantum theory. All applications of quantum theory involve filling in details of unitary evolutions and specifics of preparations and measurements. Such unitary evolutions can be seen as approximated arbitrarily closely by representations of the Artin braid group. The key to the anyonic models of quantum computation via topological quantum field theory, or via deformed spin networks, is that all unitary evolutions can be approximated by a single coherent method for producing representations of the braid group. This beautiful mathematical fact points to a deep role for topology in the structure of quantum physics. The future of knots, links, and braids in relation to physics will be very exciting. There is no question that unitary representations of the braid group and quantum invariants of knots and links play a fundamental role in the mathematical structure of quantum mechanics, and we hope that time will show us the full meaning of this relationship.

Acknowledgments It gives the author pleasure to thank the National Science Foundation for support of this research

Kontsevich Integral

under NSF Grant DMS-0245588. Much of this effort was sponsored by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement F30602-01-2-05022. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Defense Advanced Research Projects Agency, the Air Force Research Laboratory, or the US Government. See also: Chern–Simons Models: Rigorous Results; Functional Integration in Quantum Physics; Knot Homologies; Knot Invariants and Quantum Gravity; Large-N and Topological Strings; Loop Quantum Gravity; Mathematical Knot Theory; Schwarz-Type Topological Quantum Field Theory; The Jones Polynomial; Topological Knot Theory and Macroscopic Physics; Topological Quantum Field Theory: Overview; Two-Dimensional Conformal Field Theory and Vertex Operator Algebras; von Neumann Algebras: Introduction, Modular Theory, and Classification Theory; Yang–Baxter Equations.

Further Reading Alexander JW (1923) Topological invariants of knots and links. Transactions of the American Mathematical Society 20: 275–306. Atiyah MF (1990) The Geometry and Physics of Knots. Cambridge: Cambridge University Press. Ashtekar A, Rovelli C, and Smolin L (1992) Weaving a classical geometry with quantum threads. Physical Review Letters 69: 237. Bar-Natan D (1995) On the Vassiliev knot invariants. Topology 34: 423–472.

231

Crane L (1991) 2-d physics and 3-d topology. Communications Mathematical Physics 135(3): 615–640. Freedman MH, Kitaev A, and Wang Z (2002) Simulation of topological field theories by quantum computers. Communications in Mathematical Physics 227: 587–603 (quant-ph/0001071). Jones VFR (1985) A polynomial invariant for links via von Neumann algebras. Bulletin of the American Mathematical Society 129: 103–112. Kauffman LH (1987) State models and the Jones polynomial. Topology 26: 395–407. Kauffman LH (1994) Temperley–Lieb Recoupling Theory and Invariants of Three-Manifolds, Annals Studies, vol. 114. Princeton: Princeton University Press. Kauffman LH (1991) Knots and Physics (2nd edn., 1993, 3rd edn., 2002). Singapore: World Scientific Publishers. Kauffman LH (1995) Functional integration and the theory of knots. Journal of Mathematical Physics 36(5): 2402–2429. Kauffman LH (2002) Quantum computation and the Jones polynomial. In: Lomonaco S, Jr. (ed.) Quantum Computation and Information, AMS CONM/305. Providence, RI: American Mathematical Society pp. 101–137. Kauffman LH and Lomonaco SJ, Jr. (2002) Quantum entanglement and topological entanglement. New Journal of Physics 4: 73.1–73.18 (http://www.njp.org). Kauffman LH and Lomonaco SJ, Jr. (2006) Spin networks and anyonic topological quantum computing (in preparation). Marzuoli A and Rasetti M (2002) Spin network quantum simulator. Physics Letters A 306: 79–87. Penrose R (1969) Angular momentum: an approach to combinatorial spacetime. In: Bastin T (ed.) Quantum Theory and Beyond. Cambridge: Cambridge University Press. Reshetikhin NY and Turaev V (1991) Invariants of three manifolds via link polynomials and quantum groups. Inventiones Mathematicae 103: 547–597. Wilczek F (1990) Fractional Statistics and Anyon Superconductivity. Berlin–Heidelberg–New York: Springer-Verlag. Witten E (1989) Quantum field Theory and the Jones Polynomial. Communications in Mathematical Physics 121: 351–399.

Kontsevich Integral S Chmutov and S Duzhin, Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, Russia ª 2006 Elsevier Ltd. All rights reserved.

Introduction The Kontsevich integral was invented by Kontsevich (1993) as a tool to prove the fundamental theorem of the theory of finite-type (Vassiliev) invariants (see BarNatan (1995a)). It provides an invariant exactly as strong as the totality of all Vassiliev knot invariants. The Kontsevich integral is defined for oriented tangles (either framed or unframed) in R3 ; therefore, it is also defined in the particular cases of knots, links, and braids (see Figure 1). As a starter, we give two examples where simple versions of the Kontsevich integral have a

straightforward geometrical meaning. In these examples, as well as in the general construction of the Kontsevich integral, we represent 3-space R3 as the product of a real line R with coordinate t and a complex plane C with complex coordinate z. Example 1 The number of twists in a braid with two strings z1 (t) and z2 (t) placed in the slice 0  t  1 (see Figure 2) is equal to Z 1 1 dz1  dz2 2i 0 z1  z2

Figure 1 A tangle, a braid, a link, and a knot.

232 Kontsevich Integral

+

– z1(t )

z2(t )

Figure 2 Counting the number of twists.

zj (t )

–

zj′(t )

Figure 3 Counting the linking number.

Example 2 The linking number of two spatial curves K and K0 (see Figure 3) can be computed as Z X dðzj ðtÞ  z 0j ðtÞÞ 1 lkðK; K0 Þ ¼ "j 2i m 2 – this

Validity of the KdV Approximation While the above derivation of the KdV equation is simple and intuitive one may wonder how accurate an approximation it actually provides to the true solutions of [3] (or to the evolution of water waves, probably the most important physical situation in which the KdV approximation is used). In particular, note that in the notation of [9] the phenomena intrinsic to the KdV equation occur on timescales T = O(1). However, this corresponds to a very long timescale t = O(1="3 ) in the original FPU model and it could easily be the case that although the error made in derivation of the KdV approximation at any given time is quite small, over these very long timescales the errors could accumulate in such a way as to destroy the accuracy of the approximation. The KdV and other modulation equations have been used since the nineteenth century but only relatively recently have rigorous estimates of the accuracy of this approximation been proved. In fact, the first estimates demonstrating that the KdV equation actually provided an accurate approximation to the true motion of water waves over the timescales expected from the heuristic derivation were not proved until Craig (1985). More recently, powerful general methods have been developed to justify not just the KdV equation but other modulation equations like the nonlinear Schro¨ dinger equation and Ginzburg– Landau equation as well. For instance, the following method, introduced in Kirrmann et al. (1992), has been used to justify the use of modulation equations in the water-wave problem, the evolution of Taylor–Couette patterns in viscous fluids, and a number of other

242 Korteweg–de Vries Equation and Other Modulation Equations

circumstances. We will explain it in the context of a general, abstract evolution equation to indicate its generality. Suppose that one wishes to approximate the small-amplitude solutions of a general evolution equation (or system of such equations) of the form @t u ¼ Lu þ N ðuÞ

½10

where L is a linear operator and N represents the nonlinear terms. Suppose that via some formal analysis like that in the previous section we have derived a function "2 that is believed to be a good approximation to a true solution of [10]. In that example, for instance, "2 would be the sum of the solutions of the two KdV equations in [9], and in general it will be given by the solution of the modulation equation that is expected to approximate [10]. We must show that the difference between "2 and a true solution of [10] remains small over the timescales of interest. We write this difference as u  "2 = " R so that if  > 2, and if R = O(1), "2 does provide the leading-order approximation to the true solution. We can make Rjt = 0 small by choosing the initial conditions of our modulation equation appropriately and thus we need to follow how R evolves in time. If we use the equation satisfied by u we see that R evolves as  @t R ¼ LR þ " N ð"2 þ " RÞ  N ð"2 Þ þ " Resð"2 Þ ½11 where Res("2 ) = L("2 ) þ N ("2 )  @t ("2 ), the ‘‘residual’’ of our approximation is simply the amount by which the approximation fails to satisfy the original equation at any given time. In the example in the previous section the residual would include the terms O("8 ) that we ignored in our expansion. One must now, in any given example consider three points: 1. The linear evolution of R: @t R ¼ LR þ DN ð"2 ÞR

½12

Controlling the solutions of this linear, but nonconstant coefficient partial differential equation is often the most difficult step in proving that solutions of the modulation equation give accurate approximations to the true solution. One can frequently find norms that are preserved by solutions of the leading-order equation @t R = LR. However, the term DN ("2 ) = O("2 ) if N is a quadratic nonlinearity. Over the very long timescales (i.e., O("3 )) of interest in these approximation problems this O("2 ) term can

cause uncontrolled growth of R, leading to a breakdown in the approximation. In order to control [12] one must typically make use of some special features of the problem under consideration. For instance, it is sometimes possible to make a coordinate transformation which eliminates the terms of O("2 ) on the right-hand side of [12], after which relatively standard methods suffice to control the solutions of [12]. 2. The nonlinear terms in [11]: these terms are of the form " [N ("2 þ " R)  N ("2 )]  DN ("2 )R. From Taylor’s theorem we see that, if the nonlinear term is reasonably smooth, these terms are of O(" ). If  > 3, these terms are small and can be controlled over the timescales of interest by a straightforward application of Gronwall’s inequality or standard ‘‘energy estimates.’’ 3. Finally, one must consider the influence of the inhomogeneous terms " Res("2 ). Note that if this term is small enough, say O(" ), with   3 this term can also be controlled over the relevant timescales by an application of the Gronwall inequality. In order to make this term small, we need to be sure that our approximation "2 fails to solve the true equation at any given time by a small amount. In doing so, we can exploit the fact that we can add to our leading-order approximation terms of higher order without affecting the fact that to leading order the true solution is still approximated by the solution of the modulation equation. This is the role of the term "4 ’ in the approximation [6] in the previous section. The leading-order approximation is given by the functions U and V which solve the KdV equations but by adding the additional term "4 ’ to the approximation we cancel the remaining terms of O("6 ) in [7], thereby reducing the size of the residue in that example to O("8 ). This method works in other examples as well so that the inhomogeneous term in [11] can usually be treated by this means. However, in each case, we must prove that the additional terms one adds to the approximation remain bounded over the timescales of interest and demonstrating this fact may not be as easy as it was in the case of the FPU model where the additional term satisfied a simple wave equation. Using this approach one can show that the approximation derived heuristically in the previous section does accurately model the behavior of solutions of the FPU model over the expected timescales. More precisely, if r(j, t) is the solution of [3] and if U and V are the solutions of the modulation equations [9] (with appropriately chosen, small-amplitude, long-wavelength initial

Korteweg–de Vries Equation and Other Modulation Equations

conditions), one can prove (see Schneider and Wayne (1999)) that for any T0 > 0 there is an "0 > 0 and C > 0 such that for all 0 < " < "0 , sup t2½0;T0

krð; tÞ  ð"2 Uð"ð þ ctÞ; "3 tÞ

="3 

þ ð"2 Vð"ð  ctÞ; "3 tÞÞk‘1  C"7=2 One can also use this method to show that the solution of the water-wave problem with general small-amplitude, long-wavelength, initial data can be approximated by the sum of the solutions of a pair of uncoupled KdV equations (Schneider and Wayne 2000), one representing the left-moving part of the solution and one representing the rightmoving part of the solution, though in this context the technical difficulties associated with the existence theory for the water-wave problem mean the details are quite a lot more complicated.

Integrability of the KdV Equation One reason that normal forms for systems of ordinary differential equations are so useful is that they are frequently integrable – that is, they possess sufficiently many integrals, or constants of motion, that essentially explicit formulas for their solutions can be obtained. Remarkably, the same is true for the KdV equation and for many other modulation equations. An argument for why this is so has been put forth by Calogero and Eckhaus based on the universality of these equations – see Calogero and Eckhaus (1987) and references therein, as well as the article Integrable Systems: Overview for more on this point. Recall that Boussinesq and Korteweg and de Vries introduced the KdV equation to study solitary traveling waves on a fluid surface. For [1], one has an explicit family of such solutions given by: uðx; tÞ ¼ 2A2 sech2 ðA½x þ 4A2 tÞ;

A0

Note that from this formula one sees that waves of large amplitude are narrower and travel faster than waves of small amplitude. In a famous numerical study, Zabusky and Kruskal made a remarkable discovery. They considered solutions of the KdV equation in which a solitary wave of large amplitude overtook one of smaller amplitude. They found that after a highly nonlinear interaction the two solitary waves reemerged with their original amplitudes and speeds and the only reminder of their interaction was a phase shift in their relative positions. Their discovery began a search for a mathematical explanation of this remarkable ‘‘nonlinear superposition principle’’ which culminated with the solution of the KdV

243

equation via the method of inverse scattering and the identification of the KdV equation as an infinitedimensional, completely integrable Hamiltonian system. We begin by describing how a transformation discovered by Miura (1968) and then generalized by Gardner et al. (1974) leads very easily to the conclusion that there are infinitely many conserved quantities for the KdV equation. The basic idea is that given a transformation which maps solutions of one equation to solutions of a second, the existence of simple or ‘‘obvious’’ conserved quantities for the first equation may lead, via the transformation, to more complicated conserved quantities for the second. Given u = u(x, t), define w(x, t) implicitly via the formula uðx; tÞ ¼ wðx; tÞ þ i"@x wðx; yÞ þ "2 ðwðx; tÞÞ2

½13

Note that if w is smooth enough and " is small, we can invert this relation recursively to obtain w in terms of u via the formula w ¼ u  i"@x u  "2 ðu2 þ @x2 uÞ þ i"3 ð@x3 u þ 4u@x2 uÞ þ "4 ð2u3 þ 5ð@x uÞ2 þ 6u@x2 u þ @x4 uÞ þ Oð"5 Þ

½14

Now compute @t u  @x3 u  6u@x u ¼ f@t w  6w@x w  6"2 w2 @x w  @x3 wg þ 2"2 wf@t w  6w@x w  6"2 w2 @x w  @x3 wg þ i"@x f@t w  6w@x w  6"2 w2 @x w  @x3 wg

½15

From this we see immediately that if w satisfies the modified KdV equation @t w ¼ 6ðw@x w þ "2 w2 @x wÞ þ @x3 w

½16

then u, defined by [13] satisfies the KdV equation. However, one also sees immediately that the integral of w is a conserved quantity Rof [16] for all values of ", that is, if we define I " (t) = w(x, t) dx, then I " is a constant for all values of ". (We will assume here that w is defined on the real line, and that w and its derivatives go to zero as jxj tends to infinity. Similar results hold for x running over a finite interval with periodic boundary conditions.) But this in turn immediately implies that if we use [14] to expand I " in powers of " the coefficients in this expansion must also be constants in time. Since these coefficients will be expressed as integrals of u and its derivatives, they will give us (infinitely many)

244 Korteweg–de Vries Equation and Other Modulation Equations

conserved quantities for the KdV equation! Looking at the first few of these we find: R 1. K0 = u(x, t) dx. The conservation of this quantity follows immediately from the form of the KdV Requation. 2. K1 = @x u(x, t) dx = 0, if we assume that u and its derivatives tend to zero as jxj tends to infinity. Thus, we gain no new information from this quantity and in fact, all the integrals coming from the odd powers of " turn out to be ‘‘trivial’’ so we ignore them and focus just on the even powers R of ". R 3. K2 = (u2 þ @x2 u) dx = u2 dx. That this is a conserved quantity is again easy to see directly from the KdV equation, just by multiplying the equation by u and integrating with respect R R to x. 4. K4 = (3u2 þ 5(@x u)2 þ 6u@x2 u þ @x4 u)dx= (3u2  (@x u)2 )dx. The origin of this integral is not so obvious and we comment further on its meaning below. Clearly by continuing this procedure we can generate an infinite number of conserved quantities for the KdV equation. Indeed, if one chose another conserved for the modified KdV equation, [16], say Rquantity w2 (x, t) dx one could generate another sequence of conserved quantities via this same procedure. However, Kruskal, Miura, Gardner, and Zabusky proved that in fact, all of the conserved quantities that can be written as polynomials in u and its derivatives are already obtained by the procedure above. The constant of the motion K4 found above is of particular interest because one can write the KdV equation as

K4 u t ¼ @x u

½17

where =u denotes the variational derivative of K4 with respect to u(x). One can interpret this equation as a Hamiltonian system where @x defines the (nonstandard) symplectic structure and remarkably, Zhakarov and Faddeev (1971) proved that the KdV equation is actually a completely integrable Hamiltonian system. In particular, there exists a canonical transformation such that with respect to the new coordinates the Hamiltonian is a function only of the action variables (and hence in particular, the action variables remain constant in time). The transform which brings the Hamiltonian into its action-angle form is known as the inverse spectral transform and its details would take us beyond the limits of this article. However, very briefly, by observing that the Miura transformation [13] defines a Ricatti differential equation, and using the transformation that converts the Ricatti

equation to a linear ordinary differential equation one can relate the solution of the KdV equation to an eigenvalue problem for a linear Schro¨dinger operator. The potential term in the Schro¨dinger operator is given by the solution u(x, t) of the KdV equation. Remarkably, it turns out that the eigenvalues of this Schro¨dinger operator are constants of the motion if u is a solution of the KdV equation and are very closely related to the action variables for the Hamiltonian system. For more details on the inverse-scattering method and its use in solving the KdV equation we refer the reader to the monographs of Ablowitz and Segur (1981), Newell (1985), or the recent book by Kappeler and Po¨schel (2003) which develops the theory for the KdV equation on a finite interval with periodic boundary conditions in a particularly elegant fashion.

Other Mathematical Aspects of the KdV Equation In addition to the inverse-scattering transform approach, more traditional approaches to the existence and uniqueness of solutions have also been studied, starting with Temam’s proof of the wellposedness of solutions of the KdV equation with periodic boundary conditions in the Sobolev space H 2 . Noting that the Hamiltonian for the KdV equation described in the preceding section is closely related to the H 1 norm, this might seem a natural space in which to study well-posedness, but surprisingly Kenig, Ponce, and Vega, and Bourgain showed that the equation is also well posed in Sobolev spaces H s , with s < 1 and more recent work has extended the global well-posedness results to Sobolev spaces of small negative order. Aside from their intrinsic interest, these results have other physical implications. If one wishes to study statistical aspects of the behavior of ensembles of solutions of these equations, statistical mechanics suggests that the natural invariant measure for these equations is given by the Gibbs’ measure. However, the Gibbs’ measure is typically supported on functions less regular than H 1 , so that in order to define and study this measure one needs to know that solutions of the equation are well behaved in such spaces. Another natural mathematical question arises from the fact that the KdV equation is only an approximation to the original physical equation. Viewed from another perspective, the original system can be seen as a perturbation of the KdV equation. It then becomes natural to ask whether the special features of the KdV equation are preserved under perturbation. Viewing the KdV equation as a completely integrable Hamiltonian system this is

Korteweg–de Vries Equation and Other Modulation Equations

very analogous to the questions studied by the Kolmogorov–Arnol’d–Moser (KAM) theory and has led to a development of KAM-like results for a number of different partial differential equations like the KdV equation. The results are somewhat technical in nature but roughly speaking they say that if one considers the KdV equation with periodic boundary conditions, temporally periodic or quasiperiodic solutions will persist under small perturbations. The situation is more complicated and less well understood for the equation on the whole line due to the presence of a continuum of scattering states. For a very thorough review of the problem with periodic boundary conditions see Kappeler and Po¨schel (2003).

Other Modulation Equations As we stressed in its derivation, the KdV equation is an appropriate modulation equation for smallamplitude, long-wavelength solutions in dispersive nonlinear partial differential equations. However, as mentioned in the section ‘‘Derivation of the KdV equation’’ the method of multiple scales does not give a unique modulation equation even in this specific physical regime. Already in his original studies Boussinesq derived at least three different model equations for small-amplitude, long-wavelength water waves and a variety of such models continue to be studied today. For instance, an easy variation in the derivation of the KdV equation leads to the regularized long wave, or Benjamin–Bona–Mahoney equation in which the @x3 u term in the KdV equation is replaced by the term @x2 @t u. The validity of these alternatives to the KdV equation can also be studied with the aid of the methods described in the section ‘‘Validity of the Kdv approximation.’’ There have been many discussions of which of these modulation equations is the ‘‘correct’’ one. while they may all yield equivalent approximations to the original physical problem the KdV equation has at least two advantages: it is independent of the expansion parameter ", and it is completely integrable. None of the other equations that have been proposed as approximations to these small-amplitude, long-wavelength phenomena share both of these properties. If we think in terms of the Fourier transforms of the long-wavelength functions studied above they are solutions whose Fourier transform is concentrated near zero. One can also ask about modulation equations for solutions whose Fourier transform is concentrated about nonzero wave numbers. Such solutions represent a wave train with some fixed underlying wavelength, c , modulated on a much longer length scale, c =".

245

If we make the ansatz that the solution has the form uðx; tÞ "Að"ðx  cg tÞ; "2 tÞei2ðxcp tÞ=c þ complex conjugate

½18

and insert this hypothesized form of the solution into the original equation, then under mild assumptions on the form and properties of the original equation, similar to those under which we derived the KdV equation in an earlier section we find that to the lowest, nontrivial order in ", the amplitude A evolves according to the nonlinear Schro¨dinger equation i@T A ¼ c1 @X2 A þ c2 AjAj2

½19

If c1 and c2 are both real, the nonlinear Schro¨dinger equation can also be solved via the inverse-scattering method and it represents another completely integrable modulation equation. In this article, we have discussed modulation equations only for Hamiltonian, or conservative systems. However, similar equations have also played an important role in the study of dissipative equations like the Navier–Stokes equation. The most common modulation context in that setting is the Ginzburg–Landau equation, which can be derived as a modulation equation for Taylor–Couette rolls or for the convection rolls in the Rayleigh–Be´nard problem. Like the nonlinear Schro¨dinger equation, the Ginzburg–Landau equation describes how slow variations of the amplitude of an underlying periodic pattern evolve and as such it arises in a host of other situations in addition to the fluid dynamics examples mentioned above. For an extensive review of the applications of the Ginzburg–Landau equation, as well as its mathematical properties and some special solutions, see the recent article of Mielke (2002). See also: Bi-Hamiltonian Methods in Soliton Theory; Central Manifolds, Normal Forms; Hamiltonian Fluid Dynamics; Infinite-Dimensional Hamiltonian Systems; Integrable Systems and the Inverse Scattering Method; Integrable Systems: Overview; KAM Theory and Celestial Mechanics; Multiscale Approaches; Partial Differential Equations: Some Examples; WDVV Equations and Frobenius manifolds.

Further Reading Ablowitz MJ and Segur H (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics vol. 4. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). Calogero F and Eckhaus W (1987) Nonlinear evolutions equations, rescalings, model PDE’s and their integrability, i. Inverse Problems 3: 229–262.

246 K-Theory Craig W (1985) An existence theory for water waves and the Boussinesq and Korteweg–de Vries scaling limits. Communications in Partial Differential Equations 10(8): 787–1003. Gardner CS, Greene JM, Kruskal MD, and Miura RM (1974) Korteweg–deVries equation and generalization, VI. Methods for exact solution. Communications on Pure and Applied Mathematics 27: 97–133. Kappeler T and Po¨schel J (2003) KdV and KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics. vol. 45 (Results in Mathematics and Related Areas. 3rd Series, A Series of Modern Surveys in Mathematics). Berlin: Springer. Kirrmann P, Schneider G, and Mielke A (1992) The validity of modulation equations for extended systems with cubic nonlinearities. Proceedings of the Royal Society of Edinburgh Sect. A 122(1–2): 85–91. Mielke A (2002) The Ginzburg–Landau equation in its role as a modulation equation. In: Handbook of Dynamical Systems, vol. 2. pp. 759–834. Amsterdam: North-Holland. Miura RM (1968) Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. Journal of Mathematical Physics 9: 1202–1204.

Newell AC (1985) Solitons in Mathematics and Physics. CBMSNSF Regional Conference Series in Applied Mathematics, vol. 48. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). Pego RL and Weinstein MI (1997) Convective linear stability of solitary waves for Boussinesq equations. Studies in Applied Mathematics 99(4): 311–375. Schneider G and Wayne CE (1999) Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi–Pasta– Ulam model. In: Fiedler B, Gro¨ger K, and Sprekels J (eds.) International Conference on Differential Equations, (Berlin, 1999) vols. 1, 2, pp. 390–404. River Edge, NJ: World Scientific. Schneider G and Wayne CE (2000) The long-wave limit for the water wave problem. I. The case of zero surface tension. Communications on Pure and Applied Mathematics 53(12): 1475–1535. Zakharov VE and Faddeev LD (1971) The Korteweg–de Vries equation is a fully integrable Hamiltonian system. Functional Analysis and its Applications 5: 280–287.

K-Theory V Mathai, University of Adelaide, Adelaide, SA, Australia ª 2006 Elsevier Ltd. All rights reserved.

K-theory was invented in the category of algebraic vector bundles over algebraic varieties by A Grothendieck, who was directly motivated by the Hirzebruch–Riemann–Roch theorem which he subsequently greatly generalized. He also defined K-homology in terms of coherent sheaves and established the basic properties of K-theory and K-homology including Poincare´ duality for nonsingular varieties. The origin for the choice of the letter K in K-theory was apparently the German word ‘‘Klasse.’’ Using the formalism of Grothendieck, M F Atiyah and F Hirzebruch (cf. Karoubi 1978), developed topological K-theory in the category of topological (complex) vector bundles over topological spaces. It is this theory that will be the first principal focus of this article. A topological (complex) vector bundle over a compact topological space X is a topological space E together with a continuous map p : E ! X that is onto, such that p1 (x) is a vector space that is isomorphic to Cn for all x 2 X, and there is an open cover {U} of X together with homeomorphisms hU : p1 (U) ! U Cn called ‘‘local trivializations’’ n with the property that hV  h1 U :U \ V C ! U \ V n

C is of the form (Id, gUV ), where gUV : U \ V ! GL(n, C) are continuous maps satisfying the

following cocycle condition on triple overlaps, gUV gVW gWU = 1. X Cn is called the trivial vector bundle. Two vector bundles p : E ! X and q : F ! X over X are said to be isomorphic if there is a homeomorphism : E ! F with the property that p = q  , and which is a linear isomorphism when restricted to each fiber. The direct sum and tensor product of vector spaces carries over to vector bundles. There are canonical isomorphisms E F ffi F E and E  F ffi F  E, making the set Vect(X) of isomorphism classes of complex vector bundles over X into a commutative semiring. Vect(X) can be made into the commutative ring K0 (X) as follows. K0 (X) is generated by pairs ([E], [F]), together with the relation ([E], [F]) = ([E0 ], [F0 ]) if E F0 G ffi E0 F G for some [G] 2 Vect(X). Also K1 (X) is defined to be the group of homotopy classes of continuous maps from X to the infinite unitary group. Around the same time, R Bott proved his celebrated periodicity theorem, which says that the odd homotopy group of the (infinite) unitary group is the integers, whereas the even homotopy groups are all trivial. Incorporating Bott’s periodicity theorem for the unitary group into K-theory, Atiyah and Hirzebruch proved that topological K-theory K (X) = K0 (X) K1 (X) is a periodic generalized cohomology theory, and in what follows, the notation Kn (X) means n modulo 2. If M is not compact, then we can compactify M by adding to it a point þ ‘‘at infinity,’’ and denote it by Mþ . Let

: þ ! Mþ be the inclusion, inducing the pullback

K-Theory

map ! : K (Mþ ) ! K (þ) ffi Z. Then K (M) is defined to be ker(! ), also called the reduced K-theory. If X1 is a closed subset of X, the K-theory of the pair (X, X1 ) is defined as the reduced K-theory of the quotient space X=X1 . A fundamental computation of Bott is the computation of the K-theory of Euclidean space, Kn (R n ) ffi Z with canonical generator called the Bott class b 2 Kn (R n ), and Kn1 (Rn ) = {0}. Some of the basic properties of K-theory are listed as follows. Details can be found in Karoubi (1978). 1. Pullback If f : N ! M is a continuous map, then given a vector bundle  : E ! M over M, the pullback vector bundle is defined as f (E) = {(x, v) 2 N  E : f (x) = (v)} over N. This induces a pullback homomorphism, f ! : K (M) ! K (N). 2. Push-forward Let f : N ! M be a smooth proper map between compact manifolds which is K-oriented, that is, TN  f TM is a spinC vector bundle over N. Then there is a pushforward homomorphism, also called a Gysin map, f! : K (N) ! Kþd (M). where d = dim M  dim N, whose construction will be explained in the next section. 3. Homotopy If f : N ! M and g : N ! M are homotopic maps, then the pullback maps f ! = g! are equal. If in addition, f and g are K-oriented, proper maps which are homotopic via proper maps, then the Gysin maps f! = g! are equal. 4. Excision Let M1 be a closed subset of M and U be an open subset of M such that U is contained in the interior of M1 . Then the inclusion of pairs (MnU, M1 nU) ,! (M, M1 ) induces an isomorphism in K-theory, K (M, M1 ) ffi K (MnU, M1 nU). 5. Exactness Let M1 be a closed subset of M. Then there is a six-term exact sequence in K-theory, K0 ðM; M1 Þ

!

K0 ðMÞ !

" K1 ðM1 Þ

K0 ðM1 Þ

# 

K1 ðMÞ



K1 ðM; M1 Þ

6. Cup product There is a canonical map given by external tensor product, Ki (M)  Kj (N) ! Kiþj (M  N). When N = M, one can compose this with the homomorphism induced by the diagonal map M ! M  M given by x ! (x, x), to get a cup product, Kp (M)  Kq (M) ! Kpþq (M). 7. Bott periodicity This is arguably the most important property of K-theory. It says that the zerosection embedding M : M ,! M  Rn induces a ffi Gysin isomorphism, M ! : K (M)! Kþn (M  Rn ), which is given as follows. Let M : M  R n ! M and Rn : M  Rn ! Rn denote the projections

247

onto the factors, and b = ! 1 2 Kn (R n ) the Bott element, where  : {0} ,! Rn is the inclusion of the origin. Then the Bott periodicity isomorphism is given by M ! (x) = !M (x) [ !Rn (b) 2 Kþn (M  Rn ) for all x 2 K (M). Using the fact that any vector bundle over a contractible space is trivial, together with Bott’s periodicity theorem, one deduces the calculation of the K-theory of spheres. The calculation for the odd-dimensional spheres given, K0 (S2n1 ) ffi Z ffi K1 (S2n1 ), and for the even-dimensional spheres K0 (S2n1 ) ffi Z2 and K1 (S2n ) ffi {0}, for all n 1. There is a natural homomorphism of rings called the Chern character, Ch : K (X) ! H  (X, Q) which is characterized by the following axioms: 1. Naturality If f : N ! M is a smooth map, and if E is a vector bundle over M, then Ch(f ! (E)) = f (Ch(E)). 2. Additivity Ch(E  F) = Ch(E) þ Ch(F). 3. Normalization If L is the canonical line bundle over CPn which restricts to the Hopf line bundle over CP1 , then Ch(L) = exp (x), where x is the generator of H 2 (CPn , Z) ffi Z. Atiyah and Hirzebruch, cf. Karoubi (1978), also proved that the Chern character induces an isomorphism of the rings K (X)  Q and H  (X, Q). The Chern–Weil representative of the Chern character is tr(exp((i=2)E )), where E is the curvature of a Hermitian connection on E. There are many variants of K-theory, such as KO-theory, where the unitary group is replaced by the orthogonal group, which is periodic of order eight, and G-equivariant K-theory, where G is a compact Lie group. K-theory and its variants have many interesting applications such as determining the maximum number of linearly independent vector fields on spheres, which is due to Adams, cf. Karoubi (1978). We will content ourselves with the description of two important applications.

Grothendieck–Riemann–Roch Theorem for Smooth Manifolds Recall that an oriented real vector bundle E over M is said to be a spinC vector bundle if the bundle of oriented frames on E, SO(E) has a circle bundle SpinC (E) such that the restriction to each fiber yields the central extension 0 ! U(1) ! SpinC (n) ! SO(n) ! 0 that defines the group SpinC (n), where n is the rank of E. It turns out that the obstruction to the existence of a spinC structure on E is the third integral Stieffel–Whitney class of E, W3 (E) 2 H 3 (M, Z).

248 K-Theory

A generalization of Bott periodicity is the Thom isomorphism in K-theory. It says that if  : E ! M is a rank-n spinC vector bundle over M, then the zerosection embedding M : M ,! E induces a Gysin isomorphism, M ! : K (M) ffi Kþn (E), which is given as n follows. There is a canonical element M ! 1 2 K (E) called the Thom class in K-theory, which is characterized by the property that ! 1 restricts to give the Bott class on each fiber. Then the Thom isomorphism in Kþn theory is given by M ! (x) = ! (x) [ M (E) for ! 12 K all x 2 K (M). For canonical representatives of the Thom class, cf. Mathai–Quillen Formalism, or Mathai and Quillen (1986). Recall the definition of the Gysin map for smooth embeddings. Let X be a smooth, compact manifold, and Y a smooth manifold. Let h : X ! Y be a smooth embedding that is K-oriented. Since TX  TX has a canonical almost-complex structure, it follows that the normal bundle NY X = h (TY)=TX is a spinC vector bundle. If X : X ,! NY X is the zero-section embedding, then we have the Thom isomorphism  ffi þn X (NY X), where n = dim(Y) dim(X) ! : K (X) ! K is the codimension of the embedding. Upon choosing a Riemannian metric on Y, there is a diffeomorphism  from a tubular neighborhood U of h(X) onto a neighborhood of the zero section in the normal bundle ffi (X). That is, ! : K (NY X) ! K (U). For any open subset j : U ,! Y, the extension by zero defines a homomorphism j : K (U) ! K (Y). Then the Gysin map of the embedding h is defined as h! = j  !   þn X (Y), which turns out to be inde! : K (X) ! K pendent of the choices made. Next recall the definition of the Gysin map for smooth submersions. Let  : Y ! Z be a smooth submersion of smooth manifolds, which is Koriented and a proper map. Since every smooth compact manifold can be smoothly embedded in R2q for q sufficiently large, a parametrized version yields an embedding  : Y ,! Z  R2q that is spinC. Therefore the Gysin map is a homomorphism ! : K (Y) ! Kþa (Z  R2q ), where a = dim(Z) þ 2q dim(Y). Let Z : Z ,! Z  R 2q denote the zero-section embedding. Then we have the Thom isomorphism  ffi þ2q Z (Z  R2q ). Then the Gysin map ! : K (Z) ! K 1 of the submersion  is defined as ! = !  (Z ! ) :  þb K (Y) ! K (Z), where b = dim(Y)  dim(Z), and turns out to be independent of the choices made. Let f : N ! M be a smooth proper map that is K-oriented. Then f can be canonically factored, first into the smooth embedding gr(f ) : N ,! N  M, which is the graph of the function, that is, gr(f )(x) = (x, f (x)), and which is K-oriented. The Gysin map is gr(f )! : K (N) ! Kþdim(M) (N  M). Second, the projection pM : N  M ! M is a K-oriented proper submersion, when restricted to

the image of gr(f). The Gysin map is pM ! : K (M  N) ! Kþb (M), where b = dim(N). The Gysin map of f is defined as f! = pM !  gr(f )! : K (N) ! Kþd (M), where d = dim(M) þ dim(N). Given such a smooth proper map f : N ! M that is K-oriented. Then there are Gysin maps in cohomology, f : H  (N, Q) ! Hþd (M, Q) (where we consider the Z2 -grading given by even and odd degree), and in K-theory, f! : K (N) ! Kþd (M) which increases the degree by d = dim(M) þ dim(N). The Grothendieck–Riemann–Roch theorem due to Atiyah and Hirzebruch, cf. Karoubi 1978, in the smooth category can be phrased as the commutativity of the diagram, K ðNÞ ToddðTNÞ[Ch

f!

!

#

H  ðN; QÞ

Kþd ðMÞ

ToddðTMÞ[Ch

f

!

#

H þd ðM; QÞ

That is, Chðf! ðÞÞ [ ToddðTMÞ ¼ f ðChðÞ [ ToddðTNÞÞ for all  2 K (N), where Todd(E) is the Todd genus characteristic class of a Hermitian vector bundle E over M. The Chern–Weil representative of the Todd genus is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi ði=2ÞE det tanhðði=2ÞE Þ where E is the curvature of a Hermitian connection on E. There are many useful variants of this beautiful formula.

The Atiyah–Singer Index Theorem The 2004 Abel Prize citation mentions the Atiyah– Singer (1971) index theorem as being one of the greatest achievements of twentieth-century mathematics. It has stimulated considerable interaction between mathematicians and mathematical physicists. We content ourselves here with a rudimentary description of the results. Let F be the space of all Fredholm operators on an infinite-dimensional complex Hilbert space H. Recall that an operator A is said to be Fredholm if both the kernel and cokernel of A are finite dimensional. The index of such a Fredholm operator is index(A) = dim(ker(A))  dim(coker(A)) 2 Z. The index map is continuous, so it induces a map on the connected components of F , which turns out to be an isomorphism.

K-Theory

K-theory is naturally related to the space of all Fredholm operators F endowed with the norm topology. Any continuous map A : X ! F from a compact space to F has an index in K0 (X), which is given by index(A) = ker (A)  coker(A) in the special case when dim(ker(A))(x) is constant in x 2 X. In general, one uses the fact that the index is stable under compact perturbation, and shows that one can always achieve the special case after a compact perturbation. It is again the case that the index map is continuous, and so induces a map, index : [X, F ] ! K0 (X), which turns out to be an isomorphism, thanks to a fundamental theorem of Kuiper which proves that the group of all invertible operators on an infinite-dimensional complex Hilbert space is contractible in the norm topology. Now let  : N ! Z be a fiber bundle with typical fiber a smooth compact manifold M, where N and Z are also smooth compact manifolds. Consider a smooth family of elliptic operators D = {Dz }z2Z along the fibers of , parametrized by Z, where Dz : C1 (1 (z), E j 1 (z) ) ! C1 (1 (z), F j 1 (z) ) and E, F are vector bundles over N. Such a family of elliptic operators has a symbol ðDÞ :  ðEÞ !  ðFÞ where  : T (N=Z) ! N is the projection and T (N=Z) is the vertical cotangent bundle. Ellipticity for the family is the condition that (D) is an isomorphism outside the zero section, so that the triple ( (E),  (F), (D)) determines an element in K0 (T (N=Z)) denoted by (D). The analytic index of the family D is index(D) 2 K0 (Z), and it turns out that it only depends on the class of the symbol (D) 2 K0 (T (N=Z)), so the analytic index can be viewed as a homomorphism, index : K0 ðT ðN=ZÞÞ ! K0 ðZÞ Consider an embedding  : N ,! Z  Rn that is compatible with the projection  : N ! Z. The fiberwise differential is an embedding d : T(N=Z) ! Z  R 2n , which induces a Gysin map d! : K0 ðTðN=ZÞÞ ! K0 ðZ  R2n Þ upon identifying T (N=Z) with T(N=Z). Let j : Z ! Z  R2n be the inclusion j(z) = (z, 0). It induces the Bott isomorphism j! : K0 (Z) ffi K0 (Z  R2n ). The topological index of the family D is, by definition, 0 0 indext ¼ j1 !  d! : K ðT ðN=ZÞÞ ! K ðZÞ

The Atiyah–Singer (1971) index theorem for families of elliptic operators D asserts the

249

equality of the analytic index and the topological index, indexðDÞ ¼ indext ððDÞÞ 2 K0 ðZÞ Combined with the Grothendieck–Riemann–Roch theorem, one has the following exquisite formula in H  (Z, Q): ChðindexðDÞÞ ¼   fToddðTC ðN=ZÞÞ [ ChððDÞÞg where  : TC (N=Z) ! N is the projection. The map sending a complex vector bundle E over Z to its determinant line bundle det(E) = max E induces a homomorphism, det : K0 (Z) ! 0 (Pic(Z)), where 0 (Pic(Z)) denotes the isomorphism classes of complex line bundles over Z. Then c1 ðdetðindexðDÞÞÞ  ½2 ¼   fToddðTC ðN=ZÞÞ [ ChððDÞÞg where [2] denotes the degree-2 component, and the left-hand side denotes the first Chern class of the determinant line bundle of the index class. This formula is often used in the study of anomalies in physics.

K-Theory of C -Algebras The Gelfand–Naimark theorem asserts that unital abelian C -algebras A can be identified with the space of continuous functions C(X), where X is the compact Hausdorff space known as the spectrum of A, consisting of characters of A. Conversely, given a compact Hausdorff space X, the characters of C(X) consist of the evaluation maps at points of X. Let E be a vector bundle over X. Then there is a vector bundle F over X such that E  F ffi X  Cn . Setting A = C(X), M = C(X, E), N = C(X, F), we see that M  N ffi An , showing that each vector bundle E over X determines a canonical finite projective module M over A. The converse is also true and is a result of Serre and Swan, cf. Blackadar (1986), which asserts that every finite projective module M over A is the space of all continuous sections of a vector bundle over X. So we have an equivalence of the category of vector bundles over X and the category of finite projective modules over A. This motivates the following generalization of topological K-theory for a general unital C -algebra A. Let Proj(A) denote the isomorphism classes of finite projective modules over A. It is a commutative semigroup under the operation of direct sum, which can be made into the commutative group K0 (A) as follows: K0 (A) is generated by pairs ([M], [N ]), 0 together with the relation ([M], [N ]) = ([M0 ], [N ]) 0 0 if M  N  G ffi M  N  G for some [G] 2 Proj

250 K-Theory

(A). Also K1 (A) = 0 (GL(1, A)) where GL(1, A) denotes the direct limit of GL(n, A) where ðGLðn; AÞ embeds in GLðn þ 1; AÞ as 1  GLðn; AÞ: Then, defining Kj (A) = j1 (GL(1, A)) for j 1, together with generalized Bott periodicity which asserts that there is a canonical isomorphism j1 (GL(1, A)) ffi jþ1 (GL (1, A)), we see that K (A) = K0 (A)  K1 (A) is a generalized periodic cohomology theory. If A is a C -algebra without unit, then consider Aþ = A  C, with product given by (a, )(b, ) = (ab þ a þ b , ) with unit (0, 1). The projection p : Aþ ! C defined as p(a, ) = induces a map p! : K (Aþ ) ! K (C). In the nonunital case, K (A) is defined as ker(p! ). Observe that K1 (A) = K1 (Aþ ), but this is often not the case with K0 . It is easy to see that when A has a unit, then the two definitions of K0 agree. An important caveat in the case of noncommutative C -algebras is that the K-theory is often not a ring as there is no analog of the tensor product operation. Some of the basic properties of K-theory are listed as follows. Details can be found in Blackadar (1986). 1. Cup product A continuous bilinear map of C -algebras, A  B ! C, induces a cup product, Ki (A)  Kj (B) ! Kiþj (C). In particular, the continuous product A  A ! A induces a cup product homomorphism, Ki (A)  Kj (A) ! Kiþj (A). 2. Induced homomorphism If f : A ! B is a homomorphism of C -algebras, then there is an induced homomorphism, f! : K (A) ! K (B). 3. Homotopy If f : A ! B and g : A ! B are homomorphisms of C -algebras that are homotopic, the induced homomorphisms on K-theory f = g are equal. 4. Excision If I is a closed two-sided ideal in A, then there is a six-term exact sequence in K-theory, K0 ðIÞ

!

K0 ðAÞ !

" K1 ðA=IÞ

K0 ðA=IÞ

# 

K1 ðAÞ



K1 ðIÞ

5. Morita invariance The inclusion homomorphism of A into the top left of the diagonal in Mn (A) induces an isomorphism in K-theory, K (A) ffi K (Mn (A)). 6. Continuity Let A = limn ! 1 An be a C -direct limit. Then, K (A) = limn ! 1 K (An ). 7. Stability Let K be a C -algebra of all compact operators on an infinite-dimensional complex Hilbert space. Then since K = limn ! 1 Mn (C) is

a C -direct limit, we see that K (A  K) = limn ! 1 K (A  Mn (C)) = K (A). 8. Bott periodicity The continuous product A  C ! A induces the cup product Ki (A)  Kj (C) ! Kiþj (A). The computation by Bott asserts that there is a canonical element b 2 K2 (C) that gives an isomorphism K2 (C) ffi Z, and Bott periodicity asserts that the cup product with b gives rise to an isomorphism Ki (A) ffi Kiþ2j (A). We mention in passing that Connes has defined a Chern character homomorphism, Ch : K (A) ! HE (A), mapping into the entire cyclic homology of A, having similar properties as the ordinary Chern character. Due to space constraints, it will not be defined here.

A C -Algebra Generalization of the Atiyah–Singer Index Theorem and the Baum–Connes Conjecture We content ourselves here with a rudimentary account of the C -algebra generalization of the Atiyah–Singer index theorem and the Baum–Connes conjecture, and its relevance to the quantum Hall effect and strict deformation quantization. Let A be a C -algebra. Let HA = A  H, which is the analog of a Hilbert space. Let F A be the space of all A-Fredholm operators on HA . Recall that an operator T is said to be A-Fredholm if both the kernel and cokernel of T þ K are closed and finitely generated projective modules, where K is an A-compact operator. The space of A-compact operators is by definition the closure of the A-finite rank operators. The index of T is indexðTÞ ¼ ½kerðT þ KÞ  ½cokerðT þ KÞ 2 K0 ðAÞ The index map turns out to be well defined and independent of the choice of A-compact perturbation K. It is continuous, so it induces a map on the connected components of F A , which turns out to be an isomorphism, by a theorem of Mingo (cf. Rosenberg (1983, 1989)). Now let M be a smooth compact manifold. An A-vector bundle over M is a locally trivial Banach vector bundle E over M whose fibers have the structure of finitely generated left A-modules, with morphisms respecting the A-module structure. The isomorphism classes of A-vector bundles over M form a commutative semigroup under direct sums, and the associated commutative group is easily identified with K0 (C(M)  A). Let D : C1 (M, E) ! C1 (M, F) be an elliptic A-operator acting between smooth sections of A-vector bundles E, F over M. It

K-Theory

turns out that by elliptic regularity, such an operator is A-Fredholm, and has an analytic index, indexðDÞ 2 K0 ðAÞ Associated to each such operator is a symbol ðDÞ :  ðEÞ !  ðFÞ where  : T M ! M is the projection. Ellipticity is the condition that (D) is an isomorphism outside the zero section, so that the triple ( (E),  (F), (D)) determines an element in K0 (C0 (T M)  A) denoted by (D). It turns out that the analytic index of D depends only on the class (D) 2 K0 (C0 (T M)  A). Therefore, the analytic index can be viewed as a homomorphism, index : K0 ðC0 ðT MÞ  AÞ ! K0 ðAÞ Consider an embedding  : M ,! Rn , which induces an embedding d : TM ! R2n . The associated Gysin map is d! : K0 (C0 (T M)  A) ! K0 (C0 (R2n )  A). Let j : {0} ! R2n denote inclusion of the origin in R 2n . It induces a Gysin map j! : K0 (A) ! K0 (C0 (R2n )  A) which is the Bott periodicity isomorphism. Then the topological index is the homomorphism indext ¼ j1 !  d! : K0 ðC0 ðT MÞ  AÞ ! K0 ðAÞ

The C -generalization of the Atiyah–Singer index theorem due to Mishchenko–Formenko, cf. Kasparov (1988), asserts the equality of the analytic index and the topological index, indexðDÞ ¼ indext ððDÞÞ 2 K0 ðAÞ Now let M be a compact even-dimensional spinC manifold. Then there is a spinC Dirac operator D : C1 (M, Sþ ) ! C1 (M, S ), where S is the bundle of half-spinors on T M  L, where L is a line bundle over M with the property that the first Chern class of L modulo 2, c1 (L)mod 2 is equal to the second Stieffel–Whitney class of M, w2 (M). Let  be a torsion-free discrete group, and B be its classifying space. It is a paracompact space with the property that it is the quotient of  acting freely on a contractible space E. Let C r () denote the reduced group C -algebra, and consider the canonical flat C r () bundle V over B defined as follows: V ¼ fE  C r ðÞg= where  acts on the left on C r () and on the right on E. Let f : M ! B be a continuous map. Then f V is a flat C r ()-bundle over M. Upon choosing a flat connection on f V, we can couple the spinC Dirac

251

operator DV to act on sections of S  f V. The ellipticity of DV ensures that it is a C r ()-Fredholm operator, so it has an analytic index, index(DV ) 2 K0 (C r ()) by the earlier discussion, which is also equal to the topological index indext ((DV )) 2 K0 (C r ()). By Baum, Connes, and Douglas, the K-homology of B, K0 (B), is generated by the triples (M, E, f ) as described above, modulo relations that we will not present here because of space constraints. The assembly map

: K0 ðBÞ ! K0 ðC r ðÞÞ is a homomorphism given by ([(M, E, f )]) = index(DV ). The Baum–Connes conjecture asserts that is an isomorphism. There are variants of this conjecture when  has torsion. The Baum– Connes conjecture has been verified when  is an amenable group or, for instance, a word hyperbolic group. There are also variants of this conjecture for certain foliations and groupoids, and is an extremely active area of research. The injectivity of the assembly map is related to the Novikov conjecture on the homotopy invariance of the higher signatures (Kasparov 1988), and the obstructions to the existence of Riemannian metrics of positive scalar curvature on compact spin manifolds (Rosenberg 1983, 1989). A variant of the Baum–Connes conjecture, where the reduced group C -algebra is replaced by the twisted reduced group C -algebra, is used in the analysis of the noncommutative geometry approach to the integer and fractional quantum Hall effect, and also the gaps in the spectrum of magnetic Schro¨dinger operators (Bellissard et al. 1994, Marcolli and Mathai 2001).

Twisted K-theory and the Chern Character We begin by reviewing some results due to Dixmier and Douady (1963). Let M be a smooth manifold, let H denote an infinite-dimensional, separable, Hilbert space and let K be the C -algebra of compact operators on H. Let U(H) denote the group of unitary operators on H endowed with the strong operator topology and let PU(H) = U(H)=U(1) be the projective unitary group with the quotient space topology, where U(1) consists of scalar multiples of the identity operator on H of norm equal to 1. Since U(H) is contractible in the operator norm topology, it follows that PU(H) = BU(1) is an Eilenberg–MacLane space K(Z, 2). Therefore, BPU(H) is an Eilenberg– MacLane space K(Z, 3). That is, principal PU(H) bundles P over X are classified up to isomorphism by

252 K-Theory

the Dixmier–Douady class DD(P) in H 3 (X, Z) and conversely. For g 2 U(H), let Ad(g) denote the automorphism T ! gTg1 of K. As is well known, Ad is a continuous homomorphism of U(H), given the strong operator topology, onto Aut(K) with kernel the circle of scalar multiples of the identity where Aut(K) is given the point-norm topology. Under this homomorphism we may identify PU(H) with Aut(K). Define an Azumaya bundle to be a locally trivial bundle E over X with fiber K and structure group Aut(K). They are of the form KP = {P  K}= PU(H) and isomorphism classes of Azumaya bundles are also parametrized by their Dixmier–Douady class DD(P) in H 3 (X, Z) and conversely. Since K  K ffi K, the isomorphism classes of locally trivial bundles over X with fiber K and structure group Aut(K) form a group under the tensor product, where the inverse of such a bundle is the conjugate bundle. This group is known as the infinite Brauer group and is denoted by Br1 (X). So, a restatement of the Dixmier–Douady theorem is that Br1 (X) ffi H 3 (X, Z) . H 3 (X, Z) can also be described in terms of bundle gerbes (Murray 1996). The twisted K-theory, K (X, P), is defined as the K-theory of the C -algebra of continuous sections of the Azumaya bundle KP , K (C(X, KP )). It was studied in the torsion case by Donovan and Karoubi, where one can replace the compact operators K by finite-dimensional matrices, and was studied in the general case by Rosenberg (1983, 1989). Let F be the space of all Fredholm operators endowed with the norm topology. Then, one can form the bundle of Fredholm operators F P = {P  F }=PU(H), where PU(H) acts on F via the adjoint action. Consider the fibration KP ! F P ! GL(CP ), where CP = {P  C}= PU(H) and C = B(H)=K is the Calkin algebra. Since 0 (C(X, KP )) = {0}, we see that 0 (C(X, F P )) = 0 (C(X, GL(CP ))). Consider the short exact sequence of C -algebras, 0 ! CðX; KP Þ ! CðX; BP Þ ! CðX; CP Þ ! 0 where BP = {P  B(H)}=PU(H) and where PU(H) acts on B(H) via the adjoint action. It gives rise to a six-term exact sequence K0 ðCðX;KP ÞÞ ! K0 ðCðX;BP ÞÞ ! K0 ðCðX;CP ÞÞ index"

K1 ðCðX;CP ÞÞ

#

 K1 ðCðX;BP ÞÞ

 K1 ðCðX;KP ÞÞ

By definition, K1 (C(X, CP )) ffi 0 (C(X, GL(1, CP ))) and a standard argument shows that this is also equal to 0 (C(X, GL(CP ))). By Kuiper’s theorem, it is

not difficult Therefore,

to

see

that

K (C(X, BP ))= {0}.

index : 0 ðCðX; F P ÞÞ ! K0 ðX; PÞ is an isomorphism. Let X1 be a closed subset of X, and IX1 be the closed ideal of sections of KP that vanish on X1 . Then K (X, X1 , P) is by definition K (IX1 ). A geometric description of twisted K-theory in terms of modules for bundle gerbes is described in Bouwknegt et al. (2002). Some of the basic properties of twisted K-theory are listed as follows. Many of these properties follow from the corresponding properties for the K-theory of C -algebras. See Atiyah and Segal and Bouwknegt et al. (2002). 1. Normalization If P is trivial, then K (M, P) = K (M). 2. Module property K (M, P) is a module over K0 (M). 3. Pullback If f : N ! M is a continuous map, and P a principal PU(H) bundle over M, then there is a pullback homomorphism f : K (M, P) ! K (N, f(P)). 4. Push-forward Let f : N ! M be a smooth proper map between compact manifolds which is Koriented, that is, TN  f TM is a spinC vector bundle over N. Let P be a principal PU(H) bundle over M. Then there is a pushforward homomorphism, also called a Gysin map, f : K (N, f ! (P)) ! Kþd (M, P), where d = dim M  dim N. 5. Homotopy If f : N ! M and g : N ! M are homotopic maps, then the pullback maps f ! = g! are equal. If in addition, f and g are K-oriented, then the pushforward maps f! = g! are equal. 6. Excision Let M1 be a closed subset of M and U be an open subset of M such that U is contained in the interior of M1 . Then the inclusion of pairs (MnU, M1 nU) ,! (M, M1 ) induces an isomorphism in K-theory, K (M, M1 , P) ffi K (MnU, M1 nU, P j MnU ). 7. Exactness Let M1 be a closed subset of M and  : M1 ! M be the inclusion. Let P be a principal PU(H) bundle over M. Then the short exact sequence   0 ! IM1 ! CðM; KP Þ ! C M1 ; KPjM ! 0 1

gives rise to the six-term exact sequence in K-theory, K0 ðM; M1 ; PÞ ! K0 ðM; PÞ ! K0 ðM1 ; ! ðPÞÞ " # K1 ðM1 ; ! ðPÞÞ  K1 ðM; PÞ  K1 ðM; M1 ; PÞ

K-Theory

8. Cup product Let P be a principal PU(H) bundle over M and Q be a principal PU(H) bundle over N. An identification H  H ffi H gives rise to a principal PU(H) bundle P  Q over M  N whose Dixmier– Douady invariant is DD(P  Q) = p 1 (DD(P)) þ p 2 (DD(Q)), where pj denote projections onto the jth factor, j = 1, 2. Then there is a canonical map given by external tensor product, Ki ðM; PÞ  Kj ðN; QÞ ! Kiþj ðM  N; P  QÞ called the cup product. 9. Bott periodicity Let P be a principal PU(H) bundle over M. Bott periodicity says that there is a canonical isomorphism K ðM; PÞ ffi Kþn ðM  Rn ; ðPÞÞ

characterized by its first Chern class c1 (E) 2 H 2 (M, Z), in the presence of (possibly nontrivial) H-flux H 2 H 3 (E, Z). We will argue that the T-dual of E is again an oriented S1 -bundle over M, denoted ^ by E, ^S1 ! E ^ ^# M ^ 2 H 3 (E, ^ Z), such that supporting H-flux H ^ ¼  H; c1 ðEÞ

E×M Ê

E

Ê π

ChP : K ðM; PÞ ! H  ðM; HÞ

It turns out that the twisted Chern character induces an isomorphism of the rings K (M, P)  Q and H  (M, H). The Chern–Weil representative of the twisted Chern character is derived in Bouwknegt et al. (2002).

Twisted K-Theory and Duality in Type II String Theories Let E be an oriented S1 -bundle over M, S1 ! E # M

p

p

There is a natural homomorphism of rings called the twisted Chern character, which depends both on a choice of P and a de Rham representative H of DD(P),

1. Naturality If f : N ! M is a smooth map, and if x 2 K (M, P), then Chf(P) (f ! (x)) = f (ChP (x)). 2. Additivity If x, y 2 K (M, P), then ChP (x  y) = ChP (x) þ ChP (y). 3. ChP respects the K0 (M)-module structure of K0 (M, P). 4. Normalization If P is trivial, then ChP reduces to the ordinary Chern character Ch.

^ c1 ðEÞ ¼ ^ H

where  : H k (E, Z) ! Hk1 (M, Z) and, similarly,  denote the pushforward maps. Then we can form the following commutative diagram:

where  : M  Rn ! M is the projection onto the first factor. Let b 2 Kn (Rn ) be the Bott element. Then the isomorphism above is given by ! (x) [ b 2 Kþn (M  Rn , ! (P)) for all x 2 K (M, P).

Here H  (M, H) denotes the twisted cohomology, which is by definition the cohomology of the complex ( (M), d  H^). The twisted Chern character is characterized by the following axioms:

253

π M

^ is a circle bundle The correspondence space E M E ^ and it is also over E with first Chern class  (c1 (E)), ^ a circle bundle over E with first Chern class  (c1 (E)), by the commutativity of the diagram ^ = E or if E ^ = M  S1 , then the corresponabove. If E ^ dence space E M E is diffeomorphic to E  S1 . T-duality gives an isomorphism of the twisted ^ as well as an isomorphism K-theories of E and E ^ and between the twisted cohomologies of E and E, can be expressed in the following commutative diagram: K ðE; PÞ ChP

T

!

#

H  ðE; HÞ

^ PÞ ^ Kþ1 ðE;

#ChP^ T

^ HÞ ^ ! H þ1 ðE;

where the horizontal arrows are isomorphisms. Here P is a principal PU(H) bundle over E such that ^ is a principal PU(H) bundle over DD(P) = H and P ^ ^ = H. We refer to Bouwknegt E such that DD(P) et al. (2004) for details. The T-duality isomorphism above gives compelling evidence that a type IIA string theory A on a circle bundle of radius R in the presence of a background H-flux, and a type IIB string theory B on a ‘‘T-dual’’ circle bundle of radius

254 K-Theory

1/R in the presence of a ‘‘T-dual’’ background H-flux, are equivalent in the sense that the string states of string theory A are in canonical one-to-one correspondence with the string states of string theory B. We briefly mention two other applications of twisted K-theory. Consider the adjoint action of a compact connected simple Lie group G on itself, and the corresponding twisted G-equivariant K-theory, twisted by a multiple of the generator of H 3 (G, Z). The relevance of the equivariant case to conformal field theory was highlighted by the result of Freed, Hopkins and Teleman (see Freed (2002)) that it is graded isomorphic to the Verlinde algebra of G, with a shift given by the dual Coxeter number. Here the Verlinde algebra consists of equivalence classes of positive-energy representations of the loop group of G which was originally shown to be a ring in a rather nontrivial way. On the other hand, the ring structure of the twisted G-equivariant K-theory of G is just induced by the product on G, which makes this result all the more remarkable. Fractional analytic index theory, developed in Mathai et al. is a generalization of Atiyah–Singer index theory, assigning a fractional-valued analytic index to each projective elliptic operator on a compact manifold, where the fraction need not be an integer. These projective elliptic operators act on projective vector bundles, where the usual compatibility condition on triple overlaps to give a global vector bundle, may fail by a scalar factor. These are the geometric objects in twisted K-theory, when the twist is torsion. In Mathai et al., a fractional index theorem is proved, computing the fractional-valued analytic index of projective elliptic operators essentially in terms of topological data. The Dirac operator in the absence of a spin structure is also defined there for the first time resolving a long standing mystery, and its index is computed. Some topics not covered in this brief account of K-theory include: KK-theory, cf. Blackadar (1986) and Kasparov (1988), which is natural setting for the Atiyah–Singer index theorem and its generalizations, as well as higher algebraic K-theory.

See also: C*-Algebras and Their Classification; Characteristic Classes; Cohomology Theories; Equivariant Cohomology and the Cartan Model; Gerbes in Quantum Field Theory; Index Theorems; Intersection Theory; Mathai–Quillen Formalism; Spectral Sequences.

Further Reading Atiyah MF and Segal G Twisted K-theory, math.KT/0407054. Atiyah MF and Singer IM (1971) The index of elliptic operators, IV. Annals of Mathematics 93: 119–138. Bellissard J, van Elst A, and Schulz-Baldes H (1994) The noncommutative geometry of the quantum Hall effect. Journal of Mathematical Physics 35: 5373–5451. Blackadar B (1986) K-Theory for Operator Algebras. In: Mathematical Sciences Research Institute Publications, vol. 5, viiiþ338 pp. New York: Springer. Bouwknegt P, Carey A, Mathai V, Murray MK, and Stevenson D (2002) Twisted K-theory and the K-theory of bundle gerbes. Communications in Mathematical Physics 228(1): 17–49. Bouwknegt P, Evslin J, and Mathai V (2004) T-duality: topology change from H-flux. Communications in Mathematical Physics 249: 383 (hep-th/0306062). Bouwknegt P, Evslin J, and Mathai V (2004b) On the topology and flux of T-dual manifolds. Physical Review Letters 92: 181601. Dixmier J and Douady A (1963) Champs continues d’espaces hilbertiens at de C -alge`bres. Bull. Soc. Math. France 91: 227–284. Freed DS (2002) Twisted K-theory and loop groups. Proceedings of the International Congress of Mathematicians (Beijing, 2002) vol. III pp. 419–430. Karoubi M (1978) K-theory. An Introduction Grundlehren der Mathematischen Wissenschaften, Band 226, xviiiþ308 pp. Berlin: Springer. Kasparov GG (1988) Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91(1): 147–201. Marcolli M and Mathai V (2001) Twisted index theory on good orbifolds, II: fractional quantum numbers. Communications in Mathematical Physics 217(1): 55–87. Mathai V, Melrose RB, and Singer IM, The fractional analytic index, math.DG/0402329. Mathai V and Quillen DG (1986) Superconnections, Thom classes and equivariant differential forms. Topology 26: 85–110. Murray MK (1996) Bundle gerbes. J. London Math. Soc. 54: 403–416. Rosenberg J (1983) C -algebras, positive scalar curvature, and the Novikov conjecture. Inst. Hautes tudes Sci. Publ. Math 58: 197–212. Rosenberg J (1989) Continuous-trace algebras from the bundle theoretic point of view. J. Austral. Math. Soc. Ser. A 47(3): 368–381.

L Lagrangian Dispersion (Passive Scalar) G Falkovich, Weizmann Institute of Science, Rehovot, Israel ª 2006 Elsevier Ltd. All rights reserved.

Introduction To describe transport by a random flow, one needs to apply the statistical methods to the motion of fluid particles, that is, to the Lagrangian dynamics. We first present the propagators describing evolving probability distributions of different configurations of fluid particles. We then use those propagators to describe decay and steady states of a passive scalar field transported by random flows. Consider an evolution of a passive scalar tracer (r, t) in a random flow. The mean value of the scalar tracer at a given point is an average over values brought by different trajectories: Z ½1 hðr; sÞi ¼ Pðr; s; R; 0Þ ðR; 0Þ dR Here, P(r, s; R, t) is the probability density function (PDF) to find the particle at time t at position R given its position r at time s. That PDF is called the propagator or the Green function. Multipoint correlation functions of the tracer CN ðr; sÞ  hðr 1 ; sÞ . . . ðr N ; sÞi Z ¼ P N ðr; s; R; 0Þ ðR1 ; 0Þ . . . ðRN ; 0Þ dR

½2

are expressed via the multiparticle Green functions P N which are the joint PDFs of the equal-time positions R = (R1 , . . . , RN ) of N fluid trajectories. The trajectory of the fluid particle that passes at time s through the point r is described by the vector R(t; r, s) which satisfies R(t; r, t) = r and the stochastic equation R_ ¼ vðR; tÞ þ uðtÞ

½3

Here, u(t) describes the molecular Brownian motion with zero average and covariance hui (t)uj (t0 )i = 2ij (t  t0 ). We also consider macroscopic velocity v as random with various statistical properties

in space and time. There is a clear scale separation between macroscopic velocity v and molecular diffusion u that allows one to treat them separately. Using [3], one can write the Green’s function as an integral over paths that satisfy q(s) = r and q(t) = R:  Z t Z _ DpDq exp  {pðÞ  ½qðÞ Pðr; s; R; tÞ ¼ s



 vðqðÞ; Þ  uðÞ d

½4 v;u

¼

Z

 Z t _ DpDq exp  ½{pðÞ  ðqðÞ s



2

 vðqðÞ; ÞÞ þ p ðÞ d

½5 v

¼

Z

 Dq exp



 vðqðÞ; Þ2 d

1 4 

Z

t

_ ½qðÞ

s

v

¼ hPðr; s; R; tjvÞiv

½6

The integration over the auxiliary field p in [4] enforces the delta function of [3]. One passes from [4] to [5] by averaging over the Gaussian Brownian noise, and from [5] to [6] by calculating Gaussian integral over p. Generally, exact calculations are only possible for Gaussian random processes short-correlated in timelike in [5]. The simplest case is the Brownian motion when the advection is absent. One then obtains from [6] the Gaussian PDF of the displacement: 2

PðR; tÞ ¼ ð4tÞd=2 eR

=ð4tÞ

½7

which satisfies the heat equation (@t  r2 ) P(r, t) = 0. The short-correlated case is far from being an exotic exception but rather presents a long-time limit of an integral of any finite-correlated random function. Indeed, such an integral can be presented as a sum of many independent equally distributed random numbers,

256 Lagrangian Dispersion (Passive Scalar)

the statistics of such sums is a subject of the central limit theorem. One can move beyond the central limit theorem considering the correlation time finite (yet small comparing to the time of evolution). Such generalization is the subject of the large deviation theory. Consider some quantity X which is an integral of some random function over time t much larger than the correlation time . At t  , X behaves as a sum of many independent P identically distributed random numbers yi : X = N 1 yi with N / t=. The generating function hezX i of the moments of X is the product, hezX i= eNS(z) , where we have denoted hezy i  eS(z) (assuming that the generating function hezy i exists for all complex z). The PDF P(X) R is given by the inverse Laplace transform (2i)1 ezXþNS(z) dz with the integral over any axis parallel to the imaginary one. For X / N, the integral is dominated by the saddle point z0 such that S0 (z0 ) = X=N and PðXÞ / eNHðX=NhyiÞ

½8

Here H = S(z0 ) þ z0 S0 (z0 ) is the function of the variable X=N  hyi; it is called entropy function as it appears also in the thermodynamic limit in statistical physics. A few important properties of H (also called rate or Crame´r function) may be established independently of the distribution P(y). It is a convex function which takes its minimum at zero, that is, for X equal to the mean value hXi = NS0 (0). The minimal value of H vanishes since S(0) = 0. The entropy is quadratic around its minimum with H 00 (0) = 1 , where  = S00 (0) is the variance of y. We thus see that the mean value hXi = Nhyi grows linearly with N. The fluctuations X  hXi on the scale O(N 1=2 ) are governed by the central limit theorem that states that (X  hXi)=N 1=2 becomes for large N a Gaussian random variable with variance hy2 i  hyi2   as in [7]. Finally, its fluctuations on the larger scale O(N) are governed by the large deviation form [8]. The possible non-Gaussianity of the y’s leads to a nonquadratic behavior of H for (large) deviations from the mean, starting from X  hXi=N ’ =S000 (0). Note that if y is Gaussian, then X is Gaussian too for any t, but the universal formula [8] with H = (X  Nhyi)2 =2N is valid only for t  .

Single-Particle Diffusion For the pure advection without noise, the displacement ofR the single Lagrangian trajectory is t R(t)  R(0) = 0 V(s) ds, with V(t) = v(R(t), t) being the Lagrangian velocity. One can show that V(t) is statistically stationary in the frame of reference with no mean flow and under statistical homogeneity and

stationarity of the incompressible Eulerian velocities. For  = 0, the mean square displacement satisfies the equation Z t d 2 h½RðtÞ  Rð0Þ i ¼ 2 hVð0Þ  VðsÞi ds ½9 dt 0 The behavior of the displacement is crucially dependent on the Lagrangian correlation time  of V(t) defined by Z 1 hVð0Þ  VðsÞi ds ¼ hv2 i ½10 0

No general relation between the Eulerian and the Lagrangian correlation times has been established, except for the case of short-correlated velocities. For times t  , the two-point function in [9] is approximately equal to hV(0)2 i = hv2 i. The fluid particle transport is then ballistic with h[R(t)  R(0)]2 i ’ hv2 it2 and the PDF P(R, t) is determined by the whole single-time velocity PDF. When the correlation time of V(t) is finite (a generic situation in a turbulent flow where  is of order of a large-scale turnover time), an effective diffusive regime is expected to arise for t   with h(R(t)  R(0))2 i ’ 2hv2 it. Indeed, the particle displacements over time segments much larger than  are almost independent. At long times, the displacement R(t) behaves then as a sum of many independent variables and falls into the class of stationary processes treated in the previous section. In other words, R(t) for t   becomes a Brownian motion in d dimensions, normally distributed with hRi (t)Rj (t)i ’ Dije t, where the so-called eddy diffusivity tensor is as follows: Z 1 1 Dije ¼ hVi ð0ÞVj ðsÞ þ Vj ð0ÞVi ðsÞi ds ½11 2 0 The symmetric second-order tensor Dije is the only characteristics of the velocity which matters in this limit of t  . The trace of the tensor is equal to hv2 i, that is, equal to the large-time value of the integral in [9], while its tensorial properties reflect the rotational symmetries of the advecting velocity field. If the latter is isotropic, the tensor reduces to a diagonal form characterized by a single scalar value De . The main problem of turbulent diffusion is to obtain the effective diffusivity tensor given the velocity field v and the value of the molecular diffusivity .

Two-Particle Dispersion in Smooth Flows Even when velocity v(R, t) is a smooth function of the coordinates, Lagrangian dynamics can be quite

Lagrangian Dispersion (Passive Scalar)

complicated. Indeed, d ordinary differential equations R_ = v(R, t) generally produce chaotic dynamics (for d  3 already for steady flows and for d = 2 for timedependent flows). The tools for the description of what is called chaotic advection are similar to those of the theory of dynamical chaos. The description consistently exploits two simple ideas: to single out the variables that can be represented by the sum of a large number of independent random quantities and to separate variables that fluctuate on different timescales. The distance, R12 = R1  R2 , between two fluid particles with trajectories Ri (t) = R(t; r i ) passing at t = 0 through points r i satisfies the equation R_ 12 ¼ vðR1 ; tÞ  vðR2 ; tÞ

½12

If the velocity field can be considered smooth on the scale R12 , then one expands v(R1 , t)  v(R2 , t) = (t, R1 )R12 , introducing the strain matrix  which can be treated as independent of R12 . The distance thus satisfies locally a linear system of ordinary differential equations (we omit subscripts replacing R12 by R) _ RðtÞ ¼ ðtÞRðtÞ

½13

This equation, with the strain treated as given and R(0) = r, may be explicitly solved for arbitrary (t) only in the 1D case Z t ln½RðtÞ=r ¼ ln WðtÞ ¼ ðsÞ ds  X ½14 0

When t is much larger than the correlation time  of the strain, the variable X is a sum of N independent equally distributed random numbers with N = t= and one can apply [8]. In the multidimensional case, to use the large deviation theory, one introduces the evolution matrix W such that R(t) = W(t)R(0). The modulus R is expressed via the positive symmetric matrix W T W. In almost every realization of the strain, the matrix t1 ln W T W stabilizes at t ! 1, that is, its eigenvectors tend to d-fixed orthonormal eigenvectors f i . To understand that intuitively, consider some fluid volume, say a sphere, which evolves into an elongated ellipsoid at later times. As time increases, the ellipsoid is more and more elongated and it is less and less likely that the hierarchy of the ellipsoid axes will change. The limiting eigenvalues i ¼ lim t1 ln jWf i j t!1

½15

are called Lyapunov exponents. The major property of the Lyapunov exponents is that they are realization independent if the flow is ergodic (i.e., spatial and temporal averages coincide). The relation [15] states that two fluid particles separated initially by r

257

pointing into the direction f i will separate (or converge) asymptotically as exp (i t). ThePincompressibility constraints det (W) = 1 and i = 0 imply that a positive Lyapunov exponent will exist whenever at least one of the exponents is nonzero. Consider indeed EðnÞ ¼ lim t1 lnh½RðtÞ=rn i t!1

½16

whose derivative at the origin gives the largest Lyapunov exponent 1 . The function E(n) obviously vanishes at the origin. Furthermore, E(d) = 0, that is, incompressibility and isotropy make that hRd i is time independent as t ! 1. Apart from n = 0, d, the convex function E(n) cannot have other zeroes if it does not vanish identically. It follows that dE=dn at n = 0, and thus 1 , is positive. A simple way to appreciate intuitively the existence of a positive Lyapunov exponent is to consider the saddle-point 2D flow vx = x, vy = y with the axes randomly rotating after time interval T. A vector initially at the angle with the x-axis will be stretched after time T if cos  [1 þ exp (2T)]1=2 , that is, the measure of the stretching directions is larger than 1=2. A major consequence of the existence of a positive Lyapunov exponent for any random incompressible flow is the exponential growth of the interparticle distance R(t). In a smooth flow, it is also possible to analyze the statistics of the set of vectors R(t) and to establish a multidimensional analog of [8]. The idea is to reduce the d-dimensional problem to a set of d scalar problems for slowly fluctuating stretching variables excluding the fast fluctuating angular degrees of freedom. Consider the matrix I(t) = W(t)W T (t), representing the tensor of inertia of a fluid element such as the above-mentioned ellipsoid. The matrix is obtained by averaging Ri (t)Rj (t)d=‘2 over the initial vectors of length ‘ and I(0) = 1. Introducing the variables that describe stretching as the lengths of the ellipsoid axis e2 1 , . . . , e2 d , one can deduce similarly to [8] the asymptotic PDF: Pð 1 ; . . . ; d ; tÞ / exp½t Hð 1 =t  1 ; . . . ; d1 =t  d1 Þ ð 1  2 Þ . . . ð d1  d Þ ð 1 þ    þ d Þ

½17

The entropy function H depends on the statistics of . In the -correlated case, H is everywhere quadratic: HðxÞ / d1

d X i¼1

x2i ;

i / dðd  2i þ 1Þ

½18

258 Lagrangian Dispersion (Passive Scalar)

Two-Particle Dispersion in Nonsmooth Flows To consider dispersion in the inertial interval of turbulence, one should assume v(r, t)j / r , where generally  < 1. Rewriting then eqn [12] for the distance between two particles as R_ = v(R, t), we infer that dR2 =dt = 2R  v(R, t) / R1þ . It suggests RðtÞ1  Rð0Þ1 / t 1=(1)

½19

For large t, R(t) / t , with the dependence of the initial separation quickly forgotten. Of course, for the random process R(t), relation [19] is of the mean-field type and should pertain (if true) to the large-time behavior of the averages ðhR(t)p i / tp=(1) , for p > 0Þ implying their super-diffusive growth, faster than the diffusive one / tp=2 . The power-law scaling may be amplified to the scaling behavior of the PDF of the interparticle distance, P(R, t) = P(R, 1 t). The power-law growth of the second moment, hR(t)2 i / t3 , is the celebrated Richardson dispersion relation, which was the first quantitative phenomenological prediction in developed turbulence. It seems to be confirmed by experimental data and the numerical simulations. It is important to remark that, even assuming the validity of the Richardson relation, it is impossible to establish general large-time properties of the PDF P(R; t) such as those for the single-particle PDF of the distance between two particles. This is because the correlation time of the Lagrangian velocity difference, R=v(R) / hR2 i1=3 / t, is comparable with the total time of the process. It is instructive to contrast the exponential growth [16] of the distance between the trajectories with the power-law growth [19]. In a smooth flow, the closer two trajectories are initially, the more time is needed to effectively separate them. In a nonsmooth turbulent flow, the trajectories separate in a finite time independent of their initial distance R(0), provided that the latter is also in the inertial range. This explosive separation of trajectories results in a breakdown of the deterministic Lagrangian flow since the trajectories cannot be labeled by the initial conditions. That agrees with the fundamental theorem stating that the ordinary differential equation R_ = v(R, t) does not have unique solution if v(r, t) is non-Lipschitz. As shown by the example of the equation x_ = jxj with two solutions x = [(1  )t]1=(1) and x = 0 both starting at zero, one should expect multiple Lagrangian trajectories starting or ending at the same point for velocity fields with  < 1. Even though the deterministic Lagrangian description breaks down, the statistical description is still possible and one can make

sense of propagators like P(r, s; R, tjv). They are expected to be weak solutions of the equation [@t  r  v(R, t)]P(r, s; R, tjv) = 0 in the nonsmooth case. According to this assumption, the Lagrangian trajectories behave stochastically already in a given velocity field and for negligible molecular diffusivity – and not only due to a random noise or to random fluctuations of the velocities. The general conjecture about the existence and diffuse nature of propagators is known to be true for the Gaussian ensemble of velocities decorrelated in time (Kraichnan 1968): hvi ðr; tÞvj ðr 0 ; t0 Þi ¼ 2ðt  t0 ÞDij ðr  r 0 Þ

½20

Here the Lagrangian velocity v(R, t) has the same white noise temporal statistics as the Eulerian velocity v(r, t) for fixed r and the displacement along a Lagrangian trajectory R(t)  R(0) is a Brownian motion for all times. To model nonsmooth velocity field of turbulence, we choose Dij (r) = D0 ij  (1=2)dij (r) and dij ðrÞ ¼ D1 ½ðd  1 þ Þij r  r i r j r 2 

½21

Here D0 gives the eddy diffusivity of a single fluid particle (discussed earlier), whereas dij (r) describes the statistics of the velocity differences. For 0 < < 2, the Kraichnan ensemble is supported on the velocities that are Ho¨lder continuous in space with a fixed exponent  arbitrarily close to =2. It mimics this way the main property of turbulent velocities. The rough (distributional) behavior of Kraichnan velocities in time, although not very physical, is not expected to modify essentially the qualitative properties of propagators (it is the spatial regularity, not the temporal one, of a vector field that is crucial for the uniqueness of its trajectories). In exactly the same way as one derives [6] and [7] ^ 1=2 (4t)d=2 e ij Ri Rj =4t , from [4], one gets P(R, t) = j j 1 ^ where ( )ij = Dij (0) þ ij . In much the same way one can examine the two-particle PDF. The PDF P 2 (r, s; R, t) of the distance R between two particles satisfies the equation ð@t  M2 ÞP 2 ðr; s; R; tÞ ¼ ðt  sÞðr  RÞ

½22

where M2 = D1 (d  1)r1d @r r d1þ @r and [22] can be readily solved: lim P 2 ðr; s; R; tÞ /

Rd1

jt  sjd=ð2 Þ   R2 exp const: jt  sj

r!0

½23

Lagrangian Dispersion (Passive Scalar)

That confirms the diffusive character of the limiting process describing the Lagrangian trajectories in fixed non-Lipschitz velocities: the endpoints of the process stay at finite distance when the initial points converge. The PDF [23] changes from Gaussian to log–normal when changes from 0 to 2. The Richardson dispersion hR2 (t)i / t3 is reproduced for = 4=3.

The long-time asymptotics of the propagators in the nonsmooth case can be found explicitly for the Kraichnan ensemble of velocities: ð@t þ MN ÞP rel N ðr; s; R; tÞ ¼ ðt  sÞðR  rÞ

MN ¼

X n 2 > 0 > 3 . During the time less than td = j3 j1 ln (L=rd ), diffusion is unimportant and  inside the spot does not change. At larger time, the dimensions of the spot with negative Lyapunov exponents are frozen at rd , while the rest keep growing exponentially, resulting in an exponential growth of the total volume exp ( 1 þ 2 ). That leads to an exponential decay of scalar moments averaged over velocity statistics: h[(t)]N i / exp (N t). The decay rates N can be expressed via the PDF [18] of stretching variables i . Since  decays as the inverse volume, Z h½ðtÞN i / d 1 d 2 exp½tHð 1 =t  1 ; 2 =t  2 Þ  Nð 1 þ 2 Þ

½30

At large t, the integral is determined by the saddle point. At small N, the saddle point lies within the parabolic domain of H so N increases with N quadratically. At large N, the main contribution is due to the realization with smallest possible spot of size L so N saturates. For the decay in incompressible nonsmooth flow, using the Kraichnan model one gets Z

 h2n ðtÞi ¼ P 2n ð0; R; 1Þ C2n t1=ð2 Þ R; 0 dR ½31 R When J0 = C2 (r, t) dr 6¼ 0, the function td=(2 ) C2 (t1=(2 ) r, 0) tends to J0 (r) in the long-time limit and [31] is reduced to h2n ðtÞi ð2n  1Þ!! J0n tnd=ð 2Þ Z P 2n ð0; R1 ; R1 ; . . . ; Rn ; Rn ; 1Þ dR ½32 The decay is self-similar: P(t, ) = td=2(2 ) pffiffi Q(td=2(2 ) ). That means that the PDF of =  is asymptotically time independent, with (t) = h(r)2 i being time-dependent (decreasing) dissipation rate. This should be contrasted with the lack of self-similarity for the smooth case.

One can also consider steady state of  pumped by a source (r, t): @t  þ ðv  rÞ þ  ¼

½33

Assuming that pumping is white Gaussian with a zero mean and variance (r 1 , t1 ) (r 2 , t2 ) = (r 12 ) (t2  t1 ), r ij = r i  r j , one can express the correlation functions via the multiparticle propagators. For example, assuming zero conditions at the distant past and space homogeneity, one gets Z Z t 0 PðR; r; t0 ÞðRÞ dR C2 ðr; tÞ ¼ dt ½34 1

The function (R) is nonzero within the correlation scale L of the pumping which restricts integration to R(t) < L. For smooth velocity, this gives F2 (r) = j3 j1 (0) ln (L=r) at r < L. For nonsmooth velocity, the statistics of scalar fluctuations at small scales is described by the set of structure functions SN (r)  h[(r)  (0)]N i / rN with the scaling exponents determined by the zero modes (see Falkovich et al. (2001)). Therefore, existence of Lagrangian statistical invariants explains the anomalous scaling of passive scalar (here, anomaly means that scale invariance broken by pumping is not restored even when the pumping scale goes to infinity). See also: Anomalies; Intermittency in Turbulence; Large Deviations in Equilibrium Statistical Mechanics; Lyapunov Exponents and Strange Attractors; Random Walks in Random Environments; Stochastic Differential Equations; Turbulence Theories.

Further Reading Ellis R (1995) Entropy, Large Deviations, and Statistical Mechanics. Berlin: Springer. Falkovich G, Gawedzki K, and Vergassola M (2001) Particles and fields in fluid turbulence. Reviews of Modern Physics 73: 913–975. Kraichnan RH (1968) Small-scale structure of a scalar field convected by turbulence. Physics of Fluids 11: 945–963. Majda A and Kramer PR (1999) Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Physics Reports 314: 237–574. Monin A and Yaglom A (1979) Statistical Fluid Mechanics. Boston: MIT Press. Pope SB (1994) Lagrangian PDF methods for turbulent flows. Annual Review of Fluid Mechanics 26: 23–63. Zinn-Justin J (1989) Quantum Field Theory and Critical Phenomena. Oxford: Science Publishing.

Large Deviations in Equilibrium Statistical Mechanics

261

Large Deviations in Equilibrium Statistical Mechanics The ‘‘grand canonical Gibbs measure’’ , , T in  with boundary condition  at inverse temperature  = T 1 is given by

S Shlosman, Universite´ de Marseille, Marseille, France ª 2006 Elsevier Ltd. All rights reserved.

;;T ðÞ ¼ Z1 ;;T expðH; ðÞÞ

½3

where

Introduction Large deviation theory (LDT) deals with the study of probabilities of extremely rare events. As an example, consider the case of independent identically distributed random variables 1 , . . . , N with the mean value E(i ) = m. Then the typical deviations of the sum MN = 1 þ  p   ffiffiffiffi þffi N from its mean value Nm are of the order of N , while in LDT we study the probabilities of the deviations which are linear in N. In ‘‘good’’ cases we know that for b > 0 PrfMN  Nm  bNg  expfIðbÞNg as N ! 1

½1

where I() > 0 is the ‘‘rate’’ function. Questions of LDT are very natural in statistical mechanics, and they have deep physical meaning, notwithstanding the fact that the corresponding events are rare. One reason is that (some) rare events in the grand canonical ensemble become typical events in the canonical ensemble. An interesting feature of LDT in statistical mechanics is that the behavior [1] of LD is not universal, and sometimes is replaced by a nonclassical one:   PrfMN  Nm  bNg  exp ~IðbÞN  ½2 with  < 1. That usually happens in the ‘‘phase transition’’ regime, and then the quantity ~I(b), as well as the exponent , have very much to do with the geometry of a droplet of one phase formed inside the other. Below, we will illustrate all these features on the example of the Ising model.

The Ising Model in the Finite Box Our random variables x will take values 1, with x 2 Zd . They are called spins. For every finite box   Zd , we will define Gibbs states in . To do this we need the Hamiltonians X X H; ðÞ ¼  x y  x y x;y n:n: x;y2

x;y n:n: x2;y62

Here,  is some spin configuration on Zd , which is called ‘‘boundary condition,’’ while  2  is any spin configuration in .

Z;;T ¼

X

expðH; ðÞÞ

2

is called ‘‘partition function’’; it makes the measure [3] to be a probability distribution. The boundary condition  þ1(1) will be denoted by þ(). For every value of T, the Gibbs measures (l),, T with ()-boundary condition in the cubic box (l) of size l converge, as l ! 1, to the probability measures that we will denote by , T . If the two happen to be different, then þ, T is called the (þ)-phase, and , T the ()-phase. That happens to be the case iff the temperature T is lower than the critical temperature Tc = Tc (d). The critical temperature depends on dimension; Tc (1) = 0, while Tc (d) > 0 for d  2. The expectation Eþ;T ð0 Þ mðÞ is called spontaneous magnetization; m() > 0 iff  > Tc1 .

LD Properties of the Gibbs States (l), , T In what follows, we will discuss the LD properties of the sum M = 1 þ    þ jj , where the spins x , x 2 , are distributed according to the Gibbs state , , T . Note that E, T (0 ) =  m(). Classical Case

If we look on the LDs of the sum M when the temperature T is high enough (in which case the limiting states þ, T and , T coincide), or else if the temperature is low, and the deviations are negative – that is, we consider the events M þ jjm(T 1 ) bjj with b < 0 – then their probabilities behave classically: There exists a (high) temperature T0 such that if T > T0 , then    1  Pr M þ jjm T 1 bjj !Zd jj lim

¼ IT ðbÞ

for

b 0

½4

   1  Pr M þ jjm T 1  bjj !Zd jj lim

¼ IT ðbÞ

for

b0

½5

262 Large Deviations in Equilibrium Statistical Mechanics

where the function IT (b)  0 is strictly concave on the segment (m(T 1 )  1, m(T 1 ) þ 1). It vanishes at only one point b = 0. There exists a (low) temperature T1 such that if T < T1 , then the relation [4] holds with the function IT (b) > 0 strictly concave on the segment (m(T 1 )  1, 0). The limit [5] also does exist, but it can vanish once we are in the phase transition region. In order to see some nontrivial behavior, we have to change the normalization 1=jj in [5]. Nonclassical Case

The proper normalization happens to be the surface term, 1=jj(d1)=d : There exists a temperature T1 such that if T < T1 , then lim

  1   Pr M þ  T  b  j jm j j  ðd1Þ=d 1

jj ¼ W T ðbÞ for

!Zd

b>0

½6

The function W T (b) obeys W T (b) = b(d1)=d wT , with wT > 0, provided the value b > 0 is not too large: b b(d), where b(d) is some constant, depending on the dimension and temperature; one can show that b(d)  1=2d . For larger b’s the dependence is more complex. The key object here is the constant wT . To obtain it, one has to solve the following variational problem. Let T (),  2 Sd1 be the surface tension between the (þ)-phase and the ()-phase of the Ising model at the temperature T. Then, for every closed compact (hyper)surface Md1  Rd , we define its surface energy as Z W T ðM Þ ¼ T ðs Þ ds M

Moderate Deviations and the Droplet Condensation The reason behind the different order of the probabilities of the events M þ jjm(T 1 )

bjj, b < 0, and M þ jjm(T 1 )  bjj, b > 0, at low temperatures is the following. A typical configuration contributing to the first event contains many small droplets of ()-spins, of size lnjj, floating in the sea of (þ)-spins. On the contrary, in the case of the second event a typical configuration contains, in addition to small droplets, one large droplet of the size of . It has a random shape, but in the limit  ! Zd that shape converges to a nonrandom one, which happens to be the Wulff shape WT . (The precise meaning of that statement depends on dimension; in case d = 2 the convergence holds in the Hausdorff metrics, while in higher dimensions it is known only in L1 sense.) That statement makes the following question natural: consider the event M  EðM Þ  jj ;

0< d=(d þ 1), then any typical configuration contains, in addition to microscopic droplets, one large droplet of volume  jj . Therefore, the condensation happens at the value = d=(d þ 1). This picture has its counterpart in the behavior of the probabilities of ‘‘moderate deviations’’ (MD), that is, events when M þ jjm(T 1 )  jj : independent fluctuations of sizes of many small droplets, and the usual Gaussian behavior holds: PrfM  EðM Þ  jj g ( ) 2 n o ðjj Þ  exp  ¼ exp cjj2 1 2VarðM Þ

Large-N and Topological Strings 263

 if > d=ðd þ 1Þ, then the deviation is due to the formation of a large droplet, and so n o PrfM  EðM Þ  jj g  exp c0 jj ððd1Þ/dÞ Note that the two estimates match at = d=(d þ 1).

Other Questions There are many related questions; some are partially solved, others are widely open, if considered on a rigorous mathematical level. One can ask about the asymptotic behavior of probabilities of the events like M  EðM Þ ¼ b where the values b lie in the LD or MD region. The difference between such questions and those treated above is of the same nature as the difference between the integral and the local limit theorems. Partial answers to them are given in Dobrushin and Shlosman (1994). Many results about the Wulff shape and its relation to the Ising model are known, starting by Dobrushin et al. (1992). Some are still challenging. One such question concerns the so-called roughening phase transition. It is known rigorously that the

Wulff shape WT in the d  3 Ising model has flat facets at low temperatures T. It is believed that such a feature holds true only for T < TR , where the roughening temperature TR is strictly less than the critical temperature Tc (d) for d = 3. At the temperatures T 2 (TR , Tc (3)), the Wulff shape WT does not have facets. This conjecture seems to be very difficult. The question about the typical behavior of the MD of the Ising model at the threshold value M  E(M )  jjd=(dþ1) was recently answered in Biscup et al. (2003).

Further Reading Biscup M, Chayes L, and Kotecky R (2003) Critical region for droplet formation in the two-dimensional Ising model. Communications in Mathematical Physics 242: 137–183. Dobrushin RL and Shlosman SB (1994) Large and moderate deviations in the Ising model. In: Dobrushin RL (ed.) Probability Contributions to Statistical Mechanics, Advances in Soviet Mathematics, vol. 18, pp. 91–220. Providence, RI: American Mathematical Society. Dobrushin RL, Kotecky R, and Shlosman SB (1992) Wulff Construction: A Global Shape from Local Interaction. AMS Translations Series. Providence, RI: American Mathematical Society.

Large-N and Topological Strings R Gopakumar, Harish-Chandra Research Institute, Allahabad, India ª 2006 Elsevier Ltd. All rights reserved.

Introduction Topological strings have been well studied since they were introduced in the early 1990s. Essentially, they are simplified string theories that capture the information about a sector of the full (or ‘‘physical’’) string theory. Thus, while sharing many of the structural features of usual string theory, they hold out the possibility of being amenable to explicit calculations. This is especially true with regard to stringy quantum corrections (the higher genus contributions from the point of view of the string world sheet), which are normally rather intractable in the full physical string theory. This has allowed them to play a useful role in enhancing the understanding of string theory and many of its mysterious quantum properties, such as the various dualities.

In particular, in the last several years, topological strings have served as an important laboratory for testing and understanding the connection between the large-N expansion of gauge theories and closedstring theories. In this article we will sketch how this connection is illustrated in a duality between large-N Chern–Simons gauge theory and closed topological string theories. We will survey the origin and current status of these developments and indicated some of its remarkable mathematical ramifications.

Background In order to appreciate the conjecture relating the Chern–Simons theory and topological string theories, we need to go back to the seminal work of ’t Hooft, who pointed to the connection between the large-N expansion of gauge field theories and string theories. The starting point is a gauge field theory (with, say, gauge group U(N)), where we take the limit of the rank N of the gauge group to infinity (see Brezin

264 Large-N and Topological Strings

and Wadia (1993) for a collection of papers on the topic). The idea is then to make an expansion in inverse powers of N for various observables such as the free energy and correlation functions. For definiteness, let us take a gauge theory containing only gauge fields A in the adjoint representation of U(N). The quantum theory is (schematically) defined by the path integral Z Z ¼ ½DAeiSðAÞ ½1 For now, the action S(A) for the gauge fields is left unspecified. It could be either the usual Yang–Mills functional or of the Chern–Simons form which we describe below. S(A) is normalized in such a way that the gauge coupling constant, denoted by , only appears via an overall multiplicative factor of 1=. Then the expression, for instance, for the free energy F = ln Z has an expansion in a power series in , whose individual terms are given by the usual Feynman diagrammatic rules. Namely, we have is a sum over connected vacuum diagrams (those without any external legs) formed from the vertices determined by the action S(A). Even without going into the details of the action, we can write down the dependence on N and  coming from a diagram with h faces, V vertices, and E edges. Every edge is associated with a propagator (arising from the inverse of the quadratic term in S(A)) and thus comes with a weight of . Every vertex, coming from the cubic and higher-order terms in S(A), comes with a factor of 1 . There is a factor of N coming from summing over the color indices that circulate in every loop (face). We thus get a weight of N h EV and so the total contribution to the free energy can be organized as F¼

1 X

Cg;h N h 2g2þh

g¼0;h¼1

¼

1 X

Cg;h N 22g 2g2þh

½2

g¼0;h¼1

Here we have defined   N, the ’t Hooft coupling, as the combination that will be kept fixed when taking the limit of large N. We have also used the fact that V  E þ h = 2  2g, where g is the number of handles of the closed twodimensional surface one can associate with the Feynman diagram. (It is best to visualize the Feynman diagram as a ‘‘fatgraph’’ which forms the skeleton of a closed Riemann surface.) The coefficients Cg,h represent the sum of the

contributions from all genus g diagrams with h boundaries and depend on the details of the theory. We note that the reorganization of the contributions to the free energy is reminiscent of the genus expansion in a string theory. In fact, eqn [2] as it stands looks like an open-string expansion on world sheets with g handles and h boundaries. Indeed, in many cases the gauge theory arises as a limit of an open-string theory. (Recall that a massless nonabelian gauge boson is one of the low-lying excitations of an open-string theory.) So the double expansion in terms of g and h is not too surprising. However, the interesting conjecture of ’t Hooft is in the relation to closed-string theory. Note that the expansion in inverse powers of N depends only on the number of handles g. In fact, 1=N seems to play the role of closed-string coupling in that it suppresses higher genus diagrams. The total contribution to a given genus g comes from summing over all the holes h in eqn [2], for example, F¼

1 X

N22g Fg ðÞ

½3

g¼0

The conjecture is to identify this with a closed-string expansion in which Fg () is a closed-string amplitude on a genus g Riemann surface. (In carrying out the sum over the holes, we have assumed the existence of a radius of convergence. This is plausible since the number of planar diagrams (g = 0), for instance, grows only exponentially with the number of holes.) The question, since ’t Hooft, has been: what is this closed-string theory? In other words, what is the background on which the closed string propagates? A breakthrough came from Maldacena’s identification of the background for the particular case of U(N) N = 4 supersymmetric Yang–Mills theory. His conjecture was that this theory is dual to type IIB closed-string theory on AdS5  S5 with a curvature scale set by  and with closed-string coupling / =N. This proposal passed a number of nontrivial checks and is widely held to be true. It also stimulated the search for closed-string duals to other large-N gauge theories. In what follows, we explain how the conjecture of ’t Hooft has a nice realization in the case of threedimensional U(N) Chern–Simons gauge theory on S3 . The dual closed-string theory, obtained by summing over the holes, turns out to be the A-model topological string on the (six-dimensional) resolved conifold background. The parameter  maps into a Kahler parameter in the closed-string

Large-N and Topological Strings 265

geometry and once again the closed-string coupling is / =N.

The coefficents Fg,h are nonzero only for even h and are given by F0;h ¼ 

The Large-N Expansion of Chern–Simons Theory Nonabelian Chern–Simons theory is based on the following action functional for the U(N) gauge connection A: Z   k SCS ðAÞ ¼ tr A ^ dA þ 23 A ^ A ^ A ½4 4 M Here M is a three-dimensional manifold. k is called the level and is integer quantized for the path-integral equation [1] to be single valued. Note that, classically,  as defined earlier is proportional to 1=k. One of the nice properties of SCS (A) is that it is independent of the metric on M, unlike the Yang–Mills functional. Thus, it is a prototype of a topological field theory. In fact, the observables in this theory capture topological information about the 3-manifold M. Witten succeeded in quantizing the Chern– Simons theory by relating its Hilbert space to the space of conformal blocks in the two-dimensional U(N) WZW theory. (for more details on the quantization, see Chern-Simons Models: Rigorous Results). Here, merely the answers for various observables in the theory will be quoted. In particular, the free energy for the theory on S3 can be written in a completely explicit form: 3

3

ZðS ; N; kÞ ¼ exp FðS ; N; kÞ ¼

N1 Y

1 ðN þ kÞN=2

j¼1

Nj

j 2 sin Nþk

½5

One of the features one observes in the quantization is the shift (‘‘finite renormalization’’) of the effective level from k to k þ N. This can also be seen in perturbation theory. Consequently, while taking the large-N limit, the natural quantity to be held fixed as the ’t Hooft coupling is  = 2N=(k þ N). We can then carry out the ’t Hooft expansion in powers of  and 1=N, of expressions, for example, for the free energy in eqn [5]:   N2 3 1 F¼ log    log N þ  0 ð1Þ 2 12 2 þ

þ

1 X

1

g¼2

N2g2

1 X 1 X g¼0 h¼2

B2g 2gð2g  2Þ

Fg;h 2g2þh

½6

F1;h ¼ Fg;h

2ðh  2Þ h2

ð2Þ ðhÞ

ðh  2Þhðh  1Þ

6ð2Þh h   2ð2g  2 þ hÞ 2g  3 þ h ¼ h ð2Þ2g2þh B2g  2gð2g  2Þ

½7

where the last line is for g > 1. B2g are the Bernoulli numbers. The first few terms in eqn [6] are nonperturbative contributions which do not have a Feynman-diagram interpretation. The power series in  is, on the other hand, of the same form as eqn [2]. In fact, there is an open-string interpretation for these terms which will be considered later. Given the explicit form of the answer, we can carry out the summation over the holes h. Using some resummation techniques, we find 1  X t 2g2 F¼ i Fg ðtÞ ½8 N g¼0 with t  i and Fg ðtÞ ¼

ð1Þg jB2g B2g2 j 2gð2g  2Þð2g  2Þ! 1 X jB2g j þ n2g3 ent 2gð2g  2Þ! n¼1

½9

(This expression is for g > 1. There are very similar expressions for genus 0 and 1 as well.) With the identification of the string coupling gs =  it=N, the Fg (t) actually turn out to be the genus g amplitudes of a closed topological string, in line with the general expectation of the previous section. This is explained in the following.

Topological Strings Physical strings are defined in terms of a twodimensional sigma model (the theory on the world sheet) made reparametrization invariant by coupling to two-dimensional gravity. Topological strings are simpler versions of this, where the world-sheet theory is a two-dimensional topological sigma model. The latter is defined in terms of a sigma model (usually with N = 2 superconformal symmetry) with an additional twist which drastically cuts down the physical states to a subset of the low-lying modes. There are actually two inequivalent twists

266 Large-N and Topological Strings

denoted by A and B, respectively, but we will restrict to the A twist in this article. One of the simplifications of the A twisted sigma model is that the path integral localizes to contributions from only holomorphic maps from the world sheet to the target space (which will be taken to be a Calabi–Yau 3-fold). Also, all the observables in the theory depend only on the Kahler parameters of the target space and not the complex structure parameters (see Topological Sigma Models as well as the book by Hori et al. (2003) for more details). The topological string theory is defined by an appropriate integration of the observables of the topological sigma model over the moduli space of the world-sheet Riemann surface. For instance, the free energy of the string theory at genus g is given by Z 6g6 Y Fgtop ðtÞ ¼ < ðb; i Þ >X ½10 Mg

i¼1

Here b is one of the reparametrization ghost fields on the world sheet and i are Beltrami differentials. The averaging is with respect to the world-sheet sigma model for the Calabi–Yau target X, as the subscript indicates. We have also shown the dependence of Fg on the Kahler parameters of X, collectively denoted by t. The localization to the holomorphic maps in the path integral implies that Fgtop (t) takes the generic form X Y n Ng; q q  qi i ½11 Fgtop ðtÞ ¼ i



Here qi = eti and ni are the integer coefficents labeling the element  2 H 2 (X). This is in the same basis of two cycles of H 2 (X) in terms of which the complex Kahler parameters ti are expressed. (Recall that in string theory the Kahler parameters are complexified because of the presence of an additional 2-form field.) The Ng, are the Gromov– Witten invariants for X and are in general rational numbers. For nonzero , the corresponding terms are often called world-sheet instanton contributions since they correspond to topologically nontrivial maps from the world sheet to 2-cycles in the target space. The all-genus free energy of the topological string is also defined to be Ftop ðt; gs Þ ¼

1 X

gs2g2 Fgtop ðtÞ

½12

as shown by Antoniadis, Gava, Narain, and Taylor as well as Bershadsky, Cecotti, Ooguri, and Vafa, observables such as Fgtop (t) are related to special superpotential terms in the type II string compactification on the Calabi–Yau X. Using duality to M-theory, these answers were reinterpreted by Gopakumar and Vafa in terms of contributions coming from BPS states of wrapped D-branes. This gives a completely different perspective on topological strings. For instance, the all-genus free energy can naturally be reorganized as Ftop ðt; gs Þ   1 XX 1 X 1 dgs 2g2 d 2 sin ¼ ng q d 2 g¼0  d¼1

½13

g

where the n are integer invariants (Gopakumar– Vafa) since they count the number of BPS states. This will prove to be useful in extracting all-genus answers for topological string amplitudes, which is normally quite difficult using the perturbative definition given earlier.

The Large-N Dual to Chern–Simons Theory We are now in a position to state the duality (Gopakumar and Vafa 1999) between large-N Chern–Simons theory and topological strings in a precise way. The conjecture is that the closed topological string theory on the S2 resolved conifold geometry is exactly dual to the U(N) Chern–Simons theory on S3 . The resolved conifold geometry is a noncompact Calabi–Yau 3-fold described by the equation xy  zw ¼ 0

½14

where the singularity is resolved by a 2-sphere x = z, w = y. The resulting space can thus be characterized as an O(1) þ O(1) bundle over P1 . It has a single Kahler parameter t for the nontrivial 2-cycle of the S2 . In addition, the string theory is characterized by the string coupling gs . These parameters map on the gauge theory side to the ’t Hooft parameter  and N via the dictionary t ¼ i;

gs ¼

 N

½15

g¼0

with gs being the string coupling. Since topological strings are related to physical strings by a twist on the world sheet, it is natural that topological string computations are related to computations in the physical string theory. In fact,

This conjecture can be checked by comparing various exact calculations in the Chern–Simons theory with corresponding calculations in the topological string on this conifold background. The use of the duality to M-theory enables us to make exact computations on this side as well. One of the

Large-N and Topological Strings 267

nontrivial checks of this duality comes from a comparison of the free energies. In eqns [8] and [9], we already have carried out the sum over the holes in the Chern–Simons theory and organized it as a closed-string genus expansion. Note that these expressions are already of the form [11] expected of a closed topological string. One simply has to check that it is indeed that on the S2 resolved conifold. g In the language of the integer invariants n , the S2 resolved conifold is particularly simple. The only nonzero invariant is n01 = 1. Physically, this corresponds to a single brane wrapped on the genus-zero S2 . Putting this into eqn [13], and making the expansion in powers of gs , we find exactly eqn [9] for the genus-g contribution to the free energy. This is quite a remarkable agreement and represents a triumph for the ideas of large-N duality.

Geometric Transitions and Large-N Duality To understand the reason for this duality a bit better, we utilize an old observation of Witten that Chern–Simons theory is an open topological string theory. As mentioned earlier, the expansion [2] (or [6]) is suggestive of an open-string expansion in terms of handles and holes. Witten observed that open topological strings on the noncompact 3-fold T M (with Dirichlet boundary conditions on M for the end points of the string) is Chern–Simons theory on M. In fact, in the modern language of D-branes, we would say that U(N) Chern–Simons theory is the world-volume theory of N D-branes wrapped on M, for the topological A-model on T M. In particular, Chern–Simons theory on S3 is the theory of branes wrapped on S3 inside T S3 . The latter is the conifold geometry but now deformed by a nonzero size S3 . It is described by the equation xy  zw ¼ 

½16

where  is the deformation which parametrizes the size of the S3. The above large-N duality can be considered as an open–closed string duality. Namely, that the theory of open A-model topological strings on the S3 resolved conifold (with N D-branes) is dual to closed A-model topological strings on the S2 resolved conifold. Cast in this way, we see that the duality involves a transition in the background geometry in going from the open-string to the closed-string description. The sum over the holes changes the background. The S3, as it were, shrinks to zero size and a transverse S2 opens up. This geometric transition makes the connection between the

Chern–Simons theory and the closed topological string somewhat less mysterious. Maldacena’s conjecture for super Yang–Mills involves a similar passage from D-branes in flat space to a closed-string theory on anti-de Sitter space. In fact, it appears as if the best way to understand ’t Hooft’s idea in generality is to think of it as an open–closed string duality.

Further Checks and Consequences The free energy is not the only gauge-invariant observable in Chern–Simons theory. One important class of observables, which played an important role in the connection with knot invariants, are the Wilson loop expectation values. Given a knot K in S3 , we can define, in terms of an arbitrary representation R of U(N), the trace of the holonomy around the knot averaged with respect to the Chern– Simons path-integral measure:  I  WR ðKÞ ¼< trR P exp i A > ½17 K

P denotes path ordering. Similarly, we can also define the expectation values of links: products of traces of holonomies around various interlinked paths. The nonperturbative solution of Chern– Simons theory gives exact answers for the expectation values of these Wilson loops. The discussion below is, however, confined to knots. Since the trace of holonomies is being considered in different representations, it makes sense to study the generating functional X ZðU; VÞ ¼ trR ðUÞtrR ðVÞ R

"

¼ exp

1 X 1 n¼1

n

# tr Un trV n

½18

The source V here is a U(M) matrix, unrelated to the U(N) holonomy U around K. The second equality in [18] follows from use of the Frobenius formula. It was shown by Ooguri and Vafa that this generating functional is the natural object from the point of view of the open–closed string duality. We have already mentioned that the U(N) Chern– Simons theory can be thought of as the theory of N topological D-branes wrapped on the Lagrangian S3 cycle inside T S3 . For a knot K in the S3, we consider another Lagrangian 3-cycle Cˆ K in T S3 which intersects the S3 exactly in K. A canonical construction for Cˆ K is n o X ^ K ¼ ðqðsÞ; pÞ 2 T S3 j C pi q_i ¼ 0 ½19 i

268 Large-N and Topological Strings

where the knot K is parametrized by the closed curve ˆ K intersects the S3 in K. q(s). By construction, C Now consider M D-branes wrapped on Cˆ K . One now has to consider the fields coming from the strings stretching between the two sets of branes. One can show that integrating out these fields (which are in the bifundamental of the product group U(N)  U(M)) modifies the original Chern–Simons action to Seff ðAÞ ¼ SCS ðAÞ þ

1 X 1

n n¼1

trUn trV n

½20

Here V is the holonomy around K of the U(M) ˜ Thus, this configuration of M probe gauge field A. branes gives rise exactly to the generating function eqn [18] for Wilson loops of K. The geometric transition which relates the Chern– Simons theory to the closed-string theory now suggests what one needs to do to compute this generating function on the closed-string side. We have to follow the configuration of the M probe ˆ K through the conifold transition in branes on C 3 which the S shrinks and one blows up the S2 . It is not easy in general to figure out the Lagrangian cycle CK which results from following Cˆ K through the transition. It has only been done in a class of knots including the simple unknot. But assuming we know CK , the generating function for Wilson loops is given by the free energy on the S2 resolved conifold in the presence of M probe branes on CK . This requires one to know more than the closedstring partition function computed earlier. We now also need to compute amplitudes for world sheets with boundary on CK . These are called open-string Gromov–Witten invariants and the study of this subject is in its infancy. For simple knots such as the unknot, for which CK is known, these can be computed. One finds again a remarkable agreement with the nonperturbative answers of Chern–Simons theory. Thus, the computation of knot invariants gets related to open-string Gromov–Witten invariants. There have been a number of other tests involving more general knots and links. One also has to be careful of subtleties such as in the choice of framing. The reader is referred to the articles by Marino (2002, 2004) for these topics.

Conclusions The large-N duality of ’t Hooft is realized in Chern– Simons theory in a very explicit way. Thanks to the analytic control we have over both Chern–Simons theory as well as closed topological strings, the conjecture passes very nontrivial checks that extend to all-genus case. This is more than we can do in the

AdS/CFT conjecture where most computations are at tree level in the supergravity limit. In contrast, here we see the essential stringiness of the closedstring dual to Chern–Simons theory. Also, by viewing it as an open–closed string duality, many aspects of the correspondence were clarified. It, therefore, provides a useful toy model for a general understanding of open–closed string duality. Indeed, a proof of this duality using world sheet techniques has been proposed by Ooguri and Vafa. One would like to carry over some of the intuition that operates in this duality to the case of other physically interesting gauge theories. From the mathematical point of view, as already indicated, this duality leads to previously unsuspected relations between Gromov–Witten invariants and invariants of 3-manifolds, including those of knots. In fact, by considering more general geometric transitions and using this duality locally, one can learn about all-genus topological string amplitudes for a wide class of noncompact toric geometries. This line of development culminated in the formulation of the topological vertex by Aganagic, Klemm, Marino, and Vafa, which captures the essence of the topological closed-string amplitudes for noncompact toric geometries. As in the case of the general correspondence between the gauge theory and gravity, this duality sheds new light on both sides of the equation. We learn to see new integrality properties in knot and 3-manifold invariants which have an interpretation in terms of enumerative problems in 3-folds. The surprises that such a deep connection presages have not yet been exhausted. See also: AdS/CFT Correspondence; Chern–Simons Models: Rigorous Results; Duality in Topological Quantum Field Theory; Free Probability Theory; The Jones Polynomial; Knot Theory and Physics; Large-N Dualities; Quantum 3-Manifold Invariants; Schwarz-Type Topological Quantum Field Theory; String Field Theory; Topological Gravity, Two-Dimensional; Topological Quantum Field Theory: Overview.

Further Reading Brezin E and Wadia SR (eds.) (1993) The Large N Expansion in Quantum Field Theory and Statistical Physics. Singapore: World Scientific. Gopakumar R and Vafa C (1999) On the gauge theory/geometry correspondence. Advances in Theoretical and Mathematical Physics 3: 1415 (arXiv:hep-th/9811131). Hori K, Katz S, Klemm A, Pandharipande R, and Thomas R (2003) Mirror Symmetry. New York: AMS Publishing. Marino M (2002) Enumerative geometry and knot invariants (arXiv:hep-th/0210145). Marino M (2004) Chern–Simons theory and topological strings (arXiv:hep-th/0406005).

Large-N Dualities

269

Large-N Dualities A Grassi, University of Pennsylvania, Philadelphia, PA, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction Gopakumar and Vafa (1999) conjectured that U(N) Chern–Simons gauge theory on S3 is dual, for large values of N, to a closed topological string theory on a suitable Calabi–Yau 3-fold X. They suggested that this duality is realized by a geometric ‘‘transition,’’ a topological surgery which can be realized by birational contractions followed by the complex deformations of Calabi–Yau varieties. Here we will give some general comments on the history of this conjecture and then present some of its mathematical implications; we will focus on the geometric transition and the novel mathematics that it has generated. A duality relating gauge theories and string theories (with gravity) was first conjectured by ’t Hooft (1974). In 1998 Maldacena conjectured a duality between Yang–Mills gauge theory with N = 4 SUSY on a four-dimensional manifold M and IIB type closed string on the anti-de Sitter space AdS5  S5 . Chern–Simons string theory is a threedimensional theory and purely topological, hence it is in principle simpler than four-dimensional Yang– Mills theory, which also involves a metric. In this survey, we discuss the IIA open/closed dualities: we will mostly be concerned with the partition function, that is we will be working in the context of ‘‘topological strings.’’ The duality has been extended to a duality of strings, adding fluxes on the closed sector and branes on the open sector. There is much mathematical evidence supporting the conjecture.

Overview The conjecture says that U(N) Chern–Simons gauge theory on S3 is dual, for large values of N, to type IIA closed topological string theory on a suitable Calabi–Yau manifold X. A starting point for the geometry, and its mathematical implications, is that S3 can be thought of as a vanishing cycle in a local Calabi–Yau manifold Y = T  S3 , which deforms to a singular Calabi–Yau Y0 ; X is a Calabi–Yau birational resolution of Y0 . X are Y are related by a geometric transition. In fact, Witten showed that quantum Chern–Simons theory on S3 can be thought of as open IIA (with U(N) branes) on Y = T  S3 ; thus, a more general conjecture says, loosely speaking,

that open IIA theory on a Calabi–Yau manifold Y is dual, for large N, to closed IIA on a Calabi–Yau X which is related to Y via a geometric transition. A consequence of a physics ‘‘duality’’ is a matching of the free energies of the dual theories. In this particular case, if the conjecture is true, the Chern– Simons free energy Z(S3 , U(N)) should determine, and be determined by, the closed prepotential F cl (X, t). Note that Z(S3 , U(N)) is purely topological, and that F cl (X, t) includes all genera, as we will discuss later. A mathematical application is computing Gromov–Witten invariants for higher genus via large-N dualities (Marin˜o 2004). Another consequence involves the matching of the observable in S3 and X. This conjecture is now supported by a vast amount of evidence. Vafa, Gopakumar and Ooguri noted, via a string-theory analysis, that topological and knot invariants of S3 (computed through U(N) Chern–Simons theory on S3 ) determine and are determined by, for large N, the Gromov–Witten invariants of X in a neighborhood of the exceptional locus of the birational contraction X ! Y0 . The extension to the full string theory would say that open string of type IIA compactified on a Calabi–Yau manifold Y with branes is conjectured to be dual to closed string of type IIA compactified on a Calabi–Yau manifold X with fluxes, if X and Y are related by a geometric transition. A mathematical consequence of this statement is that the closed Gromov–Witten invariants of X agree, with a suitable identification of the parameters, with combinations of open Gromov–Witten invariants and knot invariants of Y. This has been shown to hold for some classes of examples. This circle of ideas has stimulated much work in physics and mathematics on the nature of the mathematical correspondence behind this duality, as well as the property of the enumerative and topological invariants involved. The ‘‘mirrors’’ of the above transitions have been studied in a series of papers, starting with the work of Dijkgraaf and Vafa (2002). The mathematics behind the open/closed dualities is still not understood: it is reasonable to speculate that the natural setup is a framework of symplectic field theory. We shall start by discussing the principal topics of this large-N duality: Chern–Simons quantum field theory, IIA closed prepotential (and Gromov–Witten invariants), and Chern–Simons as open string (and IIA open prepotential). Next we shall study the geometric transitions and conclude with some mathematical predictions of the duality.

270 Large-N Dualities

We shall not discuss some other interesting implications of this duality. For example, we shall not discuss its mirror IIB duality: it is known that the part of the closed prepotential in IIA corresponding to rational curves can be expressed as its IIB mirror dual with periods over certain suitable cycles; the IIA open contribution corresponding to open discs is expressed in terms of integrals over chains and the Abel–Jacobi map. We only remark that this large-N duality has also been interpreted as a duality between seven-dimensional manifolds with G2 holonomy.

History The chronology of various important contributions in the field of large-N duality is as follows:

 1976: ’t Hooft’s conjecture  1988: Clemens introduces transitions  1988: Witten introduces quantum Chern–Simons theory on 3-manifolds

 1992: Witten discusses Chern–Simons theory as open string

 1998: Gopakumar–Vafa–Ooguri  2001: Verification for unknot, Katz–Liu, Li, and Song

 2001: Lift to manifolds with G2 holonomy  2002: The conjecture verified for many examples of conifold transitions, including compact case; the topological vertex is introduced  2003: Relations with Donaldson–Thomas invariants

Background The varieties of interest in the physical theory must satisfy certain ‘‘supersymmetry’’ conditions; in particular, a complex algebraic manifold is required to be Calabi–Yau, a real seven-dimensional Riemannian manifold is required to have G2 holonomy group. Also of particular interest are the Lagrangian real submanifolds of the Calabi–Yau 3-folds. By a Calabi–Yau manifold X we mean a manifold with c1 (X) = 0, h0 (k ) = 0, where k is the sheaf of holomorphic k-forms, and 0 < k < dim (X). If dim X  2, we also assume that X is simply connected, but not necessarily compact. For example, if dim (X) = 1, X is a torus, if dim (X) = 2, X is a K3 surface, if dim (X)  3, X is simply called a Calabi–Yau manifold. A compact Ka¨hler manifold (M, g, J) of complex dimension m  3 is a Calabi– Yau variety if and only if its holonomy is SU(m). A subvariety L of a symplectic manifold (X, !) is Lagrangian if !jL = 0 and dim L = (1=2) dim X. Sometimes we consider noncompact manifolds,

thought of as neighborhoods of a compact projective Calabi–Yau manifold. Typically, our symplectic manifold is a Calabi–Yau 3-fold (X, !) together with its Ka¨hler form !. If there exists an antiholomorphic involution, then the fixed locus is a Lagrangian submanifold.

The Dualities We will take the point of view that dualities in physics imply relations between geometric invariants, without dwelling on the physics of the dualities themselves. A consequence of a physics ‘‘duality’’ is the matching of the prepotential of two dual string theories. A Few Comments on Chern–Simons Theory: Free Energy (Partition Function)

Let L be a closed oriented manifold together with a principal G-bundle. The classical Chern–Simons R action is defined as S(L, A) = L (A), where  is a 3-form on L which depends on a connection A and a suitable bilinear invariant form on the Lie algebra g. It is well defined under gauge transformations modulo the integers; e2iS(L, A) is well defined. In the large-N dualities considered here, the groups of interests are SU(N) and U(N). The first check of the duality was found with G = SU(N) and M = S3 ; later it was discovered that the correct group for the matching of the observables must be U(N), while both can be used for the free energies. We shall consider G = SU(N) and M = S3 . Without loss of generality, the bundle can be taken to be the product U(N)  S3 ; any bilinear invariant form on the Lie algebra su(N) is necessarily an integer multiple k of the Cartan–Killing form on the Lie algebra. Then S = S(k, A) and Z   k Sðk; AÞ ¼  2 tr A ^ dA þ 23 A ^ A ^ A 8 S3 where k is the ‘‘level’’ of the theory. Witten defines the quantum Chern–Simons theory by taking the integral of the Chern–Simons action over all possible connections A modulo gauge equivalence G: ZðS3 ; SUðNÞÞ Z ¼ ðDAÞe2iSðAÞ A=G   Z Z   ki ¼ ðDAÞexp  tr A ^ dA þ 23 A ^ A ^ A 4 S3 A=G Witten shows how to calculate the free energy Z(S3 , SU(N)) through topological surgery, assuming Z(S2  S1 )= 1. Witten also defines the partition

Large-N Dualities

function of knots and links in L (the ‘‘expectation values’’), which are knot and link invariants. The expectation values are computed by evaluating the trace of the holonomy transformation of a U(N) connection around the knot, and then taking a suitable average of the U(N) connections. These invariants depend on a choice of the framing of the knot (or link). The explicit computations involve physics, representation theory, and topology. If L = S3 , then: rffiffiffiffiffiffiffiffiffiffiffiffiffi  3  kþN N=2 Z S ; SUðNÞ ¼ ðk þ NÞ N  Nj N1 Y j  2 sin kþN j¼1 Reshetikin and Turaev, among others, described mathematically the Chern–Simons free energy and the expectation values. A Few Comments on Closed-String Theory: Free Energy (Prepotential)

In IIA closed-string theory on X, a Calabi–Yau manifold, one considers holomorphic stable maps of closed Riemann surfaces of genus g,  : g ! X, with  (g ) = [] 2 H2 (X, Z), for all genera g and homology classes  2 H2 (X, Z). Then one forms the closed prepotential F cl (X, t), which encodes the enumerative invariants of such maps to X, and which depends on the Ka¨hler parameters t of X. Sometimes the prepotential is also called ‘‘free energy’’ in the physics literature or Gromov–Witten prepotential, as it contains the Gromov–Witten invariants of X. P Setting F g (q) =  2 H2 (X, Z) Cg,  q , the closed prepotential is defined as F cl ðX; qÞ ¼

1 X

g2g2 F g ðqÞ s

g>0

Here q is a formal variable such that q1 þ2 = q1  q2 (for 1 , 2 2 H2 (X, Z)) and gs is the string coupling constant. Cg,  are the genus g Gromov–Witten invariants of X, corresponding to the class  and they have been defined as Z Cg; ¼ 1 ½Mg;0 ðX;Þvirt

It is difficult to explicitly compute the invariants Cg,  ; in particular, there is no known general method for calculating these invariants. They are computed mostly via ‘‘localization’’ methods, in the presence of a suitable torus action. In the case of g = 0 the invariants are often computed via IIA–IIB

271

duality, calculating certain periods in the mirror manifold W. Example (Faber–Pandharipande). Let X ffi OP1 (1)

OP1 (1); X is a neighborhood of a rigid rational curve, which can be thought of as a local Calabi–Yau manifold; then all the effective curves  2 H2 (X, Z) must be of the form  = d[P1 ], 8d 2 N. Faber and Pandharipande showed that F cl ðX; qÞ ¼

1 X

qd

d¼1

2 sinðdgs =2Þ2

½1

This formula was proved with localization methods after it was conjectured by Gopakumar and Vafa using large-N dualities. In fact, a consequence of a duality between two theories is the matching of the free energies of two dual string theories. In this particular case, the conjectures imply that Chern–Simons free energy determines, and is determined by, the all-genus closed prepotential of a suitable Calabi–Yau manifold X: ZðS3 ; UðNÞÞ ¸ F cl ðX; tÞ Note that the left-hand side is purely topological, as we saw in the previous section, while the righthand side is holomorphic. The trait d’union between the two prepotentials is given by the interpretation of Chern–Simons theory on S3 as open-string theory on T  S3 and the geometric transition. A Few Comments on Open-String Theory with Branes: Open Prepotential

Let Y be a Calabi–Yau manifold together with {[Li }, Lagrangian submanifolds; to each submanifold Li is assigned a gauge group Gi : Li is wrapped with Gi -branes. Here we shall focus on the case Gi = U(Ni ) and we will write (Y; Li , U(Ni )). Witten shows that the open prepotential F op (Y, , top , gs ) depends on ’t Hooft’s coupling constants i associated to Chern–Simons theory on the Lagrangian submanifolds (Li , U(Ni )), together with the open Ka¨hler parameters top 2 H2 (X; [ Li , Z), and the string coupling constant gs . To describe the open prepotential, Witten argues, we consider all maps of Riemann surfaces with boundary to Y, with the condition that the boundaries are mapped to the Lagrangian submanifolds Li ; one should also include all the ‘‘highly degenerate holomorphic maps,’’ in particular those which contract g, h to a ‘‘ribbon graph’’ on the Lagrangian [Li . The contribution of these highly degenerate maps is captured by the quantum Chern–Simons theory of the Lagrangians {Li , U(Ni )}.

272 Large-N Dualities

Application 1 (Chern–Simons free energy as open prepotential). Let us consider open IIA on Y = T  S3 with U(N)-branes wrapped on L = S3 : L is a Lagrangian submanifold with the standard symplectic structure; note that in T  L there are no nontrivial homology curves. Then, according to Witten, the corresponding open prepotential F op (Y, [ Li ) must only depend on the ‘‘highly degenerate’’ maps and must consist of the Chern– Simons term FCS on L = S3 . In particular, FCS ¼ log ZðS3 Þ ¼ F op ðY; ; gs Þ where  = 2N=(k þ N) is the ’t Hooft coupling constant. Periwal (1993) showed that, for large N, log Z(S3 ) could be expanded as a closed-string expansion: X FCS ðÞ ¼ F g ðÞg2s 2g

enumerative invariants in X can be thought as the contribution of the exceptional curve in a neighborhood of a Calabi–Yau manifold. We shall present the steps leading to this construction and the evidence for the conjecture. The Local Construction of X

Let Y = f(w1 , . . . , w4 ) 2 C4 such that

P4

j=1

w2j = g.

Proposition 1 Let  be a nonzero real positive parameter; then:

 L = S3  T  S3 is a Lagrangian submanifold of T  S3 with its standard symplectic P4 structure; 2  T  S3 ffi Y and L ffi L def = fRe( j = 1 wj = )g. In fact, we can embed T  S3 in R8 as 4 X

g0

q2j ¼ 1;

j¼1

4 X

qj pj ¼ 0

j¼1

where gs =: 2=(k þ N) is the Chern–Simons coupling constant. In 1998 Gopakumar and Vafa, using physics arguments, deduced that the expansion would have the closed form [1], which was later proved by Faber and Pandharipande.

where S3 = {pi = 0}; consider then the morphism C4 ! R 8 defined by setting

The explicit description of the open prepotential in the presence of homology classes is not known; one would need to combine the enumerative invariants of open maps together with the quantum Chern–Simons factor. We shall discuss an approach at the end of this note, but consider first the geometric transition.

which induces the diffeomorphism Y ffi T  S3 of the statement. P Remark 1 Let Y0 = f 4j = 1 w2j = 0g  C4 ; then:

The Transition The conjecture says that U(N) Chern–Simons gauge theory on S3 is dual, for large values of N, to IIA closed topological string theory on a suitable Calabi–Yau manifold X. A starting point to find such X is that S3 is a Lagrangian 3-cycle in the manifold Y = T  S3 ; performing a topological surgery by replacing S3 with S2 one obtains a (local) Calabi– Yau manifold X, on which the dual IIA theory is compactified. The key observation is that Y can be identified with the algebraic variety of equation {xy  zw = t}  C4 and that this is a complex smoothing (in fact the Milnor fiber) of Y0 with equation {xy  zw = 0}  C4 . On the other hand, X is a small resolution of this singularity, where P1 is the exceptional locus of the birational contraction. The origin is an ‘‘ordinary double point’’ singularity and the nontrivial sphere S3  Y is the vanishing cycle of the degeneration. The manifolds involved are noncompact: the exceptional curve [P1 ] = t is the only nontrivial homology class in X, and the

Reðwj Þ qj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ffi;  þ i v2i

pj ¼ Imðwj Þ

 Y0 is singular at the origin,  Y is a complex deformation of Y0 , and  L is called a ‘‘vanishing cycle.’’ With a change of coordinates we can write the equation of Y as {xy  zw = 0}; the singularity is still at the origin. This singularity is an ordinary double point, which is often referred in physics literature as ‘‘the conifold singularity.’’ Let X  C4  P1 be defined: z þ y ¼ 0;

x þ w ¼ 0

[, ] 2 P1 . Remark 2

X is smooth and the morphism

 : X ! Y0 ;

ððx; y; z; wÞ; ½; Þ 7! ðx; y; z; wÞ

is an isomorphism jXnP1 : (XnP1 ) ’ (Y0 n{0}) and P1 7! (0, 0, 0, 0, )  C4 .  is a small (nondivisorial) birational resolution of the singularity at the origin. Y is a deformation (smoothing) of Y0 . Note that topologically S3 ffi L  Y has been replaced by P1 ffi S2  X. The algebraic properties of the topological surgery between Y and X were first studied by Clemens in 1988.

Large-N Dualities Transitions in Geometry

A transition between X and Y is a birational contraction from a smooth Calabi–Yau X to a singular variety Y0 followed by a complex deformation to another smooth Calabi–Yau manifold Y:

?

Y

X # Y0

The vanishing cycles of the complex deformation [Li are always Lagrangian submanifolds of Y. The transition makes sense if dim (X) = dim (Y)  2 and it is nontrivial if dim (X) = dim (Y)  3, when the topology of X is different from the topology of Y. The possible transitions among Calabi–Yau 3-folds have been classified. Conjecture 1 Let X and Y be Calabi–Yau manifolds related by a geometric transition: then IIA open theory with U(U) branes compactified on (Y, [Li ) is dual to IIA closed theory compactified on X (with fluxes). As a consequence: Conjecture 2 Let X and Y be Calabi–Yau manifolds related by a geometric transition: then F op (Y, , gs , top ) = F cl (X, q, gs ) for a suitable identification of the parameters. The results stated in the previous section can be summarized in the the following statement, which is the proof of the above conjecture for the special case of a local conifold transition: Theorem 1 Let X ffi OP1 (1) OP1 (1) and Y = T  S3 with U(N) branes wrapped on L = S3 . Then X and Y are related by a conifold transition and log FCS (S3 ) = F op (Y, ) = F cl (X, q), with the identification ¼

2N ¼ q; kþN

gs ¼

2 kþN

This matching of the free energies is supporting evidence for the large-N conjecture. At this moment, we still do not know if Conjectures 1 and 2 hold for more general transitions.

A Few Comments on Knots and Links Later, Ooguri and Vafa extended the conjecture to the observables, that is, by adding knots and links in S3 ; the guiding principle is that a knot (or link) C  S3 should determine a noncompact Lagrangian submanifold LC  X; it is conjectured that the knot (and link) invariants, expressed as expectation

273

values, should determine and be determined by the enumerative invariants of morphisms of bounded Riemann surfaces, with boundaries mapped onto LC . We refer to these invariants as open Gromov– Witten invariants. While both statements have been verified with mathematical techniques only when C is the unknot, there is much supporting evidence for the conjecture in general. We will not describe these aspects here but only make a few remarks. The expectation values of a knot C are computed by taking first the trace of a holonomy matrix of a U(N) connection A along C and then integrating over all connections (modulo gauge equivalence). As for the case of the Chern–Simons free energy, the definition of expectation values has been worked out both in the realm of physics and of mathematics. The expectation values are knot and link invariants, and depend on a choice of the framing of the knot (or link). The open Gromov–Witten invariants have not yet been constructed, as we shall discuss in the following section; however, starting with the work of Katz and Liu, Li and Song open invariants have been successfully calculated in the presence of a torus action. The resulting invariants do depend on the choice of the torus action, which has been shown to match the choice of the framing of the knot (or link).

More on the Open Prepotential The open Gromov–Witten invariants, in analogy with the closed case, should ‘‘count’’ in an appropriate sense open morphisms; at this point, it is not known how to define this quantity. To proceed in analogy with the closed case, one would need to define the appropriate moduli space of open maps and its virtual fundamental class. On the other hand, open invariants have been successfully calculated in the presence of a torus action, assuming the existence of the moduli and virtual fundamental class and that the Atiyah–Bott localization theorems can be applied. We shall follow this approach in sketching how the IIA prepotential has been computed in many examples. Open Invariants

Let [] 2 H2 (Y; [Li , Z) be the relative homology class of Riemann surfaces in Y with boundary on the union of the Lagrangian 3-cycles [i Li and a class [ ] 2 H1 ([Li ). If g, h is a Riemann surface of genus g and h boundary components, let  : g, h ! Y be a morphism with  ðg;h Þ ¼ ½ 2 H2 ðY; [Li ; ZÞ

274 Large-N Dualities

The open generating function is Fo ðY; [Li ; top ; gs Þ ¼

1 X

g2g2þh Fg;h ðtop Þ s

g;h0

with Fg;h ðtop Þ ¼

X

Cg;h;; q y

;

Here q and y are formal variables such that q1 þ2 = q1  q2 and yh1 þh2 = yh1  yh2 , for 1 , 2 2 H2 (Y; [Li , Z), 1 , 2 2 H1 ([Li , Z); top is the open Ka¨hler parameter, gs is the string coupling constant and Cg, h, , should ‘‘count’’ in an appropriate sense the maps . Example (Ooguri–Vafa; Katz–Liu; Li–Song). If Y = OP1 (1) OP1 (1), then t is the class of the P1 ffi S2 , t=2 represents the class of the lower hemisphere in S2 . The Lagrangian L is the Lagrangian L in the previous sections, which corresponds to the unknot in S3  Y; it is the fixed locus of an antiholomorphic involution on X and it intersects S2 in an equator. Then, for a suitable choice of the torus action: Fo ðY; [Li ; top ; gs Þ ¼

X d

yd edt=2 2d sinðd=2Þ

There is a complete form for more general torus actions. The above formula was first computed by Ooguri and Vafa, using string-theory arguments, and then computed by the mathematicians, Katz and Liu, and Li and Song. More on the Open IIA Prepotential

If there is only one rigid open curve in Y, say a disk C, with boundary on L  Y, then, as Witten showed, the open prepotential is a combination of the open enumerative invariants as described above with  = d[C] and = @C and the expectation values of the unknot @C. The variable Y is changed in the trace of the holonomy of a connection. In the presence of a torus action, one can treat the fixed locus as if it were rigid and proceed accordingly.

With these techniques, Conjecture 2 has been verified for many cases of conifold transitions, with top nontrivial, for a suitable identifications of the parameters, including when both X and Y are compact manifolds (Diaconescu–Florea 2003). See also: AdS/CFT Correspondence; Chern–Simons Models: Rigorous Results; Large-N and Topological Strings; Mirror Symmetry: A Geometric Survey; String Field Theory.

Further Reading Clemens CH (1983) Double solids. Advances in Mathematics 47: 107–230. Diaconescu D-E, Florea B (2003) Large N duality of compact Calabi–Yau three-folds, hep-th/0302076. Dijkgraaf R and Vafa C (2002) Matrix models, topological strings and supersymmetric gauge theories. Nuclear Physics B 644(3): 3–20 (hep-th/0206255). Faber C and Pandharipande R (2000) Hodge integrals and Gromov–Witten theory. Inventiones Mathematicae 139: 173–199 (math.AG/9810173). Freed DS (1995) Classical Chern–Simons theory, Part 1. Advances in Mathematics 113: 237–303. Gopakumar R and Vafa C (1999) On the gauge theory/geometry correspondence. Advances in Theoretical and Mathematical Physics 3: 1415–1443. Katz S and Liu CCM (2002) Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disk. Advances in Theoretical and Mathematical Physics 5: 1–49. Li J and Song YS (2001) Open string instantons and relative stable morphisms, hep-th/0103100. Marin˜o M (2004) Chern–Simons theory and topological strings, hep-th/0406005. Ooguri HH and Vafa C (2000) Knot invariants and topological strings. Nuclear Physics B 577: 419–438. Periwal V (1993) Topological closed-string interpretation of Chern–Simons theory. Physical Review Letters 71: 1295–1298. ’t Hooft G (1974) A planar diagram theory for strong interactions. Nuclear Physics B 72: 461–473. Witten E (1989) Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121: 351–399. Witten E (1995) Chern–Simons gauge theory as a string theory. In: The Floer Memorial Volume, Progress in Mathematics, vol. 133, pp. 637–678. Basel: Birkha¨user (hep-th/9207094).

Lattice Gauge Theory

275

Lattice Gauge Theory this is done by splitting the Lagrangian L into two parts:

A Di Giacomo, Universita` di Pisa, Pisa, Italy ª 2006 Elsevier Ltd. All rights reserved.

L ¼ L0 þ LI

Introduction As a prototype of lattice gauge theory, quantum chromodynamics (QCD) will be considered in this article. All statements about QCD can easily be extended to other theories, with different gauge group and different content of particles. QCD is a gauge theory with gauge group SU(3) (color group), coupled to spin-1/2 particles (quarks) belonging to the fundamental representation of the color group. There exist in Nature six different species (flavors) of quarks, with masses ranging from mup  5 MeV to mtop  180 GeV: the values of these masses are determined by other interactions and can be treated as input parameters of the theory as well as the number of quark flavors. In standard notation, the Lagrangian reads X 1 f ði 6 D  mf Þ L ¼  trðG G Þ þ 2 f

f

½1

The sum runs over the six quark flavors f. G = @ A  @P  A þ ig[A , A ] is the field strength tensor, A = T a Aa the (gluon) gauge field, a T (a = 1, . . . , 8) are the eight generators of the gauge group in the fundamental representation, normalized as tr(T a T b ) = (1=2)ab . f is a color triplet of fields. Under a gauge transformation U(x), f ðxÞ !

0 f ðxÞ

¼ UðxÞ

f ðxÞ

½2

A ðxÞ ! A0 ðxÞ ¼ UðxÞA Uy ðxÞ þ iUðxÞ@ Uy ðxÞ

½3

D is the covariant derivative of D

f

¼ ð@   igA Þ

f

½4

and transforms like f by construction. L is invariant under the gauge transformation equations [2] and [3]. As a consequence of gauge invariance, the theory has one single coupling constant g. To make connection with the observations, one has to solve the theory, that is, one has to construct a Hilbert space on which the fields act as operators obeying the equations of motion and the canonical commutation relations. In textbook field theory,

½5

with L0 the part of L which is bilinear in the fields and LI the rest. L0 can be solved exactly since it describes free particles and the corresponding equations of motion are linear. The resulting Hilbert space is the Fock space of free particles. LI is treated as a perturbation producing scattering between the fundamental particles. This approach works well in quantum electrodynamics, where the observed particles (electrons and photons) coincide with the excitations of the fundamental fields of the Lagrangian. In QCD, the fundamental excitations (the quarks and the gluons) are observed as particles neither in Nature nor as a product of high-energy collisions between elementary particles. This feature is known as confinement of color. The conjecture is that excitations with nontrivial color are forbidden to propagate as free particles. However, if hadrons are probed at short distances by photons or by leptons, everything works as if they were composite states of quarks. The accepted explanation relies on asymptotic freedom: the effective coupling constant becomes small at short distances (high momentum transfers) and the constituents behave as free particles. At large distances, the fundamental excitations are not observed, the interaction is strong and the perturbative picture describing scattering between quarks and gluons is not adequate for the real world. An alternative quantization procedure is needed which does not rely on perturbation theory. A formally exact quantization procedure is the Feynman path integral. The solution of the theory is given in terms of a functional integral Z[J], which generates the correlators of the fields in the ground state (vacuum). Indicating symbolically the Lagrangian coordinates, namely the fields, by a single symbol , one has   Z Y Z Z½ J  ¼ dðxÞ exp S½  JðxÞðxÞdx ½6 x

The connected Euclidean vacuum correlators are given in terms of functional derivatives of Z[J] < 0jTððx1 Þðx2 Þ    ðxn ÞÞj0 >conn   1 n Z½ J   ¼ Z½0 Jðx ÞJðx Þ    Jðx Þ 1

2

n

JðxÞ¼0

½7

276 Lattice Gauge Theory

‘‘Euclidean’’ means that they are analytic continuations to imaginary times. Going to Euclidean system is necessary to isolate the vacuum state. The amplitudes can be analytically continued back to Minkowski space. The Hilbert space and all the physical observables can be constructed in terms of the correlators, a property known as reconstruction theorem. Formally (i.e., assuming that everything makes sense only if the functional integral exists), < 0jTððx1 Þðx2 Þ    ðxn ÞÞj0 >conn Z 1 Y ¼ dðxÞ Z x  expðS½Þðx1 Þðx2 Þ    ðxn Þ

½8

The continuation to imaginary time changes sign to the kinetic energy, and Z formally becomes the partition function of a four-dimensional statistical model with Hamiltonian SE [], a general fact in Feynman integrals. By definition of functional integral, Z is defined by discretizing a finite volume V of spacetime to a finite set of points and then sending their number to infinity, making a set dense in V. If the limit exists, a ZV is obtained. The volume V is then sent to infinity, to cover the whole spacetime (thermodynamical limit) and ZV eventually converges to Z. A rigorous proof of the existence of these limits does not exist for QCD, but there are qualitative arguments that this is the case, which will be presented below. In the lattice formulation of field theory, a regular lattice, usually cubic, is taken as a discretization of spacetime. From the very definition of Feynman integral, it follows that the formulation of field theory on the lattice is nothing but an approximation to the limit which defines Z. It will provide a good approximation if the lattice spacing is small enough with respect to the physical lengths involved and if the lattice is large compared to them. Perturbation theory amounts to split the action into a bilinear term S0 and an interaction term SI containing the higher powers of the fields. The Z integral is then computed by expanding the weight in a power series of SI : Z Y dðxÞ expðS0  SI Þ x

¼

Z Y x

dðxÞ expðS0 Þ

X ðSI Þn n

n!

½9

The Feynman integral thus becomes Gaussian, can be computed, and gives the usual perturbative expansion. The two limits (integral and series expansion) do not commute in general. For QCD,

there are indeed arguments that the renormalized perturbative expansion does not converge and is plagued by singularities known as renormalons.

Wilson’s Formulation For field theories of scalar particles, the lattice discretization is performed by assigning a value of the field to each site of the lattice. The Wilson formulation for gauge theories is not made in terms of the fields A , which are defined in the Lie algebra of the gauge group, but in terms of parallel transports, which are elements of the group itself. The building blocks are parallel transports along links parallel to spacetime axes connecting neighboring sites U ðxÞ

 Z  P exp ig

xþ^ 

A dx

  expðigaA ðxÞÞ

½10

x

where ˆ indicates the vector of length a in the  direction and P the ordered product. The last approximate equality is valid in the limit of small lattice spacing a. g is the coupling constant. Under a gauge transformation V(x); U ðxÞ ! VðxÞU ðxÞV y ðx þ ^Þ

½11

It follows from eqn [11] that the parallel transport along a closed path is gauge invariant. The density of action can be written in terms of the parallel transport along the elementary square of links in the hyperplanes   , known as plaquette: Y ^ y ðx þ ^ÞUy ðxÞ ¼ tr½U ðxÞU ðx þ ÞU ½12 

By expanding in powers of a, one easily finds Y 

1 ¼ Nc  a4 tr½G G  þ Oða6 Þ 2

½13

with Nc the number of colors, 3 for QCD. The lattice action can be defined as  X  1 S¼  1  ½14 Nc x with  = 2Nc =g2 , and tends to the continuum action as a ! 0, O(a2 ). An infinite number of higher-order terms in a exist, which come from the expansion of the links, but they are expected to be irrelevant in the continuum limit a ! 0. The measure of the Feynman integral is assumed to be the Haar measure of the gauge group for each link, which again can be shown to tend to the continuum measure in the continuum limit.

Lattice Gauge Theory

Everything is gauge invariant, contrary to the perturbative formulation, where a gauge fixing is required to define the vector meson propagator. By Weierstrass theorem, the integral is finite for any finite number of links, the gauge group being compact. Any other choice of the lattice action differing from the Wilson action of eqn [14] by terms of higher order in a will have the same continuum limit: there is significant freedom in the choice of the action. In the language of statistical mechanics, the Euclidean lattice formulation is a spin model. Different choices of the action correspond to different spin models. In the vicinity of a second-order phase transition, however, the correlation length becomes large with respect to the lattice spacing and all the irrelevant terms become negligible. All the spin models at the critical point belong to the same universality class and define the same field theory. This is what happens for QCD because of asymptotic freedom. By renormalization group arguments, the lattice spacing behaves as aðÞ 

1 expðb0 Þ 

The fermionic Lagrangian then reads X ðxÞ½i6 DL  m ðxÞ x



X

 ðxÞM1

ðx;x0 Þ  ðx

0

Þ

½17

x;x0 

It is convenient to indicate this expression in the form Sf = M1 , where is a large column whose elements are labeled by the site x and by the component . The functional integral over can explicitly be done by using the standard rules of integration on Grassman variables, since the action is bilinear, Z Y Z¼ dU ðxÞd ðxÞd ðxÞ  expðSE ½U  M Þ The result is Z Y Z¼ dU ðxÞ expðSE ½UÞ det M

½18

½19

½15

at sufficiently large , where b0 is the coefficient of lowest-order term of the -function, b0 is positive and  is a physical scale. As  ! 1, a tends exponentially to zero in physical units and the coarse structure of the lattice becomes unimportant, indicating that the shortdistance limit in the definition of the Feynman integral exists. The theory also develops a mass scale  which insures the existence of a finite correlation length and hence of the thermodynamical limit. In practice, when  is increased, the lattice space becomes exponentially small in physical units. As a consequence, however, the physical scale becomes exponentially large in lattice units, and an exponentially large lattice is needed to insure the large-distance convergence. This makes life difficult if the Feynman integral has to be computed numerically.

Quarks Fermion fields are defined on lattice sites. The naive lattice transcription of the fermion term in eqn [1] consists in replacing the covariant derivatives by finite differences with parallel transports to make the result gauge covariant. In principle, D (x) = Uy (x) (x þ ) ˆ  (x) is a correct definition. In practice, a more symmetric difference is used which is correct O(a2 ), namely DL ðxÞ 1 ^ ðx  Þ ^ ¼ ½UðxÞ ðx þ ^Þ  Uy ðx  Þ 2

277

½16

The effect of fermions is to multiply the weight by a functional determinant which depends on the gauge field configuration. A problem exists, however, in this procedure already at the level of free fermions, that is, putting U = 1 in the action and in the determinant of eqn [18]. The equation of motion reads, in Fourier transform,    X k  sin 2  m ~ðkÞ ¼ 0 ½20 L  With respect to the continuum, the momentum p = 2 k =L has been replaced by its sinus. At small values of p , eqn [20] coincides with the Dirac equation. However, an alternative solution exists at p  , for each  independently. The new equation differs from the other by a change of sign of  . Changing sign of one of the gammas means changing sign to  5   1  2  3  4, which is the chirality of the fermion. Instead of one fermion, we then have 24 = 16 fermion species, organized in pairs with opposite chiralities. It is impossible to have a single fermion with a given chirality. A number of recipes have been proposed to circumvent this artifact of the lattice regulation, for example, introduce by hand a term in the action which removes the spurious particles in the limit of zero lattice spacing (Wilson’s fermions); double the lattice spacing by constructing two sublattices on even and odd sites, respectively, which propagate fermions of opposite chirality (staggered fermions),

278 Lattice Gauge Theory

so that the argument of the sinus in the derivative is doubled. More recently, an idea which goes back to Ginsparg and Wilson has been implemented, which consists in replacing a strictly local equation of motion like eqn [20] by an equation with the same continuum limit which is nonlocal, but with a nonlocality falling off exponentially at large distances, a recipe which makes propagation of chiral fermions possible. This is an important improvement, even if very demanding in computer power.

Numerical Simulations Solving analytically the lattice version of QCD would allow one to follow constructively all the steps which bring to the definition of Z, that is, the ultraviolet and the infrared limit, as explained earlier. Presently that is out of reach. Also an attempt by Wilson to solve the lattice renormalization group equations by techniques of decimation is not conclusive. The problem can be attacked numerically. One way would be to compute the integral numerically. That is, however, prohibitive: it would be like solving exactly the equations of motion for the molecules of a gas. The lattice theory is in fact a four-dimensional statistical mechanics with the Boltzmann factor  = 2Nc =g2 and Hamiltonian equal to the Euclidean action. As in statistical mechanics the way out is to create a significant sample of configurations with weight exp (SE ) and to determine the field correlators which describe physics by an average on this ensemble. This is done by Monte Carlo techniques. The basic principle is to start from an arbitrary field configuration and make a sequence of random changes, normally on a single link at a time, with uniform probability in the group measure so as to converge toward the equilibrium distribution exp (SE ). For that purpose, the probability PC0 C to change from a configuration C to another C0 is constrained to obey the detailed balance relation PC0 C expðS½CÞ ¼ PCC0 expðS½C0 Þ

½21

A common algorithm is known as metropolis. The way to implement the condition (eqn [21]) is to accept the new trial configuration C0 if S[C0 ] S[C], and to accept it with probability exp ( [S(C0 )  S(C)]) if S[C0 ] S[C]. An alternative method is known as ‘‘heat-bath’’. If the probability of the configuration for one link at a fixed value of the other variables is

explicitly known, the change can be accepted with that probability. In the presence of dynamical quarks, the integral eqn [18] is converted into an integral on bosonic variables by inverting the matrix M: Z Y Z¼ dU ðxÞ d ðxÞ d ðxÞy  expðSE ½U  y ½My M1 Þ The RQ

½22

property has been used such that d (x) d y (x) exp ( y [My M]1 ) = j det Mj. A metropolis updating is then performed on the combined U and variables. To have a choice of the trial uniform in the measure, an algorithm is commonly used which is based on ergodicity, known as hybrid molecular dynamics. A fictitious conjugate momentum is associated with all variables, and a fictitious Hamiltonian is defined by adding to the action, considered as a potential energy, the sum of the squares of the conjugate momenta. A classical evolution is then performed in time by small steps which should displace the state in phase space ergodically: the evolution is called a trajectory. After a number of steps, a metropolis test is made as explained above. Typically, the computer time needed to produce a significant configuration is proportional to the volume V of the lattice for pure gauge systems, to V 5=4 in the hybrid algorithm for full QCD. As explained before, in order to have a good approximation to the Feynman integral the lattice spacing has to be small compared to the physical scales, for example, with respect to the Compton wavelength of the heaviest quark. On the other hand, to control volume effects it has to be large compared to the biggest physical length, for example, with respect to the Compton wavelength of the lightest quark. Since there is a factor mtop =mup  3  103 between these two lengths, the lattice size needed would be prohibitive from numerical point of view. In practice, lattices of size L4 are affordable with L 64  128. For this reason, only the light quarks u, d, s are kept, which have mass smaller than the typical scale of the theory, which can be identified as the square root of the string tension. In the limit in which light quark masses are small compared to QCD scale, the Lagrangian is invariant under any unitary mixing of them. A global SU(3) invariance exists, which is known as flavor symmetry, and is broken by the difference of quark masses. Heavier quarks can be described by an effective theory, since they have negligible dynamical effects at low energies.

Lattice Gauge Theory

A Selection of Physics Results String Tension

A big excitement followed the first numerical calculations by M Creutz at the beginning of the 1980s in which the static potential V(r) between a quark and an antiquark was computed in puregauge theory on the lattice. One way to measure it is to measure the correlator of two Polyakov lines at a distance r on a significant ensemble of field configurations. The Polyakov line is the parallel transport in the fundamental representation along the time axis across the lattice: with periodic boundary conditions it is a closed loop, and hence it is gauge invariant. It can be proved that the log of this correlator is equal to V(r)aLt with Lt the extension of the lattice in the time direction. It was found that VðrÞ ¼ r

½23

The parameter  is known as string tension. A potential of the form eqn [23] means confinement: an infinite amount of energy is required to pull apart the particles at infinite distance. The parameter  can be determined phenomenologically from the mass spectrum of the mesons and 2  1 GeV. What is measured on the lattice is aðÞ2 n2

½24

where n is the distance of the two Polyakov lines in lattice spacings and a() the lattice spacing in physical units. In fact, the computer only produces pure numbers. If the lattice QCD belongs to the same universality class of QCD at the critical point, that is, if the lattice really defines QCD, the dependence of a() on  is dictated by the -function of the renormalization group. At sufficiently large  = 6=g2 , aðÞ 

1 expðb0 Þ latt

½25

with b0 = (11=3)Nc =16 2 . latt is the energy scale of the theory. The measurement of the potential gives indeed a dependence of the lattice spacing on  consistent with eqn [25] and allows one to determine =2latt . The absolute value of the lattice spacing can be determined by comparison with the physical value of the string tension. The theory is able to produce a physical scale. The correlation length is finite and as a consequence the infrared limit of the Feynman integral exists. Mass Spectrum

Any operator with the quantum numbers of a particle can be used as interpolating field for it.

279

The correlator of the operator at large distances behaves like a sum of exponentials exp (mr) with m the masses of the particles with the same quantum numbers. At large distances the lightest particle dominates, especially if the operator has a good overlap, that is, if its matrix element between vacuum and the state of the particle is the biggest. From the correlators mr can be determined. On the lattice r = na() so that, by eqn [25] what is really determined is the ratio m=latt . If latt has been determined, for example, from the string tension, the mass of the particle results in physical units. Alternatively, the ratios of any two masses can be determined and the scale fixed by the value of one of them. A good agreement is obtained already in pure gauge (quenched approximation) indicating that the quark loops are relevant at the level of 10% typically. This fact supports the idea that the large Nc -limit is a good approximation to reality, quark loops being nonleading in that limit. The light particle masses are more difficult to compute, being sensitive to the masses of light quarks which cannot be taken at realistic values due to computational difficulties: large lattices are required and big fluctuations are present near the chiral point. The spectrum of particles made of heavy quarks can be computed using effective theories, and nicely fits experiment. A byproduct is a precise determination of the gauge coupling constant, competitive with phenomenological determinations from short distance perturbative QCD. Weak Interaction Matrix Elements

There exist matrix elements of currents (or products thereof) entering in weak amplitudes which involve large distances and are not computable in perturbation theory. Lattice can be used to evaluate them. Renormalization problems can appear in this approach when the cutoff is removed, which, however, are not difficulties of principle but only of technical nature. This activity is of fundamental importance to have precise predictions in order to understand the limits of the standard model. Finite-Temperature QCD and the Deconfinement Transition

The static thermodynamics of a system of fields is described by the partition function ZT ¼ tr½expðH=TÞ

½26

It is easy to show that ZT is equal to the Euclidean Feynman integral on the imaginary time interval (0, 1=T) with boundary conditions in time periodic for bosons and antiperiodic for fermions. Indeed, the

280 Lattice Gauge Theory

Boltzmann factor is formally an imaginary time evolution by 1/T. A lattice of extension Lt L3S with Ls  Lt provides the partition function at a temperature T = 1=aLt , if a is the lattice spacing in physical units. Finite-temperature simulations are important to investigate the transition from the phase in which color is confined to a phase in which quarks and gluons can propagate as free particles. This phase is called deconfined phase or quark gluon plasma. Big experiments at Brookhaven and at CERN are looking for this phase transition in high-energy collisions between heavy nuclei, but no definite evidence has yet been produced for it. Lattice simulations instead definitely prove that such a transition exists. For pure SU(3) gauge theory (quenched) at T  270 MeV, a first-order phase transition is observed, at which the string tension vanishes. In a more realistic theory with dynamical quarks, a transition is also observed at T  160 MeV, where chiral symmetry, which is spontaneously broken at zero temperature, is restored. This transition is also associated to deconfinement even if, in the presence of light quarks, the string tension does not exist. Indeed, when pulling apart a quark and an antiquark, an instability for production of quark–antiquark pairs sets in when the potential energy becomes large enough, which physically manifests itself as a production of light mesons. An alternative order parameter is needed. The possibility of defining alternative order parameters is discussed in next section. The equation of state can also be studied relating internal energy to pressure, which is useful to understand heavy ion collisions. From the features of the deconfinement transition, information can be extracted on the mechanisms by which QCD confines color. A connected issue is the behavior of QCD at nonzero baryon density or chemical potential. The corresponding thermodynamics is described by a grand canonical ensemble Z ¼ tr½exp½ðH þ NÞ=T ½27 R 3 y where N = d x is the baryon number operator and  the chemical potential. In the process of converting the partition function Z into a Feynman integral, the term H at the exponent of eqn [27] generates the Euclidean action, which is real. The term proportional to N becomes imaginary. The integral is well defined, but the analogy with a fourdimensional statistical mechanics is broken, the effective Hamiltonian being non-Hermitian and no sampling can be made. Approximate methods have been developed, but the problem is open. Exploring

numerically the region of phase space with  6¼ 0 would be interesting, since a rich structure is expected, which could be relevant to dense systems such as neutron stars. Mechanisms of Color Confinement

Understanding how QCD manages to confine color is one of the most fascinating problems in field theory. To prove confinement, one should, in principle, prove that, at zero temperature, no gauge-invariant quasilocal operator exists, carrying nontrivial color and obeying cluster property at large distances. This proof is not known. There exists evidence form lattice simulations that a string tension exists, as discussed before. In any case, a guess can be made of the physical mechanism of confinement. If confinement is an absolute property reflecting a symmetry property of the vacuum, an order parameter should exist which discriminates between confined and deconfined phase, and the transition between the two phases has to be a true transition. Observing a crossover in some part of the boundary between the two phases would disprove this view. A lattice determination of the order of the deconfining transition is therefore of fundamental importance. A possible mechanism of confinement proposed by G ’t Hooft is dual superconductivity of the vacuum: dual means interchange of electric with magnetic with respect to ordinary superconductors. In the same way as the magnetic field is constrained into Abrikosov flux tubes in an ordinary superconductor, the chromoelectric field acting between a quark and an antiquark would be constrained into flux tubes by a dual Meissner effect producing an energy proportional to the distance, or a string tension. This mechanism can be investigated by lattice simulations, by checking if any magnetically charged operator exists whose vacuum expectation value is nonzero in the confined phase signaling condensation of magnetic charges and zero in the deconfined phase. Progress has been made in this direction which, however, is not yet conclusive. Chromoelectric flux tubes between q–q¯ pairs are observed in lattice field configurations. Topology

Euclidean QCD admits classical solutions with finite action and with a nontrivial topology which makes them stable. These solutions, known as instantons or multi-instantons, realize a mapping of the threedimensional sphere at infinity on the gauge group, and the topological charge is the winding number of this mapping. The Jacobian of this mapping is the Chern

Leray–Schauder Theory and Mapping Degree

current K and its divergence @ K (x) R  Q(x) is the density of topological charge. Q = d4 x Q(x) is the topological charge which has integer values. Explicitly, QðxÞ ¼

g2 tr½G G   16 2

½28

with G  = (1=2)  G the dual field strength tensor. Q(x) plays an important role in hadron physics, being related to the P anomaly of the flavor singlet axial current J5 = f  5  f . J5 is conserved at the classical level in the chiral limit mf = 0, but this symmetry does not survive quantization. In fact, @ J5 ¼ 2Nf QðxÞ

½29

A consequence of eqn [29] is the high mass m 0  1 GeV of the flavor singlet partner 0 of the pseudoscalar flavor octet. An Nc ! 1 argument by Witten and Veneziano relates m 0 to the response of the quenched (no quark) vacuum to topological R excitation, the topological susceptibility   d4 x < 0jTQ(x)Q(0)j0 > . The relation is 2Nf  ¼ ½m2 0 þ m2  2m2K ½1 þ Oð1=Nc Þ f 2

½30

This approximate relation has been checked on the lattice.  has been determined by different methods which agree in confirming it. This is an important verification of QCD. Instantons are stable solutions in the continuum, approximately stable in the lattice discretized version. A cooling procedure which locally freezes short-distance quantum fluctuations would leave the instantons untouched if they were stable. On the lattice the instanton is stable anyhow if the

281

distance in correlation reached by the local cooling procedure is small compared to the size of the instanton: cooling is indeed a diffusion process and the distance involved grows as the square root of the number of cooling iterations. Instanton configurations can nicely be exposed by cooling. See also: Anomalies; Quantum Chromodynamics; Renormalization: General Theory; Spin Foams; Symmetry Breaking in Field Theory.

Further Reading Creutz M (1983) Quarks, Gluons and Lattices. Cambridge: Cambridge University Press. Feynman RP (1948) Space–time approach to nonrelativistic quantum mechanics. Reviews of Modern Physics 20: 367. Feynman RP and Hibbs AR (1965) Quantum Mechanics and Path Integral. New York: McGraw-Hill. Ginsparg PH and Wilson KG (1982) A remnant of chiral symmetry on the lattice. Physical Review D 25: 2649. Gottlieb S, Liu W, Toussaint D, Renken RL, and Sugar RL (1987) Hybrid-molecular-dynamics algorithms for the numerical simulation of QCD. Physical Review D 35: 2531. Kogut J and Susskind L (1975) Hamiltonian formulation of Wilson’s lattice gauge theories. Physical Review D 11: 395. ’t Hooft G (1981) Topology of the gauge condition and new confinement phases in non-abelian gauge theories. Nuclear Physics B 190: 455. Veneziano G (1979) U(1) without instantons. Nuclear Physics B 159: 213. Wilson KG (1974) Confinement of quarks. Physical Review D 10: 2445–2459. Wilson KG (1983) The renormalization group and critical phenomena (Nobel Lecture). Reviews of Modern Physics 55: 583. Witten E (1979) Current algebra theorems for the U(1) goldstone boson. Nuclear Physics B 156: 269.

Leray–Schauder Theory and Mapping Degree J Mawhin, Universite´ Catholique de Louvain, Louvain-la-Neuve, Belgium ª 2006 Elsevier Ltd. All rights reserved.

Introduction The Leray–Schauder theory gives a powerful and versatile continuation method for proving the existence, multiplicity, and bifurcation of solutions of nonlinear operator, differential and integral equations. Let X and Y be topological spaces, A X, f : X ! Y, a continuous mapping, and y 2 Y. The fundamental idea of a continuation method to solve

the equation f (x) = y in A consists in embedding it into a one-parameter family of equations Fðx; Þ ¼ zðÞ

½1

where the continuous functions F : X  [0, 1] ! Y, z : [0, 1] ! Y are chosen in such a way that F(  , 1) = f , z(1) = y and 1. equation F(x, 0) = z(0) has a nonempty set of solutions in A; 2. one of those solutions at least can be continued into a solution in A of [1] for each  2 [0, 1]. Simple examples show that Assertion 2 can be violated when all solutions of [1] leave A after some

282 Leray–Schauder Theory and Mapping Degree

 2 ]0, 1[. A way to avoid such a situation consists in ‘‘closing the boundary,’’ through the ‘‘boundary condition’’: Fðx; Þ 6¼ zðÞ for each

ðx; Þ 2 @A  ½0; 1

When this condition is satisfied, Assertion 2 can still fail when two existing solutions for  small disappear after coalescing at some 0 < 1. Losing all solutions through this process can be eliminated by reinforcing Assumption 1 into 20 . Equation F(x, 0) = z(0) has a ‘‘robust’’ nonempty set of solutions in A. This statement can be made precise through the concept of topological degree of a mapping, an ‘‘algebraic’’ count of the number of its zeros. In a finite-dimensional setting, this concept was introduced by Kronecker for smooth mappings and by Brouwer for continuous mappings. Its extension by Leray and Schauder to some classes of mappings in Banach spaces made much wider applications to nonlinear differential and integral equations possible.

Topological Degree of a Mapping If U  Rn is a bounded open set, z 2 Rn and  ! Rn is a C1 mapping such that z 62 F(@U) F:U and det F0 (x) 6¼ 0 on F1 (z), the Brouwer degree degB [F, U, z] is defined (analytically) by X degB ½F; U; z :¼ sign det F0 ðxÞ x2F1 ðzÞ

¼

X

ð1ÞðxÞ

x2F1 ðzÞ

where (x) is the sum of the multiplicities of the negative eigenvalues of F0 (x). The case of a continuous F such that z 62 F(@U) is treated by approximating F through mappings of the above type, and showing that the corresponding degrees stabilize to an unique value, defining degB [F, U, z] in the general case. This number remains the same under sufficiently small perturbations of F and/or z, which expresses the ‘‘robustness’’ mentioned above. When n = 2 and U is bounded by a closed Jordan curve, then degB [F, U, 0] is nothing but the winding number of F=kFk along @U. Leray and Schauder have extended Brouwer degree to the important class of compact perturbations of identity in a normed space. A compact mapping f : A ! B between metric spaces is a continuous mapping on A such that f(A) is relatively compact. If f : A ! B is continuous and compact on

each bounded B  A, f is called ‘‘completely continuous’’ on A. If X is a real normed space, U  X an open bounded  ! X compact, and z 62 (I  f )(@U), the set, f : U Leray–Schauder degree degLS [I  f , U, z] of I  f in U over z is constructed from Brouwer degree by  by approximating the compact mapping f over U mappings f with range in a finite-dimensional subspace X of X containing z. One shows that the values of the Brouwer degrees degB [(I  f )jX , U \ X , z] stabilize for sufficiently small positive  to a common value which defines degLS [I  f , U, z]. Again, this topological degree is an algebraic count of the number of elements of (I  f )1 (z), equal to 0 when z 62 (I  f )(U). When f is of class C1 , and I  f 0 (x) invertible at each fixed point x 2 (I  f )1 (z), (I  f )1 (z) is finite and the Leray–Schauder formula holds: X degLS ½I  f ; U; z ¼ ð1ÞðxÞ ½2 x2ðIf Þ1 ðzÞ

where (x) is the sum of the algebraic multiplicities of the eigenvalues of f 0 (x) contained in [1, þ1]. Let I = [0, 1]. For A  X  I, and  2 I, we write A = {x 2 X : (x,) 2 A}. The Leray–Schauder degree inherits the basic properties of Brouwer degree: 1. Additivity. If U = U1 [ U2 , where U1 and U2 are open and disjoint, and if z 2 = (I  f )(@U1 ) [ (I  f )(@U2 ), then degLS ½I  f ; U; z ¼ degLS ½I  f ; U1 ; z þ degLS ½I  f ; U2 ; z then 2. Existence. If degLS [I  f , U, z] 6¼ 0, z 2 (I  f )(U). 3. Homotopy invariance. Let   X  I be a  ! X be compact. bounded open set, and let F :  If x  F(x, ) 6¼ z for each (x, ) 2 @, then degLS [I  F( , ),  , z] is independent of . In particular, if a is an isolated fixed point of f, and B(a, r) denotes the open ball of center a and radius r, degLS [I  f , B(a, r), 0] is defined and independent of r for sufficiently small r > 0. Its value is called the ‘‘Leray–Schauder index’’ of I  f at a, and denoted by indLS [I  f , a].

Fixed-Point Theorems for Compact Perturbations of Identity in a Normed Space An important application of Leray–Schauder degree is the obtention of general fixed point theorems for compact mappings in normed spaces based on

Leray–Schauder Theory and Mapping Degree

continuation along a parameter. If F : A  X  I ! X, we denote by A the (possibly empty) solution set defined by A ¼ fðx; Þ 2 A : x ¼ Fðx; Þg Let   X  I be a bounded open set and  ! X be a compact mapping. The general F: Leray–Schauder fixed-point theorem goes as follows: Theorem

If the following conditions hold:



(i)  \ @ = ; (a priori estimate) (ii) degLS [I  F( , 0), 0 , 0] 6¼ 0 (degree condition),  then  contains a continuum C along which   takes all values in I. In other words,  contains a compact connected subset C connect ing  0 to 1 . If one refines Assumption (ii) into   (iii) 0 is a finite nonempty set {a1 , . . . , a } and indLS [I  F( , 0), a1 ] 6¼ 0, the conclusion takes the form of an ‘‘alternative’’: if assumptions (i) and (iii) hold, then (a1 , 0) belongs either to  a continuum in  containing one of the points  (a2 , 0), . . . , (a , 0), or to a continuum in  along which  takes all the values in I. Condition (iii) automatically holds in the following  important special case: If  \ @ = ;, F( , 0) = 0,   and 0 2 0 , then  contains a continuum C 3 (0, 0) along which  takes all values in I. When dealing with   X!X the fixed-point problem x = f (x) with f : U compact, U open and bounded, a natural choice is F(x, ) = f (x),  = U  I, giving the statement: If 0 2 U and if x 6¼ f (x) for each (x, ) 2 @U  I, then   I : x = f (x)} contains a continuum C 3 {(x, ) 2 U (0, 0) along which  takes all values in I. Condition (i) requires the a priori knowledge of  the localization of the solution set  and is in general very difficult to check. An important special case occurs when X is a priori bounded: if F is completely continuous on X  I, F( , 0) = 0, and X  B(r)  I for some r > 0, then X contains a continuum C 3 (0, 0) along which  takes all values in I. Its special case with F(, x) = f (x) can be stated as Schaefer’s alternative: Let f : X ! X be completely continuous. Then either there exists, for each  2 [0, 1], at least one x 2 X such that x = f (x), or the fixed point set {x 2 X : x = f (x), 0 <  < 1} is unbounded in X. Schaefer’s alternative is equivalent to the following Schauder fixed-point theorem: Theorem Any compact mapping f : B(r) ! B(r) has a fixed point. A simple consequence of Schauder’s theorem is that, for any continuous and bounded g : R ! R, any open bounded D  Rn , any  different from an

283

eigenvalue of  on D with Dirichlet boundary conditions, the nonlinear Dirichlet problem u þ u þ gðuÞ ¼ hðxÞ in D u¼0

on @D

has a weak solution for each h 2 L2 (D). An interesting consequence of Leray–Schauder theorem with X a priori bounded is that, for any bounded domain D  Rn with @D of class C2 , the Dirichlet problem for the equation of surfaces with constant mean curvature  ð1 þ kruk2 Þu 

n X

@i u @j u @ij2 u

i;j¼1 2 3=2

¼ nð1 þ kruk Þ

has a unique solution for arbitrary smooth boundary data if and only if the mean curvature of the boundary @D is everywhere greater than [n=(n  1)]jj. The use of auxiliary continuous functionals gives a fixed-point theorem in the absence of a priori bounds: Theorem (Capietto–Mawhin–Zanolin). Let    ! X be completely X  I be an open set and F :    continuous. If 0 is bounded, degLS [I  F( , 0), U0 , 0] 6¼ 0 for some open bounded neighborhood  U0 of  0 , and if there exists a continuous  mapping ’ : X  I ! R þ , proper on  , and c < min ’( , 0) max ’( , 0) < cþ such that  62 0 0  {c , cþ } and @ 62 [c , cþ ], then  contains a continuum C along which  takes all values in I. This result implies, for example, that for g : R ! R continuous, odd and superlinear (limjuj ! 1 g(u)= u = þ1), and p : [0, 1]  R2 with at most linear growth in u and u0 at infinity, the two-point boundary-value problem u00 þ gðuÞ ¼ pðt; u; u0 Þ;

uð0Þ ¼ uð1Þ ¼ 0

has, for all sufficiently large j, at least one solution uj having exactly j þ 1 zeros on [0, 1], and kuj kC1 ! 1 if j ! 1.

Extensions of Leray–Schauder degree Fixed-point theorems for operators between suitable nonlinear spaces can also be proved using topological continuation arguments. For example, if C  X is a nonempty convex set, one has the following extension of a result of the previous section to mappings in C: if U  C is open and bounded, F : clC U  I ! C compact and such that x 6¼ F(x, ) for each (x, ) 2 @C U  I, F( , 0) = x0 2 U, then

284 Leray–Schauder Theory and Mapping Degree

F( , ) has a fixed point in U for each  2 I. The special case where C is a wedge is useful in finding positive solutions of nonlinear differential or integral equations. For nonlinear spaces, the degree has to be replaced by the fixed-point index, which generalizes both the ‘‘Hopf–Lefschetz number’’ and Leray–Schauder degree. The Leray–Schauder degree also has been extended to other classes of operators. Compact operators can be replaced by k-set-contractive or condensing mappings f, with respect to various measures of noncompactness, and fixed-point problems can be replaced by problems of the form x 2 F(x) for multivalued mappings F. Equivariant degree theories have been developed when U is invariant and f equivariant with respect to the action of some compact Lie group G on X. The special case of G = S1 is of special importance in the study of periodic solutions of autonomous differential systems. Degree theories have also been constructed for various classes of mappings between two different Banach spaces or manifolds, which include monotone-like and nonlinear Fredholm operators. We just describe a simple but useful situation in this direction. Many differential equations, when expressed as equations in an abstract space, do not have the fixed-point form but can be written as Lx = Nx with  ! Z, X and Z real L : D(L)  X ! Z linear, N : U normed spaces. If L is invertible, the equation is trivially equivalent to the fixed-point problem x = L1 Nx, to which Leray–Schauder theory can be applied when L1 N is compact. The situation is more delicate when L has no inverse. If L is a linear Fredholm mapping of index zero (its range R(L) is closed and has a finite codimension equal to the dimension of its null space N(L)), the set F (L) of linear continuous mappings of finite rank A : X ! Z such that L þ A : D(L) ! Z is a bijection is nonempty and the compactness of (L þ A)1 G does not depend upon the choice of A 2 F (L). G is then called ‘‘L-compact’’ on E, and ‘‘L-completely continuous’’ on E when compact on each bounded set of E. The following continuation theorem for perturbed Fredholm mapping of index zero holds. Theorem Let   X  I be open and bounded, L : D(L)  X ! Z linear Fredholm of index zero,  ! Z L-compact, and let ={(x,) 2 (D(L)  I) N:  \:Lx=N(x,)}. If

then  contains a continuum C along which  takes all values in I. When dealing with equation Lx = f (x) with f L-completely continuous, an interesting special case of the above result follows from the choice N(x, ) = f (x) þ (1  )Qf (x), with Q : Z ! Z a projector such that N(Q) = R(L). In this case, the homotopy is equivalent to Lx ¼ f ðxÞ ð 20; 1Þ Qf ðxÞ ¼ 0; x 2 NðLÞ ð ¼ 0Þ An application (among many) of this result, for g : R ! R continuous such that 1 < lim supu ! 1 g(u) < lim inf u ! þ1 g(u) < þ1, D  Rn open, bounded, k an eigenvalue of the Dirichlet problem for  on D, is the weak solvability of the nonlinear problem u þ k u þ gðuÞ ¼ hðxÞ u¼0

in D

on @D

for each h 2 L2 (D) such that Z h i hðxÞ’ðxÞ dx < lim sup gðuÞ u!1 D Z h iZ þ  ’ ðxÞ dx  lim inf gðuÞ ’ ðxÞ dx u!þ1

D

D

for all eigenfunctions ’ associated to k . The addition of Rthe nonlinearity g ‘‘widens’’ the range {h 2 L2 (D) : D h’ = 0} of the corresponding linear problem.

Bifurcation Theory Leray–Schauder degree is a powerful tool in bifurcation theory, where, given a family F of solutions, one tries to detect and analyze other ones branching or bifurcating from F . Consider the equation x ¼ Lx þ Rðx; Þ

½3

in a real normed space X, where L : X ! X, linear, and R : X  R ! X are completely continuous, and R(0, ) = 0 for each  2 R. Thus, {(0, ) :  2 R} is the trivial solution set of [3]. A bifurcation point ( , 0) for [3] is the limit of a sequence (k , xk ) of solutions of [3] in Rn{0}. If kRðx; Þk ¼0 kxk uniformly on bounded -sets

lim

x!0

(i)  \ @ 6¼ ; (a priori estimate),  0  {0})  Y, with Y  R(L) = Z (transvers(ii) N( ality condition), and (iii) degB [N( , 0)jkerL , 0 \ kerL, 0] 6¼ 0 (degree condition)

½4

it is easy to prove that if ( , 0) is a bifurcation point for [3], then  is a characteristic value (reciprocal of an eigenvalue) of L. Leray–Schauder theory gives a partial

Leray–Schauder Theory and Mapping Degree

converse to this result known as Krasnosel’skii’s bifurcation theorem: Theorem For each real characteristic value  of L with odd algebraic multiplicity, ( , 0) is a bifurcation point of [3]. Of fundamental importance in the proof is the special case of [2] with f = L and N(I  L) = {0}. Another fruitful concept is Krasnosel’skii’s bifurcation from infinity. We say ( , 1) is a bifurcation point for [3] if there exists a sequence (n , xn ) of solutions of [3] such that n !  and kxn k ! 1. The corresponding bifurcation result goes as follows (Krasnosel’skii): if kRðx; Þk ¼0 kxk kxk!1 lim

uniformly on bounded -sets

½5

then, for each real characteristic value  of L with odd algebraic multiplicity, ( , 1) is a bifurcation point of [3]. Global versions of Krasnosel’skii’s theorems can be given, whose statements are reminiscent of Leray– Schauder’s alternative theorem. Let S denote the closure in R  X of the set of (, x) 2 R  (X n {0}) satisfying [3]. For bifurcation from zero, one has Rabinowitz global bifurcation theorem: Theorem If [4] holds and  is a real characteristic value of L with odd algebraic multiplicity, then S contains a component C which either is unbounded, or contains ( , 0), where  6¼  is a characteristic value of L. As an application, one can show that the nonlinear Sturm–Liouville problem ðpðxÞu0 Þ0 þ qðxÞu ¼ aðxÞu þ hðx;u;u0 ;Þ ðx 20;1½Þ a0 uð0Þ þ b0 u0 ð0Þ ¼ a1 uð1Þ þ b1 u0 ð1Þ ¼ 0 with p 2 C1 positive, q, a, h continuous, a positive, (a20 þ b20 )(a21 þ b21 ) 6¼ 0 and h(x,u,v)=o(juj þ jvj) if juj þ jvj ! 0 uniformly on compact -intervals, has, for each k 2 N, an unbounded component of solution Ck in R  C1 ([0,1]) emanating from (k ,0), with k an eigenvalue of the problem with h  0 (Rabinowitz). One has also global bifurcation from infinity: if [5] holds and if  is a real characteristic value of L with odd algebraic multiplicity, then [3] has an

285

unbounded component of solutions D which contains ( , 1). See also: Bifurcation Theory; Bifurcations in Fluid Dynamics; Bifurcations of Periodic Orbits; Minimal Submanifolds; Minimax Principle in the Calculus of Variations; Partial Differential Equations: Some Examples; Riemann–Hilbert Problem; Topological Defects and Their Homotopy Classification; Viscous Incompressible Fluids: Mathematical Theory.

Further Reading Cronin J (1964) Fixed Points and Topological Degree in Nonlinear Analysis. Providence: American Mathematical Society. Deimling K (1985) Nonlinear Functional Analysis. Berlin: Springer. Fitzpatrick P, Martelli M, Mawhin J, and Nussbaum R (1993) Topological Methods for Ordinary Differential Equations. Berlin: Springer. Fonseca I and Gangbo W (1995) Degree Theory in Analysis and Applications. Oxford: Oxford Science Publisher. Geba K and Rabinowitz P (1985) Topological Methods in Bifurcation Theory. Montre´al: Presses Univ. Montre´al. Granas A and Dugundji J (2003) Fixed Point Theory. New York: Springer. Ize J and Vignoli A (2003) Equivariant Degree Theory. Berlin: de Gruyter. Krasnosel’skii MA (1963) Topological Methods in the Theory of Nonlinear Integral Equations. Oxford: Pergamon. Krasnosel’skii MA and Zabreiko PP (1984) Geometrical Methods of Nonlinear Analysis. Berlin: Springer. Krawcewicz W and Wu J (1997) Theory of Degrees with Applications to Bifurcations and Differential Equations. New York: Wiley. Leray J and Schauder J (1934) Topologie et e´quations fonctionnelles. Annales Scientifiques de l’Ecole Normale Supe´rieure 51(3): 45–78. Lloyd NG (1978) Degree Theory. Cambridge: Cambridge University Press. Matzeu M and Vignoli A (eds.) (1995–97) Topological Nonlinear Analysis. Degree, Singularity and Variations I, II. Basel: Birkha¨user. Mawhin J (1979) Topological Degree Methods in Nonlinear Boundary Value Problems. Providence: American Mathematical Society. Petryshyn WW (1995) Generalized Topological Degree and Semilinear Equations. Cambridge: Cambridge University Press. Rothe E (1986) Introduction to Various Aspects of Degree Theory in Banach Spaces. Providence: American Mathematical Society. Schwartz JT (1969) Nonlinear Functional Analysis. New York: Gordon and Breach. Zeidler E (1986–88) Nonlinear Functional Analysis and Its Applications, vols. I–IV. New York: Springer.

Lie Bialgebras see Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups

286 Lie Groups: General Theory

Lie Groups: General Theory R Gilmore, Drexel University, Philadelphia, PA, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction Local continuous transformations were introduced by Lie as a tool for solving ordinary differential equations. In this program, he followed the spirit of Galois, who used finite groups to develop algorithms for solving algebraic equations (the general quadratic, cubic, and quartic), or else to prove that some equations (the generic quintic) could not be solved by quadrature. Lie’s work led eventually to the definition and study of Lie groups. Lie groups are beautiful in their own right – so beautiful that they have been studied independently of their origin as a tool for solving differential equations and studying the special functions determined by certain classes of these equations.

Lie Groups Lie groups exist at the interface of the two great divisions of mathematics: algebra and topology. Their algebraic properties derive from the group axioms. Their geometric properties arise from the parametrization of the group elements by points in a differentiable manifold. The rigidity of these structures arises from the continuity requirements imposed on the group composition and inversion maps. The algebraic axioms are standard. Definition A group G consists of a set gi , gj , gk , . . . 2 G together with a combinatorial operation  that satisfy the four axioms: (i) Closure. If gi 2 G, gj 2 G, then gi  gj 2 G. (ii) Associativity. If gi , gj , gk 2 G, then (gi  gj ) gk = gi  (gj  gk ). (iii) Identity. There is a unique operation e 2 G that satisfies e  gi = gi = gi  e. (iv) Inverse. Every group operation gi 2 G has an 1 inverse, denoted g1 i , that satisfies gi  gi = e = g1  g . i i Lie groups have more structure than groups. In particular, each gi 2 G is a point in an n-dimensional manifold Mn . That is, the subscript i actually identifies a point x 2 Mn , so that we can write gi = g(x) or most simply gi = x. The group multiplication can be expressed in the

form gi  gj = gk ! g(x)  g(y) = g(z), where x 2 Mn , y 2 Mn , z = (x, y) 2 Mn . The group inversion map can be expressed in the form g(x) ! g(x)1 = g(y), y = (x) 2 Mn . The topological axioms for Lie groups can be taken as: (v) Continuity of composition. The mapping z = (x, y) defined by the group composition law is differentiable. (vi) Continuity of inversion. The mapping y = (x) defined by the group inversion law is differentiable. The dimension of the Lie group is the dimension of the manifold that parametrizes the operations in the group. The most familiar examples of Lie groups consist of n  n nonsingular matrices over the fields R, C, Q of real numbers, complex numbers, and quaternions. For example, the set of 2  2 real unimodular matrices   a b ; ad  bc ¼ 1 c d is a three-dimensional submanifold embedded in 2 R2 = R4 .

Matrix Lie Groups Not every Lie group is a matrix group. Yet, it is a surprising and useful result that almost every Lie group encountered in physics is a matrix Lie group. These are all subgroups of the general linear groups GL(n; F) of n  n nonsingular matrices over the field F (R, C, Q). These groups have real dimension n2  (1, 2, 4), respectively. The special linear subgroups SL(n; F) are defined as the subgroups of n  n matrices with determinant þ1: M 2 SL(n; F) if det M = þ1. This definition is problematic for quaternions, as they do not commute. To avoid this problem, it is useful to map quaternions into 2  2 complex matrices in the same way complex numbers can be mapped into 2  2 real matrices:   a b a þ ib ! b a   q0 þ iq3 iq1 þ q2 q0 þ I q1 þ J q2 þ Kq3 ! iq1  q2 q0  iq3 Here (1, i) are basis vectors for C1 considered as a real two-dimensional linear vector space,

Lie Groups: General Theory

(1, I , J , K) are basis vectors for Q1 considered as a real four-dimensional linear vector space, and (a, b) and (q0 , q1 , q2 , q3 ) are all real. The squares of the imaginary quantities i and I , J , K are all 1: i2 = 1; I 2 = J 2 = K2 = 1 and the imaginary quaternion basis elements anticommute: {I , J } = {J , K} = {K, I } = 0. The unimodular subgroup SL(n; Q) of GL(n; Q) is obtained by replacing each quaternion matrix element by a 2  2 complex matrix, setting the determinant of the resulting 2n  2n matrix group to þ1, and then mapping each of the n2 complex 2  2 matrices back to quaternions. Many other important groups are defined by imposing linear or quadratic constraints on the n2 matrix elements of GL(n; F) or SL(n; F). The compact metric-preserving groups U(n; F) leave invariant lengths (preserve a positive-definite metric g = In ) in linear vector spaces. The matrices M 2 U(n; F) satisfy My In M = In . These conditions define the orthogonal groups O(n) = U(n; R) and the unitary groups U(n) = U(n; C). Their noncompact counterparts O(p, q) and U(p, q) leave invariant nonsingular indefinite metrics   I 0 g ¼ Ip;q ¼ p 0 Iq in real and complex n = (p þ q)-dimensional linear vector spaces: My Ip, q M = Ip, q . Intersections of matrix Lie groups are also Lie groups. The special metric-preserving groups are intersections of the special linear groups SL(n; F)  GL(n; F) (with F = Q, SL(n; Q) is defined as described above) and the metric-preserving subgroups U(n; F)  GL(n; F): SLðn; RÞ \ Uðn; RÞ¼ SOðnÞ;

nðn  1Þ=2

SLðn; CÞ \ Uðn; CÞ¼ SUðnÞ;

n2  1

SLðn; QÞ \ Uðn; QÞ¼ SpðnÞ ¼ USpð2nÞ;

nð2n þ 1Þ

The real dimensions of these groups are given in the right-hand column. Under the replacement of quaternions by 2  2 complex matrices, the group of n  n metric-preserving and unimodular matrices Sp(n) over Q is identified as USp(2n), an isomorphic group of 2n  2n matrices over C. Noncompact forms SO(p, q), SU(p, q), and Sp(p, q) = USp(2p, 2q) are defined similarly. The Lie group SU(2) rotates spin states to spin states in a complex two-dimensional linear vector space. It leaves lengths, inner products, and probabilities invariant. If an interaction is spin independent, only an invariant (‘‘Casimir invariant’’) constructed from the spin operators can appear in the Hamiltonian. The same group can act

287

in isospin space, rotating proton to neutron states. The Lie group SU(3) similarly rotates quark states or color states into quark states or color states, respectively. The Lie group SU(4) rotates spin– isospin states into themselves. The conformal group SO(4, 2) leaves angles but not lengths in spacetime invariant. It is the largest group that leaves the source-free Maxwell equations invariant. It is also the largest group that transforms all the (bound, scattering, and parabolic) hydrogen atom states into themselves. Lie groups such as the Poincare´ group (inhomogeneous Lorentz group) and the Galilei group have the matrix structures 32 3 2 t1 x 6 7 6 Oð3; 1Þ t2 7 76 y 7 6 76 7 6 6 t3 76 z 7 76 7 6 6 7 6 t4 7 54 ct 5 4 0

0

0

2 6 6 6 6 6 60 4 0

0 v1 v2 v3

Oð3Þ 0

0

1

0

0

0

1

1

32 3 x t1 6y7 t2 7 76 7 6 7 t3 7 76 z 7 76 7 6 7 t4 7 54 t 5 1 1

respectively. In these transformations t = (t1 , t2 , t3 ) describes translations in the space (x-, y-, and z-) directions, v = (v1 , v2 , v3 ) describes boosts, and t4 resets clocks. The matrices in these defining matrix representations are reducible. The Heisenberg covering group H4 is a fourdimensional Lie group with a simple 3  3 matrix structure: 2 3 1 l d Heisenberg covering group ¼ H4 ¼ 4 0 n r 5; 0 0 1 n 6¼ 0 This matrix representation of H4 is faithful but nonunitary.

‘‘Linearization’’ of a Lie Group At the topological level, a Lie group is homogeneous. That is, every point in a manifold that parametrizes a Lie group looks like every other point. At the algebraic level, this is not true – the identity group operation e is singled out as an exceptional group element. At the analytic level, the group composition law z = (x, y) is nonlinear, and can therefore be arbitrarily complicated.

288 Lie Groups: General Theory

The study of Lie groups is enormously simplified by exploiting these three observations. Specifically, it is useful to ‘‘linearize’’ the group multiplication law in the neighborhood of the identity. The linearization leads to a local Lie group. This is a linear vector space on which there is an additional structure. Once the local Lie group properties are known in the neighborhood of the identity, they are known everywhere else in the group, since the group is homogeneous. A Lie group is linearized in the neighborhood of the identity by expressing an operator near the identity in the form g() = I þ X, where the local Lie group operator X = xi Xi , the Xi are n linearly independent vector fields on the manifold Mn , and the small coordinates xi measure the distance (in some rough sense) of g() from the point that parametrizes the identity group operation e = g(0). For another group operation g(Y) = I þ Y in the neighborhood of the identity, the following holds. 1. The product g(X)g(Y) = (I þ X)(I þ Y) = I þ (X þ Y) þ (h.o.t) is in the local Lie group. 1 2. The commutator gi  gj  g1 in the group i  gj leads to 1

gðXÞgðYÞgðXÞ gðYÞ

1

¼ I þ 12 ðXY  YXÞ þ h:o:t ¼ I þ 12 ½X; Y  þ h:o:t in the local Lie group. The first condition shows that the local Lie group is a linear vector space. The n vector fields Xi can be chosen as a set of basis vectors in this space. The second condition shows that the commutator of two vectors in this linear vector space is also in this linear vector space. The commutator endows this linear vector space with an additional combinatorial operation (‘‘vector multiplication’’) and provides it with the structure of an algebra, called a Lie algebra.

The structure of a Lie algebra, or local Lie group, is summarized by the structure constants, defined in terms of the basis vectors Xi , by   Xi ; Xj ¼ cij k Xk summation convention The structure constants cij k are components of a third-order tensor, covariant and antisymmetric in two indices (cij k = cji k ) and contravariant in the third. These components obey the Jacobi identity, which places a quadratic constraint on them: cij s csk t þ cjk s csi t þ cki s csj t ¼ 0 Linearization of a Lie group generates a Lie algebra. A Lie group can be recovered by the inverse process. This is the exponential operation. A group operation a finite distance from the origin (the point identified with the identity group operation) of the manifold that parametrizes the Lie group can be obtained from the limiting procedure ( = 1=K ! 0): K Y 1 gðXÞ ¼ lim I þ X ¼ eX ¼ EXPðXÞ K!1 K The exponential operation is well defined for real numbers, complex numbers, quaternions, n  n matrices over these fields, and vector fields. A 1:1 correspondence between Lie groups and Lie algebras does not exist. Isomorphic Lie groups have isomorphic Lie algebras. But nonisomorphic Lie groups may also possess isomorphic Lie algebras. The best known examples of nonisomorphic Lie groups and their isomorphic Lie algebras are SOð3Þ 6¼ SUð2Þ; SOð4Þ 6¼ SUð2Þ  SUð2Þ;

soð3Þ ¼ suð2Þ soð4Þ ¼ suð2Þ þ suð2Þ

SOð5Þ 6¼ Spð2Þ ¼ USpð4Þ;

soð5Þ ¼ spð2Þ ¼ uspð4Þ

There is a 1:1 correspondence between Lie algebras and ‘‘locally’’ isomorphic Lie groups. This has been extended to global Lie groups by a beautiful theorem due to E Cartan.

Definition A Lie algebra la consists of a set of operators X, Y, Z, . . . , together with the operations of vector addition, scalar multiplication, and commutation [X,Y] that satisfy the following three axioms:

Theorem (Cartan) There is a 1:1 correspondence between Lie algebras and simply connected Lie groups. Every Lie group with the same Lie algebra is either the simply connected (‘‘universal covering’’) group or is the quotient of this universal covering group by one of its discrete invariant subgroups.

(i) Closure (linear vector space). If X, Y 2 la, X þ Y 2 la and [X, Y] 2 la. (ii) Antisymmetry. [X, Y] = [Y, X]. (iii) Jacobi identity. [X, [Y, Z]] þ [Y, [Z, X]] þ [Z, [X, Y]] = 0.

This relation is summarized in Figure 1. As a concrete example, the Lie algebra of SO(3), which is the group of real 3  3 matrices satisfying My I3 M = I3 and det(M) = þ1, is spanned by the three ‘‘angular momentum vector

Lie Groups: General Theory

289

SG /D1

SG /D2

EXP (unique)

Multiply connected Lie groups

Linearization “LOG” (unique)

Simply connected Lie group SG

SG /Dr

Lie, algebra, Figure 1 Cartan’s theorem states that there is a 1:1 correspondence between Lie algebras and simply connected Lie groups. All other Lie groups with this Lie algebra are quotients of the covering group by one of its discrete invariant subgroups Dj DMax : There is a relation between the discrete invariant subgroup Dj and the homotopy group of SG=Dj . Reproduced with permission from Gilmore R (1974) Lie Groups, Lie Algebras, and Some of Their Applications. New York: Wiley.

fields’’ Li (x) = ijk xj @k or the three angular momentum matrices 2 3 0 0 0 6 7 0 þ1 7 L1 ¼ L23 ¼ 6 40 5 0

0

1 2

3

0

0

6 L2 ¼ L31 ¼ L13 ¼ 6 4 0

0

7 07 5

þ1

0

0

2

0

þ1 0

6 L3 ¼ L12 ¼ 6 4 1

1

3

7 0 07 5

0

0 0

The Lie group SU(2) is the group of complex 2  2 matrices satisfying My I2 M = I2 and det(M) = þ1. Its Lie algebra is spanned by the three spin matrices Sj = (i=2)j , which are multiples of the Pauli spin matrices j :     i 0 þ1 i 0 i S1 ¼ ; S2 ¼ 2 þ1 0 2 þi 0  i þ1 S3 ¼ 2 0

0



1

The two Lie algebras are isomorphic as they share isomorphic commutation relations [J1 , J2 ] = J3 (and cyclic), Jj = Lj or Jj = Sj . The group SU(2) is simply connected. Its maximal discrete invariant subgroup D consists of all multiples of the identity, I2 , so that  = 1. According to Cartan’s theorem, SO(3) = SU(2)=D2 , D2 = {I2 , I2 }. The group SO(3) is doubly connected, with a two-element homotopy group.

Matrix Lie Algebras A deep theorem of Ado guarantees that every Lie algebra is equivalent to a matrix Lie algebra, even though the same is not true of Lie groups. Sets of n  n matrices that close under vector addition, scalar multiplication, and commutation (M1 2 la, M2 2 la ) [M1 , M2 ] = M1 M2  M2 M1 2 la) form matrix Lie algebras. The antisymmetry properties and Jacobi identity are guaranteed by matrix multiplication. Lie algebras for the general linear groups GL(n; F) consist of n  n matrices over F. Lie algebras for the special linear groups SL(n; F) consist of traceless n  n matrices. The Lie algebras of the unitary groups consist of anti-Hermitian matrices. The Lie algebras of U(p, q; F) consist of matrices that obey My Ip;q þ Ip;q M ¼ 0;

M 2 uðp; q; FÞ

The matrix Lie algebras of other matrix Lie groups are obtained by constructing the most general Lie group operation in the neighborhood of the identity by linearization. For example, the Lie algebra of the Heisenberg covering group H4 is 2 3 2 3 1 l d 1 l d 6 7 6 7 4 0 n r 5 ! 4 0 1 þ n r 5 0

0

0

1

0

1

! I3 þ n N þ r R þ l L þ d D 2

N ’ ay a

3

0

0

0

6 40

1

7 05

0

0

0

2

R ’ ay 0 0

6 40 0 0 0

0

3

7 15 0

290 Lie Groups: General Theory

2

L’a

0 1 6 40 0 0 0

0

3

7 05 0

D’I¼ 2 0 0 6 40 0 0 0



a; ay 3 1 7 05 0



The four 3  3 matrices N, R, L, D that span the Lie algebra h4 of H4 satisfy commutation relations isomorphic with the commutation relations satisfied by the photon operators (ay a, ay , a, I = [a, ay ]). The 3  3 matrix representations of the group H4 and the algebra h4 are faithful. The representation of H4 is nonunitary and that of h4 is non-Hermitian. There is a simple way to relate a large class of operator Lie algebras to matrix Lie algebras. If A, B, C, . . . belong to a Lie algebra of n  n matrices with [A, B] ¼ C, the matrix-to-operator mapping A ! A ¼ x i A i j @j preserves commutation relations, for   ½A; B ¼ xi Ai j @j ; xr Br s @s     ¼ xi Ai j @j ; xr Br s @s  xr Br s @s ; xi Ai j @j ¼ xi Ai j Bj s @s  xr Br i Ai j @j ¼ xi ½A; Bi j @j ¼ C This relation depends on the bilinear products xi @j satisfying commutation relations  i  x @j ; xr @s ¼ xi @s j r  xr @j s i These commutation relations are satisfied by products of creation and annihilation operators ayi aj for either bosons (byi bj ) or fermions (fiy fj ). These matrixto-operator mappings can be extended to include bilinear products such as xi xj, xi @j , @i @j and their boson and fermion counterparts ai aj , ayi aj , ayi ayj . For example, the vector fields associated with the operator J1 for SO(3) and SU(2) are xi (L1 )i j @j = x2 @3  x3 @2 and ui (S1 )i j @j = (i=2)(u1 @2 þ u2 @1 ). Boson and fermion bilinear products ayi aj (1  i, j  n) are isomorphic to u(n). Boson bilinear products bi bj , byi bj , byi byj are isomorphic to usp(2n) while fermion bilinear products fi fj , fiy fj , fiy fjy are isomorphic to so(2n).

Structure of Lie Algebras The study of Lie algebras is greatly facilitated by studying their structure. The structure is determined by the commutation properties of the Lie algebra. Invariant Subalgebra

If a Lie algebra has an invariant subalgebra, then the commutator of anything in the algebra with

anything in the subalgebra is in the subalgebra. Suppose a is a linear vector subspace of g. If [g, a] a, then a is an invariant subspace of g. In particular, [a, a] a and a is therefore also a subalgebra of g: it is an invariant subalgebra in g. Example The Lie algebra iso(3) consists of the three rotation operators Lij = xi @j  xj @i and the three displacement operators Pk = @k . The subset of displacement operators is an invariant subspace in iso(3), since it is mapped into itself by all commutators. It is also a subalgebra in iso(3). This particular invariant subalgebra is commutative. Solvable Algebra

If g is a Lie algebra, the linear vector space obtained by taking all possible commutators of the operators in g is called the ‘‘derived’’ algebra: [g, g] = g(1) g. If g(1) = g, there is no point in continuing this process. If g(1)  g, it is useful to define g = g(0) and to continue this process by defining g(2) as the derived algebra of g(1) : g(2) = [g(1) , g(1) ]. We can continue in this way, defining g(nþ1) as the algebra derived from g(n) . Ultimately (for finite-dimensional Lie algebras), either g(nþ1) = 0 or g(nþ1) = g(n) for some n. If the former case occurs, g ¼ gð0Þ gð1Þ gð2Þ gðnÞ gðnþ1Þ ¼ 0 the Lie algebra g(0) is called solvable. Each algebra g(i) is an invariant subalgebra of g(j) , i > j. Example The Lie algebra spanned by the boson number, creation, annihilation, and identity operators is solvable. The series of derived algebras has dimensions 4, 3, 1, 0. gð0Þ

gð1Þ

gð2Þ

gð3Þ

ay a ay a I

 ay a I

   I

   

Semidirect Sum Algebra

When a Lie algebra g has an invariant subalgebra a, the linear vector space of the Lie algebra g can be written as the direct sum of the linear vector subspace of the subalgebra a plus a complementary subspace b. The subspace b is generally not by itself a Lie algebra. The Lie algebra g is written as a semidirect sum of the two subspaces. The semidirect

Lie Groups: General Theory

sum structure satisfies the commutation relations shown: g¼b^a

½b; b b ^ a ½b; a a ½a; a a

291

Regular Representation

This representation assigns the structure constants to a set of n n  n matrices according to   X ; X ¼ c X

X ! RðX Þ ¼ c ;

Example The Lie algebra spanned by bilinear products of photon creation and annihilation operators ayi aj , creation operators ayi , annihilation operators aj , and the identity operator I(1  i, j  n) is nonsemisimple. The solvable invariant subalgebra is spanned by the 2n þ 2 operators consisting of the single photon operators ayi , aj , the identity P operator ^ = ni= 1 ayi ai . I, and the total number operator n

The matrices of the regular representation contain exactly as much information as the components of the structure tensor. They can be studied by standard linear algebra methods. For example, a secular equation can be used to put the commutation relations into canonical form. The structure of the matrices of the regular representation determines the structure of the Lie algebra. The identification is carried out according to the usual rules of representation theory, as shown in Figure 2. If a basis X can be found in which all the matrices of the regular representation are simultaneously reducible, the algebra possesses an invariant subalgebra. If the representation is not fully reducible, the invariant subalgebra is solvable. If the regular representation is fully reducible, the algebra consists of the direct sum of two (or more) smaller, mutually commuting subalgebras. If the regular representation is irreducible, the algebra is simple. If a Lie algebra is solvable (solv), all matrices in the regular representation can be transformed to upper triangular matrices. If the Lie algebra is nilpotent (nil  solv), the diagonal matrix elements in the upper triangular matrices are zero. The converses are also true.

Semisimple Algebra

Cartan–Killing Form

A Lie algebra is semisimple if it has no solvable invariant subalgebras.

The Cartan–Killing form is a second-order symmetric tensor that is constructed from the thirdorder antisymmetric tensor c by cross-contraction  g ¼ c c ¼ g ¼ tr RðX ÞRðX Þ ¼ X ; X  ¼ X ; X

The subspace b can be given the structure of an algebra modulo the component of the commutator in a: b = g mod a. Example The three-dimensional Lie algebra spanned by the photon operators ay , a, I has a semidirect sum decomposition where b is spanned by ay , a and a is spanned by I. The subspace b is not closed under commutation, and a is commutative. The Lie algebra iso(3) also has the structure of a semidirect sum, with b = b = so(3) and the invariant subalgebra a is spanned by the three displacement operators Pk . Nonsemisimple Algebra

A Lie algebra is nonsemisimple if it has a solvable invariant subalgebra.

Example The Lie algebra so(4) is semisimple. This Lie algebra has two invariant subalgebras, both isomorphic to so(3). The direct sum decomposition soð4Þ ¼ soð3Þ þ soð3Þ is well known to physical chemists and is responsible for the dualities that exist between rotating and laboratory frame descriptions of molecular systems. Simple Algebra

The metric g can be used to place an inner product (X , X ) on this linear vector space. This inner product is not necessarily positive definite. Reducible

Fully reducible

Irreducible

Nonsemisimple

Semisimple

Simple

A Lie algebra is simple if it has no invariant subalgebras at all. The prettiest page in the theory of Lie groups is the classification theory of the simple Lie algebras. We turn to this subject now.

Lie Algebra Tools Two powerful tools have been developed for studying the structure of a Lie algebra. These are the regular representation and the Cartan–Killing form.

Figure 2 When the regular matrix representation of a Lie algebra is reducible, fully reducible, or irreducible, the Lie algebra is nonsemisimple, semisimple, or simple.

292 Lie Groups: General Theory

The matrix g can also be treated by standard linear algebra methods. Since it is real and symmetric, it can be diagonalized. If there are n negative eigenvalues, nþ positive eigenvalues, and n0 vanishing eigenvalues (n = n þ nþ þ n0 ), the Lie algebra has a corresponding linear vector space decomposition of the form g ¼ g þ gþ þ g0 The inner product is positive definite on the subspace gþ and negative definite on g . We call g0 the singular subspace. The subspace g0 is closed under commutation and in fact is a nilpotent invariant subalgebra of g.

Decomposition of Lie Algebras The most general Lie algebra g is the semidirect sum of a semisimple Lie algebra ss and a solvable invariant subalgebra solv:

g ¼ ss ^ solv

exponentiates to a noncompact coset in G0 that is simply connected. Every element in a semisimple Lie algebra can be expressed as the commutator of two elements in the Lie algebra. In this sense, a semisimple algebra reproduces itself under commutation. To illustrate this algorithm, we tear apart the eight-dimensional Lie algebra spanned by the photon operators ayi aj , 1  i, j  2 and ay3 a3 , ay3 , a3 , I, where the photon operators obey [ai , ayj ] = ij I. The regular representative of the general linear combiP nation X = ij mij ayi aj þ nay3 a3 þ ray3 þ la3 þ I is 2

0 6 6 6 m21 6 6 m12 6 RðXÞ ¼ 6 6 6 6 6 6 4

0 m21 m12

m12 m12 þm11  m22 0

m21 m21 0 m11 þ m22

3

½ss; ss ¼ ss ½ss; solv solv ½solv; solv  solv

The decomposition of g into its component parts is accomplished by a simple two-step algorithm. 1. Compute the Cartan–Killing metric for g and determine the singular subspace. If there is none, stop. If the dimension of g0 > 0, nil = g0 is the maximal nilpotent invariant subalgebra of g. 2. Compute the structure constants of the Lie algebra g0 = g  nil = g mod nil = g=nil, the Cartan–Killing metric tensor on g0 , and the decomposition g0 = g0 þ g0þ þ g00 . Then a = g00 is abelian and invariant in g0 . In fact, a is the largest abelian invariant subalgebra in g0 . The algorithm stops here, for the algebra g00 = g0 mod a = g0 =a = g0 þ g0þ has no singular subspace under its Cartan–Killing metric. Under this algorithm, the decomposition of g into its semisimple part and its maximal solvable invariant subalgebra is   g ¼ g0 þ g0þ ^ g00 ^ g0 The maximum solvable invariant subalgebra solv in g is the semidirect sum of a and nil: solv = g00 ^ g0 = a ^ nil. In addition, ss = g mod solv = g=solv = g0 þ g0þ . The subspace g0 is closed under commutation and exponentiates into a compact subgroup of G0 . The subspace g0þ

0 n n

l r 0

ay a1 7 1y 7 a2 a2 7 y 7 a a2 7 1 7 ay a 7 2 1 7 y 7 a3 a3 7 y 7 a 7 3 5 a3 I

The Cartan–Killing inner product is the trace of the square of this matrix: ðX; XÞ ¼ tr RðXÞ2 ¼ 2ðm11  m22 Þ2 þ 8m12 m21 þ 2n2 The subspace g0 is spanned by ay1 a1 þ ay2 a2 , ay3 , a3 , I, leaving the four operators ay1 a1  ay2 a2 , ay1 a2 , ay2 a1 , ay3 a3 to span g0 . A simple calculation shows that g00 is spanned by ay3 a3 . As a result: Subspace

Spanned by

g0þ

a1y a1  a2y a2 , p1ffiffi2 (a1y a2 þ a2y a1 )

g0

y y p1ffiffi (a a2  a a1 ) 1 2 2 y a3 a3 a1y a1 þ a2y a2 , a3y , a3 , I

g00 g0

The Lie algebra is the direct sum g = sl(2; R) þ u(1) þ h4 .

Lie Groups: General Theory

Structure of Semisimple Lie Algebras The Cartan–Killing metric g is nonsingular on a semisimple Lie algebra. The metric and its inverse g , can be used to raise and lower indices. In particular, the tensor whose components are c = c  g  is thirdorder antisymmetric: c = c = c =c . . . . Classification of semisimple Lie algebras is equivalent to classifying such tensors. Another useful way to describe semisimple Lie algebras is to search for a canonical structure for the commutation relations. A useful canonical form is an eigenvalue form ½X; Y  ¼ Y In a basis Xi , with X = xi Xi and Y = yj Xj , this equation reduces to a standard eigenvalue equation for the regular representation XX  yj Rðxi Xi Þj k  j k Xk ¼ 0 j

k

293

pffiffiffiffiffiffiffiffiffi i x x. The only linear operators that commute with X are scalar multiples of X. There is one independent homogeneous operator that commutes with all generators Xi , obtained by the substitutions xi ! Li (for so(3)) or xi ! Si (for su(2)): C2 ðLÞ ¼ 2 ðxi ! Li Þ ¼ L21 þ L22 þ L23 The secular equation [1] is over the field of real numbers. This is not an algebraically closed field. There is no guarantee that the number of independent functions j (x) in the secular equation is equal to the number of (real) roots of this equation until we extend the field from R to C, which is algebraically closed. As a result, the classification of semisimple Lie algebras is done over complex numbers. After the complex extensions of the simple Lie algebras have been classified, their different inequivalent real forms can be determined.

Root Spaces

Thus, the search for a standard form for the commutation relations reduces to a study of the secular equation detðRðXÞ  IÞ ¼

n X ðÞnj j ðXÞ ¼ 0

½1

j¼0

The coefficients j (X) are homogeneous polynomials of degree j in the coefficients xi of X = xi Xi . In order to extract maximum information from this secular equation, a generic vector X 2 g is chosen. Such a choice minimizes all degeneracies. With a generic choice of X 2 g, it is useful to define the rank, l, of the Lie algebra g as: 1. the number of functionally independent coefficients j (X) in the secular equation; 2. the number of independent roots, 1 , 2 , . . . , l of the secular equation; 3. the dimension of the subspace H  g that commutes with X; and 4. the number of independent (Casimir) operators that commute with all Xi : Cj (X) = j (xi ! Xi ): [Cj (X), Xi ] = 0. For example, for equation for X = xi Xi 22 0 66 det44 x3 x2

so(3) or su(2), the secular is 3 3 x3 x2 7 7 0 x1 5  I3 5 x1 0

3

¼ ðÞ þ ðÞ2 ðxÞ ¼ 0 where 2 (x) = x21 þ x22 þ x23 . The rank is l = 1. There is one independent coefficient 2 (x)pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and one ffi independent root of this equation, 1 =  ij xi xj =

When the secular equation for the regular representation of a generic element in a Lie algebra is solved, the commutation relations can be put into a simple and elegant canonical form. This canonical form depends on the rank, l, of the Lie algebra, not the dimension, n, of the Lie algebra. This provides a very useful simplification, as n  l2 . For this canonical form, the independent roots 1 (x), 2 (x), . . . , l (x) are gathered into a single vector a with l components. The vectors a = (1 , 2 , . . . , l ) are called root vectors. The root vectors exist in an l-dimensional space on which a positive-definite inner product can be defined. The root vectors for a rank-l semisimple Lie algebra g span this Euclidean space. The basis vectors of g can be identified with the roots in the root space. The roots in a root space have the following properties: 1. A positive-definite metric can be placed on the root space. 2. The vector 0 is a root. 3. The root 0 is l-fold degenerate. 4. If a is a root and ca is a root, c = 1, 0. 5. If a and b are roots, 0

b ¼b

2a b a a a

is also a root and 2a b=a a is an integer, n1 . In 0 fact, b is the root obtained by reflecting b in the hyperplane orthogonal to a. 6. The set of reflections generated by nonzero roots itself forms a group, the Weyl group of the Lie algebra.

294 Lie Groups: General Theory

7. The angle between roots a and b is determined by cos2 ða; bÞ ¼

It is possible to build up all possible root space diagrams using an ‘‘Aufbau’’ construction. We start with a rank-1 root space. This consists of three roots in R1 : a, 0, a. To construct rank-2 root spaces, a new noncolinear root b is adjoined to the two nonzero roots. The new root and the old roots span R2 . The new root can only have a limited set of angles with the roots already present. The set of roots a, b is completed by reflection in hyperplanes orthogonal to all roots present. If any pair of roots violates the angle conditions, the result is not a root space. In this way, the rank-2 root spaces G2 (30 ),B2 =C2 (45 ),A2 (60 ), and D2 =A1 þ A1 (90 ) are constructed from A1 . Proceeding in this way, it is possible to construct rank-3 root spaces (B3 , C3 , A3 = D3 ) from the rank-2 root spaces, the rank-4 root spaces from the rank-3 root spaces, and so forth. Ultimately, there are four unending chains An , Bn , Cn , Dn and five exceptional root spaces G2 , F4 , E6 , E7 , E8 . The rank-2 root spaces are shown in Figure 3 and the rank-3 root spaces are shown in

a b a b n1 n2 1 2 3 ¼ ¼ 0; ; ; ; 1 a ab b 4 4 4 2 2

The integers n1 , n2 for noncolinear roots are constrained by jn1 n2 j < 4. 8. The relative lengths of the roots are determined by the angles between them: cos2 ( (a, b))

(a, b) 

a a=b b 

30 , 150 45 , 135 60 , 120

3/4 2/4 1/4

3 1 2 1 1

9. When the roots are normalized so that X X i j ¼ ij or a a ¼l a6¼0

a6¼0

the commutation relations can be placed in the canonical form presented in the next section. – α1 = – √3/2e1 +

– α1 – α2 = – √3/2e1 +

3 2

√3/2e1 + 23 e2 α1 + 3α2

e2 e2 = α2

1 2 e2

–2α1 – 3α2 = – √3e1 – α1 – 2α2 = – √3/2e1 –

– α1 = –e1 + e2

α1 + 2α2 = √3/2e1 – 21 e2

α2

α2 = e1 + e2

2α1 + 3α2 = – √3e1 1 2

= e2

α1 + α2 = √3/2e1 – 21 e2 –e2 = – α2

– α1 – 3α2 = – √3/2e1 –

3 2

– α2 = –e1 – e2

α1 = √3/2e1 –

e2

G2 = e1 = √1/12

3 2

– α2

α1

±e1

α1 = e1 – e2

– α2

±e1 ± e2

e2

3

1

α1

α2

B2 =

+

±e2′

1

1

α1

α2

A1

⊕

A2

α2 = 2e2 – α1 = –e1 + e2

α2 = e2

– α1 – α2 = –e1 – α1 – 2α2 = –e1 – e2

α1 + 2α2 = e1 + e2

α1 + α2 = e1

–2α1 – α2 = –2e1

α1 = e1 + e2

– α2 = –e2

α1 + α2 = e1 + e2

– α1 = –e1 + e2

2α1 + α2 = 2e1

– α1 – α2 = –e1 – e2

α1 = e1 – e2 – α2 = –2e2

±e1 ±e2; ±e1; ±e2 e1 = B2

1 6

±e1 ±e2; ± 2e1; ± 2e2 e1 = 1/12

2

1

α1

α2

Figure 3 Rank-2 root spaces: G2 30 , B2 = C2 45 , A2 60 , D2 = A1 þ A1 90 .

C2

2

1

α1

α2

Lie Groups: General Theory

295

–e,+e3

e3

–e2,+e4 –e2+e4

e2+e3

–e2,+e3

+e1+e3

–e1+e2

–e1,+e2

e1–e3

e1

e

e1–e4 –e1+e2 +e1,–e2

+e1+e2

e1

–e1,–e3

e1–e4

A3

+e2–e3

–e3

e1–e4

B3

+e1,–e3 –e1+e3

+e2 +e3

–e2+e3 –2e1

+e1 +e3

–e1+e4

–2e2 2e2 +e1–e3

+e2–e3 –e2–e3

+e2–e3

2e1

C3

+e1–e3

–e1–e3 +e2–e3

–e2–e3

D3

+e1–e3

–2e3

Figure 4 Rank-3 root spaces: A3 , B3 , C3 , D3 = A3 .

Figure 4. The normalization factors (cf. point (9) above) are shown for the rank-2 root spaces in Figure 3.

where b  (m þ 1)a and b þ (n þ 1) a are not roots (cf. Figure 5). The structure constants are 2 1 Na; b ¼ 2 nð1 þ mÞða aÞ

Canonical Commutation Relations The canonical commutation relations are expressed in terms of root vectors. The l operators in g with the l-fold degenerate root vector 0 are H1 , H2 , . . . , Hl . These l operators mutually commute. In a matrix Lie algebra, they can be taken as simultaneously commuting diagonal matrices. Associated with each nonzero root a 6¼ 0, there is exactly one basis vector, Ea , in g. The canonical commutation relations are expressed in terms of the roots as follows:   Hi ; Hj ¼ 0 1  i; j  l ½Hi ; Ea  ¼ i Ea ½Ea ; Ea  ¼ a H ( Nab Eaþb   Ea ; Eb ¼ 0

The operators H and Ea are often called diagonal and shift operators, respectively. They are generalizations of the shift operators J3 and J of angular momentum theory. The general idea is as follows. Since the operators Hi mutually commute, the matrices (Hi ) representing these operators can be chosen as diagonal in any matrix representation.

β–α

β

β+α

–α

α

a þ b a root a þ b not a root

The structure constants Nab are determined from a recursion relation derived from a chain of roots b  m a, b  (m  1)a, . . . , b þ (n  1)a, b þ na,

β + 2α

–β

Figure 5 An a chain containing .

296 Lie Groups: General Theory

The action of any of these operators on a basis vector in this representation is Hi jmi = mi jmi. The operator Ea shifts the eigenvalue of H according to

antisymmetric, of USp(2n) that are symmetric, and of SO(2n) that are antisymmetric (bosons $ symmetric, fermions $ antisymmetric).

HðEa jmiÞ ¼ ð½H; Ea  þ Ea HÞjmi ¼ ða þ mÞðEa jmiÞ In this sense the operators Ea act on basis vectors jmi in such a way that the eigenvalue m is shifted by a to m þ a. For the simple classical Lie algebras, the roots can be expressed in terms of an orthogonal Euclidean basis set as shown in Table 1 and Figures 3 and 4 for the rank-2 and rank-3 root spaces. The roots for the five remaining inequivalent simple Lie algebras (‘‘exceptional’’ algebras) are shown in Table 2. The diagonal and shift operators for several of the classical Lie algebras can be related to bilinear products of boson or fermion creation and annihilation operators. For u(n), the bilinear products ayi aj are related to Ea with a = ei  ej , 1  i 6¼ j  n, and Hi = ayi ai . This holds for either boson or fermion operators. For sp(2n; R), we have the identifications with bilinear products of boson operators as follows: þei þ ej $ byi byj , þei  ej $ byi bj , ei  ej $ bi bj , and Hi = byi bi . In particular, þ2ei $ by2 i and 2ei $ b2i . For so(2n), we have the identifications with bilinear products of fermion operators as follows: þei þ ej $ fiy fjy , þei  ej $ fiy fj , ei  ej $ fi fj , and Hi = fiy fi . In particular, fiy fiy = fi2 = 0. These identifications make it a relatively simple matter to construct unitary matrix representations of the compact Lie groups SU(n) that are symmetric or

Dynkin Diagrams Every root in a rank-l root space can be represented as a linear combination of l ‘‘basis roots.’’ These basis roots can be chosen in such a way that all coefficients are integers. In fact, the basis roots can be chosen so that all linear combinations that are roots involve only positive integers (and zero) or only negative integers and zero. This comes about because every shift operator Ed can be written as a multiple commutator    Ed  Ea ; Eb ; Eg ; d ¼ a þ b þ g One simple way to construct such a basis set of fundamental roots is to construct an (l  1)-dimensional plane through the origin of the root space that contains no nonzero roots, and choose as l fundamental roots the l roots on one side of this hyperplane that are closest to it. For the classical simple Lie algebras, the fundamental roots are: Root Space Al1 Dl Bl Dl

a1

a2

a l1

al

e1  e2 e1  e2 e1  e2 e1  e2

e2  e3 e2  e3 e2  e3 e2  e3

e l1  e l e l1  e l e l1  e l e l1  e l

e l1 þ e l þ1e l þ2e l

Table 1 Roots for the simple classical Lie groups and algebras Group

Algebra

Root space

Rank

Roots

Conditions

SU(l ) SO(2l ) SO(2l þ 1) Sp(l) = USp(2l)

su(l) so(2l) so(2l þ 1) sp(l) = usp(2l)

Al1 Dl Bl Cl

l 1 l l l

þe i e i e i e i

1i 1i 1i 1i

 ej ej e j , e k e j , 2e k

6¼ j  l 0. This implies that if c := cj = cjþ1 =    = cjþq for some q 0, then catM (Kc ) q þ 1. Assume, by contradiction, that catM (Kc )  q, let Uc be an open neighborhood of Kc such that catM (Uc ) = catM (Kc ) (Uc = ; if q = 0), " > 0 such that (1, f cþ" n Uc )  f c" , and A 2 Ajþq such that supA f  c þ ", that is, A  f cþ" . Then catM ðð1; A n Uc ÞÞ catM ðA n Uc Þ catM ðAÞ  catM ðUÞ j giving the contradiction c  sup(1, A) f  c  ". Notice that, for each j, cj = inf {c 2 R : catM (f c ) j}, which shows that the cj are precisely those levels of f where catM (f c ) changes. The presence of critical values is detected by changes in the topology of the

330 Ljusternik–Schnirelman Theory

sublevel sets f c when c varies, a common feature of many techniques for finding critical points of functions. A direct consequence is that for each even f 2 C2 (Rn , R), system [1] has at least n pairs of solutions (u, u) with kuk = 1. Indeed, the solutions of [1] are the critical points of f on Sn1 . As f takes the same values at antipodal points, it is well defined on the projective space Pn1 , and cat(Pn1 ) = n. The Ljusternik–Schnirelman theorem can be extended to the C1 -situation. The category of M gives a lower bound for the number of critical points of f on the closed manifold M. If Crit(M) denotes the minimum of the number of critical points of all C1 -functions on M, so that Crit(M) cat(M), an interesting question is to estimate the gap Crit(M)  cat(M). For M closed connected, Crit(M)  dim(M) þ 1 (Takens). If Crit(M) = 2, M is homeomorphic to a sphere, so that the equality Crit(S) = cat(S) for homotopy spheres is equivalent to Poincare´’s conjecture! Manifolds with Crit(M) = cat(M) þ 1 are known, but not with Crit(M) > cat(M) þ 1.

principle, the T-periodic solutions of [7] are the critical points of the action functional # Z T" 0 ku ðtÞk2  VðuðtÞÞ þ hðtÞuðtÞ dt f ðuÞ ¼ 2 0 on the Hilbert space HT1 obtained by completion of the space of T-periodic C1 functions for the norm associated with the inner product Z T Z T hu; vi :¼ uðtÞ  vðtÞ dt þ u0 ðtÞ  v0 ðtÞ dt 0

0

 = 0 that f is bounded It follows easily from condition h j from below and that f (u þ 2e ) = f (u) for all u 2 HT1 , with ej the jth unit vector in R n (1  j  n). Consequently, we can see f as defined on the Riemannian f1 , where H f1 = {u 2 H 1 : u  = 0}. It is manifold Tn  H T

T

T

f1 ) = cat(Tn ) = n þ 1 and easy to show that cat(Tn  H T f1 . that f satisfies Palais–Smale condition on Tn  H T

Consequently, system [7] has at least n þ 1 geometrically distinct T-periodic solutions. The same result holds for the more general systems Mu00 þ Au þ rFðuÞ ¼ hðtÞ

Ljusternik–Schnirelman Theory in Infinite-Dimensional Manifolds The main difficulty in extending the results of the previous section to functions defined on infinitedimensional manifolds lies in the lack of compactness. J T Schwartz and Palais have shown that such an extension is possible for functions f satisfying on M a compactness property (allowing an infinitedimensional deformation lemma), now referred to as the Palais–Smale condition: each sequence (uk ) with (f (uk )) bounded and limk ! 1 rf (uk ) = 0 has a convergent subsequence. Such a condition can be localized at level c by replacing the boundedness of (f (uk )) by limk ! 1 f (uk ) = c. The infinite-dimensional extension of Ljusternik–Schnirelman’s theorem goes as follows: Let M be an infinite-dimensional Riemannian (or even Finsler) connected complete manifold of class C1 without boundary. Any f 2 C1 (M, R) bounded from below and satisfying Palais–Smale condition has at least cat(M) distinct critical points. A simple application can be given to the periodic solutions of period T (T-periodic solutions) of Lagrangian systems u00 þ rVðuÞ ¼ hðtÞ

½7

where V 2 C1 (R n , R), 2-periodic in each component uj (1  j  n), h is continuous, T-periodic and  equal to zero. By the least action has mean value h

occurring in the theory of multipoint Josephson junctions or in space discretizations of the sine-Gordon equation. In particular, the classical forced pendulum equation u00 þ a sin u ¼ hðtÞ has at least two geometrically distinct T-periodic  = 0, a result solutions when h is T-periodic and h first proved, in a different way, by Mawhin and Willem. Another way to study nonlinear eigenvalue problems of the form f 0 ðuÞ ¼ g0 ðuÞ in a Hilbert or a suitable reflexive Banach space X is based upon a Rayleigh–Ritz approximation through a sequence of finite-dimensional problems, where the classical theory is applied. Conditions upon f , g 2 C1 (X, R) are given, generalizing Ljusternik–Schnirelman’s ones, which ensure the existence of infinitely many solutions. Again, some compactness is needed to justify the limit process, and expressed by some assumptions upon f and g too lengthy to be reproduced here. The following application is exemplary. Let   R N be a bounded domain and X = W01, p (), p > 1, be the Sobolev space of functions u :  ! R obtained as the completion of the smooth functions with compact support

Ljusternik–Schnirelman Theory

R in  for the norm kuk1, p = (  kru (x)kp dx)1=p . 1, p Define the functionals f and g on W0 () by Z Z p f ðuÞ ¼ kruðxÞk dx; gðuÞ ¼ juðxÞjp dx 



The critical points of f on {u 2 X : g(u) = 1} correspond to the nontrivial solutions of the Dirichlet eigenvalue problem p u ¼ jujp2 u in ;

u ¼ 0 on @

½8

for the p-Laplacian operator p defined by   p uðxÞ :¼ r  kruðxÞkp2 ruðxÞ which occurs in the modelization of various problems in a porous medium. An eigenvalue is any  2 R such that problem [8] has a nontrivial solution. The Ljusternik–Schnirelman technique implies the existence of a sequence of eigenvalues going to infinity, with the usual minimax characterization. When N = 1, direct computations show that this sequence gives all eigenvalues, but the problem remains open for N 2. The corresponding forced problem p u  jujp2 u ¼ hðxÞ in ;

u ¼ 0 on @

is always solvable (although not uniquely) when  is not an eigenvalue, but solvability conditions at the higher eigenvalues (Fredholm alternative) remain almost terra incognita.

Index Theories and Critical Points of Symmetric Functionals on a Banach Space Closely related to the Ljusternik–Schnirelman category is the concept of index associated to the action of a compact topological group G on a normed space X, that is, to a continuous map G  X ! X, [g, u] 7! gu such that 1  u = u, (gh)u = g(hu), u 7! gu is linear. The action is isometric if kguk = kuk, A  X is invariant if gA = A for all g 2 G, f : X ! R is invariant if f g = f for all g 2 G, and h : X ! X is equivariant if g h = h g for each g 2 G. Let Fix G = {u 2 X : gu = u for all g 2 G}. The aim of an index is to measure the size of invariant sets. Explicitly, an index theory associates to each closed invariant subset A of X a non-negative (possibly infinite) integer G-ind(A), its G-index, such that 1. G-ind(A) = 0 if and only if A = ;; 2. if R : A ! B is equivariant and continuous, G-ind(A)  G-ind(B); 3. G-ind(A [ B)  G-ind(A) þ G-ind(B); and

331

4. if A is compact, there is a closed invariant neighborhood U of A such that G-ind(U) = G-ind(A). A first example of index is Krasnosel’skii’s genus or Z2 -index which corresponds to the action 0  u = u, 1  u = u of G = Z2 . The invariant sets are the ones symmetric with respect to the origin and Z2 -ind(A) is defined by Z2 -ind(;) = 0 and, for A 6¼ ;, as the smallest integer k such that there exists an odd h 2 C(A, Rk n {0}). A consequence of the Borsuk–Ulam theorem in algebraic topology is that any symmetric bounded neighborhood of the origin in Rn has Z2 -index equal to n. Furthermore, for a compact A  Rn n {0} symmetric with respect e = A=Z2 (A with antipodal to the origin, and A e points identified), one has Z2 -ind(A) = catRn n{0} (A). 1 A second example, the S -index, is important in the study of periodic solutions of autonomous Hamiltonian systems. S1 -ind(;) = 0 and for a nonempty closed invariant A  X, S1 -ind(A) is defined as the smallest integer k such that there exists a positive integer n and h 2 C(A, Ck n {0}) with h g = gn h for all g 2 S1 . A Borsuk– Ulam-type theorem for S1 -equivariant mappings implies that if Z is a finite-dimensional invariant subspace of X such that Fix S1 \ Z = {0} and D is an open bounded invariant neighborhood of 0 in Z, then S1 -ind(@D) = (1=2)dim Z. As the category of a Banach space X = 1, the classical Ljusternik–Schnirelman approach does not provide any information about the multiplicity of the unconstrained critical points of f 2 C1 (X, R). If f is invariant under the action on X of a compact group G and satisfies Palais–Smale condition, a Ljusternik–Schnirelman minimax method associated to a G-index provides multiplicity results for unconstrained critical points. Letting Aj ¼ fA  X : A is compact, invariant, and G-indðAÞ jg cj ¼ inf sup f A2Aj

ðj ¼ 1; 2; . . .Þ

A

one shows as in classical Ljusternik–Schnirelman theory that if c := cj = cjþ1 =    = cjþq for some j and some q 0, then G-ind(Kc ) q þ 1. The proof uses an equivariant deformation lemma.

Z2 - and S 1 -Invariant Functionals In the case of the Z2 -action, the following multiplicity result holds for possibly unbounded even f 2 C1 (X, R) satisfying the Palais–Smale condition and having the mountain pass geometry: if Y \ {u 2 X : f (u) 0} is bounded for each finite-dimensional subspace Y of X,

332 Ljusternik–Schnirelman Theory

f (0) = 0, and f (u) a > 0 on @B(r), then f has infinitely many couples of critical points. As an application, the semilinear Dirichlet problem u þ u þ jujp1 u ¼ 0 in  u ¼ 0 on @

½9

has infinitely many solutions when   R N is bounded, 1 < p < (N þ 2)=(N  2), and  < 1 , the smallest eigenvalue of  with Dirichlet boundary conditions. The corresponding energy functional, defined on W01, 2 () by # Z " kruðxÞk2 juðxÞj2 juðxÞjpþ1   dx f ðuÞ ¼ 2 2 pþ1  satisfies the Palais–Smale condition. This condition fails in the critical case where p = (N þ 2)=(N  2), at least at some levels c, and this lack of compactness creates both difficulties and interesting phenomena. This situation, which occurs in many important problems of geometry and physics (harmonic maps, Yang–Mills connections, Yamabe problem, equations of constant mean curvature, closed geodesics problems, etc.), reveals indeed, in physical terms, ‘‘phase transitions’’ or ‘‘particle creations’’ at the levels where the Palais–Smale condition fails. In the special case of eqn [9] with p = (N þ 2)=(N  2), if N 4, a positive solution exists when  2 [0, 1 ], and, if N = 3, the same is true for  2 [ , 1 ] and some  2 [0, 1 ], with the optimal value  = 1 =4 when  is a ball. For N 4, [9] has at least cat() nontrivial solutions when  2 [0,  ] for some  < 1 . Such a lack of compactness, which can also occur for eqn [9] in RN (nonlinear Schro¨dinger equation), is associated to the invariance of f with respect to the action of some noncompact group, coming, for example, from scale or gauge invariance. P L Lions’ concentration–compactness method is useful to analyze those problems. The following multiplicity theorem holds for an S1 -invariant f 2 C1 (X, R) satisfying Palais–Smale condition. Let Fix(S1 ) = {0} and Z be a closed invariant vector subspace of X of positive finite dimension. If f is bounded from below, f (u)  c < 0 whenever u 2 Z and kuk = r, and f (0) 0 for u 2 Fix(S1 ) \ (f 0 )1 (0), then f has at least dim Z=2 distinct S1 -orbits of critical points of f with critical values less or equal to c. This abstract theorem provides multiplicity results for the periodic solutions (closed orbits) of autonomous Hamiltonian systems in R2n Ju0 þ rHðuÞ ¼ 0

½10

where J is the symplectic matrix, H 2 C1 (R2n , R), and c 2 R is such rH(u) 6¼ 0 for u 2 H 1 (c). If

such H 1 (c) bounds a strictly convex compact set C p ffiffiffi that B[r]  C  B[R] for some 0 < r < R < 2r, then [10] has at least n closed orbits on H 1 (c). The problem is reduced to finding the critical points of a suitable dual action functional acting on some space X of 2-periodic functions having mean value zero. The S1 -action on X is defined by time translations [, u] 7! u = u(  þ) for all  = ei 2 S1 . One takes, in the abstract result above, Z = {(cos t)e þ (sin t) Je : e 2 R2n }, so that dim Z = 2n. The complete proof is quite involved, and, although some improvements of Ekeland–Lasry conditions have been obtained, the problem remains open to know if some pinching condition of the energy surface between spheres or ellipsoids is necessary.

Some Extensions When dealing with unbounded functionals, it may be convenient to replace the Ljusternik–Schnirelman category catX (A) by a relative category catX, Y (A) with respect to a closed subset Y where, in the covering of A occurring in the classical definition, a set A0  Y is added, which is continuously deformable in X into a subset of Y in such a way that points of Y remain in Y during the deformation. Clearly catX, ; (A) = catX (A). This allows us to prove, under some restrictions on the coefficients and the period, the existence of at least four periodic solutions for the double pendulum with periodic forcing of mean value zero. The classical Ljusternik– Schnirelman category gives at least three periodic solutions without restrictions, and the question of their necessity to obtain four solutions is open. The relative category also gives a simpler proof of Conley–Zehnder’s version of the Arnol’d conjecture (the existence of at least 2n þ 1 geometrically distinct 1-periodic solutions for the Hamiltonian system Ju0 þ rHðt; uÞ ¼ 0 with H 1-periodic in each variable), under minimal regularity assumptions upon H. The general conjecture, namely that the minimum number of fixed points of all Hamiltonian symplectomorphisms of a closed symplectic manifold M is larger than the minimum number of critical points of smooth functions f on M, remains open. In another direction, a Ljusternik–Schnirelman theory for functionals defined on closed convex sets of a Banach space has been developed, which is specially well suited for the study of the Plateau problem for minimal surfaces, for surfaces of constant mean curvature, as well as for variational inequalities.

Localization for Quasiperiodic Potentials See also: Bifurcations of Periodic Orbits; Compact Groups and Their Representations; Floer Homology; Ginzburg–Landau Equation; Inequalities in Sobolev Spaces; Minimal Submanifolds; Minimax Principle in the Calculus of Variations; Saddle Point Problems; Sine-Gordon Equation; Spectral Theory for Linear Operators.

Further Reading Ambrosetti A (1992) Critical Point and Nonlinear Variational Problems, Cours de la Chaire Lagrange. Paris: Socie´te´ Mathe´matique de France. Cornea O, Lupton G, Oprea J, and Tanre´ D (2003) Ljusternik– Schnirelman Category. Providence: American Mathematical Society. Courant R and Hilbert D (1953–62) Methods of Mathematical Physics, vol. 2. New York: Wiley-Interscience. Ekeland I (1990) Convexity Methods in Hamiltonian Mechanics. Berlin: Springer. Ekeland I and Ghoussoub N (2002) Selected new aspects of the calculus of variations in the large. Bulletin of the American Mathematical Society (NS) 39: 207–265. Fucˇik S, Necˇas J, Soucˇek J, and Soucˇek V (1973) Spectral Analysis of Nonlinear Operators. Berlin: Springer. Ghoussoub N (1993) Duality and Pertubation Methods in Critical Point Theory. Cambridge: Cambridge University Press.

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Gould SH (1966) Variational Methods for Eigenvalue Problems, 2nd edn. Toronto: Toronto University Press. Hofer H and Zehnder E (1994) Symplectic Invariants and Hamiltonian Dynamics. Basel: Birkha¨user. Krasnosel’skii MA (1963) Topological Methods in the Theory of Nonlinear Integral Equations. Oxford: Pergamon. Ljusternik LA (1966) The Topology of the Calculus of Variations in the Large. Providence: American Mathematical Society. Ljusternik LA and Schnirelman LG (1930) Me´thodes topologiques dans les proble`mes variationnels (Russian). Moscow: Trudy Scient. Invest. Inst. Math. Mech. (French translation (1934) Actualite´s scientifiques et industrielles no. 188. Paris: Hermann). Mawhin J and Willem M (1989) Critical Point Theory and Hamiltonian Systems. New York: Springer. Palais RS (1970) Critical point theory and the minimax principle. In: Proceedings of Symposia in Pure Mathematics, vol. 15, pp. 115–132. Providence: American Mathematical Society. Rabinowitz P (1986) Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence: American Mathematical Society. Schwartz JT (1969) Nonlinear Functional Analysis. New York: Gordon and Breach. Struwe M (1996) Variational Methods, 2nd edn. Berlin: Springer. Willem M (1996) Minimax Theorems. Basel: Birkha¨user. Zeidler E (1985) Nonlinear Functional Analysis III. New York: Springer.

Localization for Quasiperiodic Potentials S Jitomirskaya, University of California at Irvine, Irvine, CA, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction Discrete Schro¨dinger operators with quasiperiodic potentials are operators acting on ‘2 (Zd ) and defined by H ¼  þ V

½1

where  is the lattice tight-binding Laplacian  1; dist ðn; mÞ ¼ 1 ðn; mÞ ¼ 0; otherwise and V(n, m) = Vn (n, m) is a potential given by Vn = f (T1n1    Tdnd ),  2 Tb , where Ti  =  þ !i , and ! is an incommensurate vector. In certain cases  may also be replaced by a long-range Laplacian L(n, m) = L(n  m) with L(n) ! 0 sufficiently fast. The questions of interest in the study of quasiperiodic and other ergodic operators are the nature and structure of the spectrum, behavior of the eigenfunctions, and the quantum dynamics: properties of the

time evolution t = eitH 0 of an initially localized wave packet 0 . Of particular importance is the phenomenon of Anderson localization which is usually referred to the property of having pure point spectrum with exponentially decaying eigenfunctions. A stronger property of dynamical localization (see the section ‘‘Dynamical localization’’) indicates the insulator behavior, while ballistic transport, which for d = 1 follows from the absolutely continuous spectrum, indicates the metallic behavior. Operators with ergodic potentials always have spectra (and pure point spectra, understood as closures of the set of eigenvalues) constant for a.c. realization of the potential. The individual eigenvalues however depend very sensitively on the phase. Moreover, the pure point spectrum of operators with ergodic potentials never contains isolated eigenvalues, so pure point spectrum in such models is dense in a certain closed set. An easy example of an operator with dense pure point spectrum is H1 which is operator [1] with 1 = 0, or pure diagonal. It has a complete set of eigenfunctions, characteristic functions of lattice points, with eigenvalues Vj . H may be viewed as a perturbation of H1 for small 1 . However, since Vj are dense, small denominators (Vi  Vj )1 make any

334 Localization for Quasiperiodic Potentials

perturbation theory difficult, for example, requiring intricate KAM-type schemes. Various methods developed for the Anderson model (where Vn are i.i.d.r.v.’s) such as Fro¨hlich– Spencer multiscale analysis and its enhancements, or Aizenman–Molchanov method, do not work for quasiperiodic potentials as, among other reasons, quasiperiodicity does not allow for nice perturbations. The situation here is more difficult and the theory is far less developed than for the random case. With a few exceptions, the results are confined to the one-dimensional setting, and also the case of one frequency (b = 1) has been much better understood than that of higher frequencies. One might expect that H with  small can be treated as a perturbation of H0 = , and therefore have absolutely continuous spectrum. It is not the case though for random potentials in d = 1, where Anderson localization holds for all . The same is expected for random potentials in d = 2 (but not higher). Moreover, in one-dimensional case, there is strong evidence (numerical, analytical, as well as rigorous) that even models with very mild stochasticity in the underlying dynamics (and sufficiently nice sampling functions) have point spectrum for all values of , like in the random case (e.g., Vn = f (n  þ ), for any  > 1). At the same time, for quasiperiodic potentials, one can in many cases show absolutely continuous spectrum for  small as well as pure point spectrum for  large (see below), and therefore there is a metal–insulator transition in the coupling constant. It is an interesting question whether quasiperiodic potentials are the only ones with metal–insulator transition in 1D.

Perturbative and Nonperturbative Approaches It is probably fair to say that much of the theory of qusiperiodic operators has been first developed around the almost-Mathieu operator, which is H;!; ¼  þ f ð þ n!Þ

½2

acting on ‘2 (Z), with f : T ! T; f () = cos (2). Several KAM-type approaches, starting with the pioneering work of Dinaburg–Sinai in 1975, were developed, in 1980s and 1990s, for this or similar models in both large and small coupling regimes. Of those, the most robust and detailed is the reducibility result of Eliasson (1998) that settled the case of small couplings for sufficiently regular potentials. The common feature of those perturbative approaches is that, besides all of them being rather

intricate multistep procedures, they rely extensively on eigenvalue and eigenfunction parametrization and perturbation arguments. The common feature of the perturbative results in the quasiperiodic setting is that they typically provide no explicit estimates on how large (or small) the parameter  should be, and, more importantly,  clearly depends on ! at least through the constants in the Diophantine characterization of !. In contrast, the nonperturbative results allow effective (in many cases even optimal) and, most importantly, independent of !, estimates on . The latter property (uniform in ! estimates on ) has been often taken as a definition of a nonperturbative result. Recently developed nonperturbative methods are also quite different from the perturbative ones in that they do not employ multiscale schemes: usually only a few (from one to three) sufficiently large scales are involved, do not use the eigenvalue parametrization, and rely instead on direct estimates of the Green’s function. They are also significantly less involved, technically. One may think that in these latter respects they resemble the Aizenman–Molchanov method for random localization. It is, however, a superficial similarity, as, on the technical side, they are still closer to and do borrow certain ideas from the multiscale analysis proofs of localization.

Lyapunov Exponents Here for simplicity we consider the quasiperiodic case, although the definition of the Lyapunov exponents and some of the mentioned facts apply more generally to the one-dimensional ergodic case. Let d = 1. For an energy E 2 R the Lyapunov exponent (E) is defined as R1 ln kMk ð; EÞkd  ðEÞ ¼ lim 0 ½3 n!1 k where Mk ð; EÞ ¼

0  Y E  f ð!n þ Þ 1

n¼k1

1 0



is the k-step transfer matrix for the eigenvalue equation H = E. In physics literature, positivity of the Lyapunov exponent is often taken as an implicit definition of localization, as Lyapunov exponent is often called the inverse localization length. Thus, we will be interested in the regime when Lyapunov exponents are positive for all energies in a certain interval intersecting the spectrum. If this condition holds for all E 2 R, there is no absolutely continuous

Localization for Quasiperiodic Potentials

component in the spectrum for all . Positivity of Lyapunov exponents, however, does not imply localization or exponential decay of eigenfunctions (in particular, neither for the Liouville ! nor for the resonant  2 Tb ). Nonperturbative methods, at least in their original form, stem to a large extent from estimates involving the Lyapunov exponents and exploiting their positivity. The general theme of the results on positivity of (E), as suggested by perturbation arguments, is that the Lyapunov exponents are positive for large . This subject has had a rich history. The strongest result in this general context up to date is the following theorem (Bourgain 2003): Theorem 1 Let f be a nonconstant real-analytic function on Tb , and H given by [1]. then, for  > (f ), we have (E) > (1=2) ln  for all E and all incommensurate vectors !. Corollaries of Positive Lyapunov Exponents

The almost-Mathieu operator On one hand the almost-Mathieu operator, while simple looking, seems to represent most of the nontrivial properties expected to be encountered in the more general case. On the other hand it has a very special feature: the duality (essentially a Fourier) transform maps H to H4= ; hence  = 2 is the self-dual point. Aubry and Andre in 1980, conjectured that for this model, for irrational ! a sharp metal–insulator transition in the coupling constant  occurs at the critical value of coupling  = 2: the spectrum is pure point for  > 2 and purely absolutely continuous for  < 2. This conjecture was modified based on later discoveries of singular-continuous spectrum in this context for frequencies or phases with certain arithmetic properties. The modified conjecture stated pure point spectrum for Diophantine ! and a.e.  for  > 2 and pure absolutely continuous spectrum for  < 2 for all !,. The spectrum at  = 2 is singular continuous for all ! and a.e. (this follows from a combination of works by Gordon, Jitomirskaya, Last, Simon Avila, and Krikoryan). As with the KAM methods, the almost-Mathieu operator was the first model where the positivity of Lyapunov exponents was effectively exploited (Jitomirskaya 1999): Theorem 2 Suppose ! is Diophantine and (E, !) > 0 for all E 2 [E1 , E2 ]. Then the almost-Mathieu operator has Anderson localization in [E1 , E2 ] for a.e. . The condition on  can be made explicit (arithmetic) and close to optimal. This, combined with the

335

mentioned results on the Lyapunov exponents, critical value  = 2, and duality, gives the following description in the Diophantine case: Corollary 3 has

The almost-Mathieu operator H!, , 

1 for  > 2, Diophantine ! 2 R and almost every  2 R, only pure point spectrum with exponentially decaying eigenfunctions. 2 for  = 2, all ! 62 Q, and a.e.  2 R purely singular-continuous spectrum. 3 for  < 2, Diophantine ! 2 R and a.e.  2 R, purely absolutely continuous spectrum. Precise arithmetic descriptions of !,  are available. Thus, the Aubry–Andre conjecture is settled at least for almost all !, . One should mention, however, that while 1 can be made optimal by existing methods, both 2 and 3 are expected to hold for all  and all ! 62 Q, and such extension remains a challenging problem (see Simon (2000)). The method in the above work, while so far the only nonperturbative method available allowing precise arithmetic conditions, uses some specific properties of the cosine. It extends to certain other situations, for example, quasiperiodic operators arising from Bloch electrons in a perpendicular magnetic field, where the lattice is triangular or has next-nearest-neighbor interactions. However, it does not extend easily to the multifrequency or even general analytic potentials. A much more robust method was developed by Bourgain–Goldstein (2000), which allowed them to extend (a measuretheoretic version of) the above localization result to the general real analytic as well as the multifrequency case. Note that essentially no results were previously available for the multifrequency case, even perturbative. Theorem 4 Let f be nonconstant real analytic on Tb and H given by [2]. Suppose (E, !) > 0 for all E 2 [E1 , E2 ] and a.e. ! 2 Tb . Then for any , H has Anderson localization in [E1 , E2 ] for a.e. !. Combining this with Theorem 1, Bourgain (2003) obtained that for  > (f ), H as above satisfies Anderson localization for a.e. !. Those results were recently extended by S Klein to potentials belonging to certain Gevrey classes. One very important ingredient of this method is the theory of semialgebraic sets that allows one to obtain polynomial algebraic complexity bounds for certain ‘‘exceptional’’ sets. Combined with measure estimates coming from the large deviation analysis of (1=n) ln kMn ()k (using subharmonic function theory and involving approximate Lyapunov exponents),

336 Localization for Quasiperiodic Potentials

this theory provides necessary information on the geometric structure of those exceptional sets. Such algebraic complexity bounds also exist for the almost-Mathieu operator and are actually sharp albeit trivial in this case due to the specific nature of the cosine. Further corollaries of positive Lyapunov exponents for analytic sampling functions f and b = 1 include Ho¨lder regularity of the integrated density of states, zero-dimensionality of spectral measures for all !, , almost Lipshitz continuity of spectral gaps, continuity of measure of the spectrum (in frequency), and vanishing of lower transport exponents for all !, . Some weaker statements are available for b > 1 or f belonging to certain Gevrey classes.

Without Lyapunov Exponents While having led to significant advances, Lyapunov exponents have obvious limitations, as any method based on them is restricted to one-dimensional nearest-neighbor Laplacians. It turns out that the above methods can be extended to obtain nonperturbative results in certain quasi-one-dimensional situations where Lyapunov exponents do not exist. For example, nonperturbative localization results extend to the strip (of arbitrary dimension). The following nonperturbative theorem deals with the case of small coupling: Theorem 5 Let H be an operator [2], where f is real analytic on T and ! is Diophantine. then, for  < (f ), H has purely absolutely continuous spectrum for a.e. . We note that an analog of this theorem does not hold in the multifrequency case (see next section). The results of this type are obtained by a method (developed by Bourgain and Jitomirskaya in 2000–02) that studies large deviations for the quantities of the form (1=n) ln j det (H  E) j and path-determinant expansion for the matrix elements of the resolvent. Those techniques apply also to certain other situations with long-range Laplacians, for example, the kicked-rotor model. Theorem 5 is a result on nonperturbative localization in disguise as it was obtained using duality from a localization theorem for a dual model which has in general a long-range Laplacian and a cosine potential, and was in turn obtained by an extension of the method of Jitomirskaya (1999). A certain measure-theoretic version of it allowing nonlocal Laplacians but leading only to continuous spectrum is also available (see Bourgain (2004)).

Multidimensional Case: d > 1 As mentioned above, there are very few results in the multidimensional lattice case (d > 1). Essentially, the only result that existed before the recent developments was a perturbative theorem – an extension by Chulaevsky–Dinaburg of Sinai’s method to the case of operator [1] on ‘2 (Zd ) with Vn = f (n  !), ! 2 Rd , where f is a cos-type function on T. This also holds nonperturbatively for any realanalytic f (see Bourgain (2004)). Note that since b = 1, this avoids most serious difficulties and is therefore significantly simpler than the general multidimensional case. We therefore have: Theorem 6 For any > 0 there is (f , ), and, for  > (f , ), (, f )  Td with mes() < , so that for !2 = , operator [1] with Vn as above has Anderson localization. This should be confronted with the following theorem of Bourgain: Theorem 7 Let d = 2 and f () = cos 2 in H = H! defined as above. Then for any  measure of ! s.t. H! has some continuous spectrum is positive. Therefore, for large  there will be both ! with complete localization as well as those with at least some continuous spectrum. This shows that nonperturbative results do not hold in general in the multidimensional case! Perturbative results, however, had been obtained, see next section. A similar (in fact, dual) situation is observed for one-dimensional multifrequency (d = 1; b > 1) case at small disorder. One has, by duality: Theorem 8 Let H be given by [2] with , ! 2 Tb and f real analytic on Tb . Then for any > 0 there is (f , ) s.t. for  < (f , ) there is (, f )  Tb with mes() < so that for ! 2 = , H has purely absolutely continuous spectrum. And also Theorem 9 Let d = 1, b = 2 and f be a trigonometric polynomial on T2 with a nondegenerate maximum. Then for any , measure of ! s.t. H! has some point spectrum, dense in a set of positive measure, is positive. Therefore, unlike the b = 1 case (see Theorem 5), nonperturbative results do not hold for absolutely continuous spectrum at small disorder.

Perturbative Localization by Nonperturbative Methods While the above demonstrates the limitations of the nonperturbative results, the nonperturbative

Localization for Quasiperiodic Potentials

337

methods have been applied to significantly simplify the proofs and obtain new perturbative results that previously had been completely beyond reach. Many such applications, that are outside the scope of this article, are described in Bourgain (2004). In particular, new results on the construction of quasiperiodic solutions in Melnikov problems and nonlinear PDEs, obtained by using certain ideas developed for nonperturbative quasiperiodic localization (e.g., the theory of semialgebraic sets), are presented there. Other results in this group contain localization for the skew-shift model by Bourgain–Goldstein–Schlag, almost periodicity for the quantum kicked-rotor model by Bourgain and Bourgain–Jitomirskaya, and localization for potentials in higher Gevrey classes by S Klein. The main goal in a nonperturbative method is to obtain exponential off-diagonal decay for the matrix elements of the Green’s function of box-restricted operators along with subexponential bounds on the distance from the spectrum of such box restrictions to a given energy. From that result one can obtain localization through elimination of energy via an argument involving complexity bounds on semialgebraic sets (see Bourgain (2004)). A nonperturbative way to achieve the desired Green’s function estimates uses Cramer’s rule to represent the matrix elements of the resolvent. Then, in the one-dimensional (in space) case it is often possible to obtain the estimates from the positivity of Lyapunov exponents: uniformly for the numerator, and from large deviation bounds for the subharmonic functions for the denominator. This is done in one step for a sufficiently large scale (see the subsection ‘‘Corollaries of positive Lyapunov exponents’’) A perturbative way consists of establishing the desired estimates in a multiscale scheme: namely, the estimates are proved outside a set of parameters of (subexponentially) decaying (in the size of the box) measure. Moreover, this set should be shown to have a semialgebraic description, in order to make possible sublinear upper bounds on the number of times a trajectory of a given phase (under the underlying rotation or other ergodic transformation of the torus) hits the ‘‘forbidden’’ set. This, plus certain subharmonic function arguments, allows passage to a larger scale through a repeated use of the resolvent identity. An application that is most relevant to the current article is localization for a ‘‘true’’ d > 1 situation. The best currently available result is the following very recent theorem (Bourgain 2005):

is a nonconstant function of i 2 T. Then for any > 0 there is (f , ) s.t. for  > (f , ) there is (, f )  Td with mes() < so that for ! 2 = operator [1] with Vn = f (n1 !1 , n2 !2 ) has Anderson (and dynamical) localization.

Theorem 10 Let d = b and let f be real analytic on Td such that for all i = 1, . . . , d and (1 , . . . , i1 , iþ1 , . . . , d ) 2 Td1 , the map

we will say that H exhibits dynamical localization if hx2 iT < const. We will say that the family strong dynamical localization if R{H }2Tb exhibits 2 b d supt hx it < const. We note that the results T mentioned below will hold with more restrictive

i 7! vð1 ; . . . ; i ; . . . ; d Þ

This result was obtained previously, for d = 2 only, by Bourgain, Goldstein, and Schlag. There were some serious purely arithmetic difficulties that prevented an extension of this result to higher dimensions. In the previous results on localization there were two major steps: estimations on the Green’s function for fixed energy and elimination of energy. The main difficulty in the multidimensional case lies in establishing the sublinear bound described above, that enters in the first step. It is for this bound that an arithmetic condition on ! was needed. The condition used was to guarantee that the number of (n1 , n2 ) 2 [1, N]2 such that (n1 !1 , n2 !2 )(mod Z2 ) 2 S is bounded from above by N  for some  < 1, uniformly for all semialgebraic sets S of degree D, with D0 =D = o(1=N) and with the measure of all horizontal and vertical sections Sx satisfying log mesSx = o( log 1=N). This condition roughly means that too many points close to an algebraic curve of a bounded degree would force it to oscillate more than it should. Such a statement is essentially two dimensional and not extendable to d  3. In Theorem 10, Bourgain circumvents it by using from the beginning the theory of semialgebraic sets to eliminate energy and the translation variable to get conditions on ! (that depend on the potential) already in the first step.

Dynamical Localization Anderson localization does not in itself guarantee absence of quantum transport, or nonspread of an initially localized wave packet, as characterized, for example, by boundedness in time of moments of the position operator. This was first observed in del Rio et al. (1996), where a rather artificial example of coexistence of exponential localization and quantum transport was constructed. However, such phenomena also happen in models of interest to physicists such as the random dimer model. Considering for simplicity the second moment Z 1 TX hx2 iT ¼ jt ðnÞj2 n2 dt T 0 n

338 Localization for Quasiperiodic Potentials

definitions of dynamical localization (involving the higher moments of the position operator) as well. Dynamical localization implies pure point spectrum by RAGE theorem so it is a strictly stronger notion. It turns out that nonperturbative methods allow for such dynamical upgrades as well. For the almostMathieu operator, strong dynamical localization holds throughout the regime of localization. It was shown by Bourgain and Jitomirskaya that in Theorems 4 and 6 as well as some other localization results, dynamical localization also holds (see Bourgain (2004)). However, methods that require elimination of certain frequencies based on implicit conditions currently do not provide sufficient information to obtain strong (i.e., averaged) dynamical localization, like what was done in the almostMathieu case.

of the Ten Martini conjecture, for all irrationals (Avila and Jitomirskaya 2005). It can be shown that proving localization for a large set of phases allows one to conclude reducibility of the transfer-matrix cocycle for the dual model, for a large set of energies, and this in turn can be shown to contradict the presence of an interval in the spectrum.

Quasiperiodic Localization and Cantor Spectrum

Further Reading

A remarkable feature of quasiperiodic operators with b = d = 1 is their tendency to have Cantor spectrum. In particular, it was conjectured that all almost-Mathieu operators (for all nonzero couplings and all irrational frequencies) have Cantor spectrum. This conjecture became known as the Ten Martini problem. In a significant recent development (Puig 2004), it was shown that for Diophantine frequencies Cantor structure of the spectrum follows from localization for phase  = 0, with corresponding eigenvalues being the boundaries of noncollapsed gaps. The key idea here is that for energies dual to eigenvalues of H0 , corresponding to localized eigenfunctions, the rotation number of the transfer-matrix cocycle is of the form k!(modZ), thus they are the ends of the gaps (possibly collapsed). However, a collapsed gap in this case would correspond to reducibility of the system to the identity which can be shown to contradict the simplicity of pure point spectrum for the dual model. Since those energies form a dense subset of the spectrum the result follows. The same idea works, thus establishing Cantor spectrum, for potentials that are generic in certain sense. Localization also played an important role in the final proof

Acknowledgement The research work of this author was partially supported by NSF grant DMS-0300974. See also: Multiscale Approaches; Quantum Hall Effect; Quasiperiodic Systems; Schro¨dinger Operators; Stability Theory and KAM.

Avila A and Jitomirskaya S (2005) The ten martini problem. Preprint. Bourgain J (2003) Positivity and continuity of the Lyapunov exponent for shifts on Td with arbitrary frequency vector and real analytic potential. Preprint. Bourgain J (2004) Green’s Function Estimates for Lattice Schro¨dinger Operators and Applications. Annals of Mathematies Studies, vol. 158. xþ173 pp. Princeton, NJ: Princeton University Press. Bourgain J (2005) Anderson localization for quasi-periodic lattice Schro¨dinger operators on Zd , d arbitrary. Preprint. Bourgain J and Goldstein M (2000) On nonperturbative localization with quasiperiodic potential. Annals of Mathematics 152: 835–879. del Rio R, Jitomirskaya S, Last Y, and Simon B (1996) Operators with singular continuous spectrum: IV. Journal d’Analyse Mathematique 69: 153–200. Eliasson LH (1998) Reducibility and point spectrum for linear quasiperiodic skew products. Doc. Math, Extra volume ICM 1998 II, 779–787. Jitomirskaya S (1999) Metal–insulator transition for the almost Mathieu operator. Annals of Mathematics 150: 1159–1175. Last Y (2005) Spectral theory of Sturm–Liouville operators on infinite intervals: a review of recent developments. In: Sturm– Liouville Theory, pp. 99–120. Basel: Birkhauser. Puig J (2004) Cantor spectrum for the almost Mathieu operator. Communications in Mathematical Physics 244: 297–309. Simon B (2000) Schro¨dinger operators in the twenty-first century. In: Mathematical Physics 2000. pp. 283–288. London: Imperial College.

Loop Quantum Gravity

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Loop Quantum Gravity C Rovelli, Universite´ de la Me´diterrane´e et Centre de Physique The´orique, Marseilles, France ª 2006 Elsevier Ltd. All rights reserved.

Introduction Loop quantum gravity (LQG) is a mathematical formalism that defines a tentative quantum theory of spacetime. Equally, the formalism provides a description of the gravitational field in regimes in which its quantum properties cannot be neglected. The distinctive feature of LQG is to be a quantum field theory consistent with general relativity. According to general relativity, the physical fields that form the world do not live on a background spacetime. Rather, these fields make up spacetime themselves (‘‘background independence’’). Accordingly, the quanta of a quantum field theory compatible with this principle – the s-knots described below – do not live on a background spacetime: rather, they themselves form physical spacetime. This physical idea is realized in the formalism by the gauge invariance under active diffeomorphisms of the manifold on which the fields are originally defined (‘‘diffeomorphism invariance’’). Such gauge invariance renders the localization of the field’s excitations on the manifold physically irrelevant. LQG implements these physical motivations by merging two traditional lines of thinking in theoretical physics. The first is the long-standing idea that gauge fields are naturally understood in terms of variables associated to lines (holonomies of the gauge connection, Wilson loops, Faraday lines, . . .). This idea can be traced to Faraday’s initial intuition that gave birth to modern field theory: physical fields are real entities formed by lines. The second is the backgroundindependent canonical or covariant quantization of general relativity developed by following the ideas of Wheeler, DeWitt, and Hawking. Each of these two lines of research has encountered serious obstructions, but the two turn out to solve each others’ difficulties: the formulation in terms of holonomies renders the old ill-defined background-independent quantum gravity well defined; conversely, background independence cures the divergences associated to the Wilson loop basis. The formalism of LQG can be separated into two parts. A kinematics, describing the quantum properties of space, and a dynamics, describing its evolution. Here we outline the LQG kinematics, and we give only the main result of the LQG dynamics.

LQG can be extended to include standard matter couplings such as fermions and Yang–Mills fields. It finds numerous applications, for instance, in early cosmology, astrophysics and black hole thermodynamics (see Black Hole Mechanics, Quantum Cosmology). So far no empirical evidence supports the physical correctness of this – nor of any other – tentative theory of quantum gravity.

General Relativity in Canonical Form Classical general relativity is the field theory describing the gravitational field and the structure of physical spacetime. It is a well-established physical theory, strongly supported empirically. In its Riemannian version, the theory can be written in canonical form in terms of two fields on a three-dimensional (3D) manifold  with coordinates xa (a = 1, 2, 3): a 2-form E = Ea abc dxa dxb , called the ‘‘triad field’’ and a 1-form A = Aa dxa , called the ‘‘gravitational connection’’ (abc is the totally antisymmetric tensor density). Both take values in the su(2) algebra, and they satisfy the three ‘‘constraint’’ equations G ¼ Da Ea ¼ 0

½1

Ca ¼ tr½Fab Ea  ¼ 0

½2

C ¼ tr½Fab Ea Eb  ¼ 0

½3

Da is the SU(2) covariant derivative defined by the connection A, Fab is the SU(2) curvature of A, and the trace is on su(2). E and A are canonically conjugate: their Poisson brackets are {Ea (x), Ab (y)} = 8Gc3 ba 3 (x, y); where G is the Newton constant, c is the speed of light, ba is the Kronecker delta, and 3 (x, y) is the Dirac-delta on , which is a scalar density in x. The Poisson brackets of G with the fields define their SU(2) gauge transformations: E transforms in the adjoint representation and A transforms as a connection. The Poisson brackets of Ca (more precisely, of an appropriate linear combination of Ca and G) with the fields determine their transformation under a diffeomorphism of : E transforms as a 2-form and A as a 1-form. The Poisson brackets of C with the fields generate their coordinate time evolution. If the t derivatives of the fields E(xa , t) and A(xa , t) are given by their Poisson brackets with (the 3D integral of) p C,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi then (assuming that the determinant ffi E = det tr[Ea Eb ] does not vanish) the metric field

340 Loop Quantum Gravity

g00 = 1, ga0 = 0, gab = tr[Ea Eb ]=E is a general solution of the Riemannian Einstein equations in a fixed gauge. The physical Lorentzian theory can be obtained in this formalism in two ways. Either by adding an appropriate term to eqn [3], or by taking A in sl(2,C) and satisfying a suitable reality condition. (For more details, see Canonical General Relativity.)

Spin Network and s-Knot States LQG can be defined as a Schro¨dinger quantization of the canonical formalism described above. The space of the quantum states is defined as a Hilbert space K of Schro¨dinger wave functionals [A] of the gravitational connection. The nontrivial aspect of this construction is the definition of a scalar product invariant under the two kinematical gauge invariances of the theory: the local SU(2) and the diffeomorphisms transformations generated by the constraints [1] and [2]. The state space K is defined as follows (see Quantum Geometry and its Applications for an essentially equivalent construction). Given an su(2) connection A and an oriented path  : s 2 [0, 1] ! xa (s) 2 , recall that the ‘‘holonomy’’ U[A, ] of A along  is the element of SU(2) defined by d U½A; ðsÞ þ _ a ðsÞAa ððsÞÞU½A; ðsÞ ¼ 0 ds U½A; ð0Þ ¼ 1;

U½A;  ¼ U½A; ð1Þ

½4

½5

where _a (s)  dxa (s)=ds is the tangent to the path. The solution of this equation is usually written in the form R A U½A;  ¼ Pe  ½6 where the path ordered P is understood as acting on the power series expansion of the exponential. Let A be the space of the smooth connections A on . (For technical reasons, it is convenient to consider smooth fields A defined everywhere in  except at most at a finite number of points, and the group Diff  of the ‘‘extended diffeomorphisms’’ defined by the continuous invertible maps  :  !  that are smooth everywhere in  except at most at a finite number of points.) A graph  is an ordered collection of smooth oriented paths, l , denoted as links, with l = 1, . . . , L, where the links overlap only at their endpoints, called nodes. Given a graph  and a smooth, Haar-integrable complex function f : U 2 (SU(2))L 7! f (U) 2 C, the couple (, f ) defines the (‘‘cylindrical’’) functional of A ; f ½A ¼ f ðU½A; Þ

½7

U½A;   ðU½A; 1 ; . . . ; U½A; L Þ

½8

Let L be the linear space of all functionals , f [A], for all  and f. L is dense (in an appropriate sense) in the space of all continuous functionals on A. An SU(2) and Diff  invariant scalar product can be defined in L as follows. If two functionals , f [A] and , g [A] are defined by the same graph , define Z h; f j;g i  dU f ðUÞ gðUÞ ½9 where dU is the Haar measure on (SU(2))L . The extension to functionals defined on different graphs is obtained by observing that (, f ) and (0 , f 0 ) define the same functional if  contains 0 and f is independent of the variables in  but not in 0 . It follows that any two given functionals 0 , f 0 and 00 , g00 can be written as functionals , f and , g with the same graph , where  is obtained from the union of 0 and 00 . Using this, the scalar product [9] is defined for any two functionals in L: h0 ; f 0 j00 ;g00 i  h;f j ;g i

½10

Standard completion in the Hilbert norm defines the kinematical Hilbert space K of LQG. L is dense in K and defines the Gelfand triple L  K  L . K carries a natural unitary representation of the group of local SU(2) representations and a natural unitary representation U of the group of the extended diffeomorphism of . These two properties are nontrivial; they represent the main physical motivation for the definition of the scalar product. The SU(2)-invariant subspace of K is a proper subspace K0 . An orthonormal basis in K0 can be defined using the Peter–Weyl theorem. The basis states are labeled by a graph , by the assignment of a nonvanishing spin j to each link  2  and by the assignment of a basis element in in the space of the intertwiners (invariant tensors in the tensor product of the representations space of the adjacent links) at each node n of . The triple S = (, j , in ) is called an imbedded spin network. The quantum state S [A] = hAjSi in K0 labeled by the spin network S = (, j , in ) is the cylindrical function obtained by contracting the representation matrices of the holonomies U(A, ), in the representations j , with the invariant tensors at the nodes. The diffeomorphism-invariant state space Kdiff is the SU(2) and diffeomorphism invariant subspace of L . It is the (closure of the) image of the map Pdiff : L ! L defined by X h00 ; 0 i 8; 0 2 K ½11 ðPdiff Þð0 Þ ¼ 00 ¼U 

Loop Quantum Gravity

The sum is over all states 00 in L for which there exists a diffeomorphism  such that 00 = U ; this is a finite sum. The scalar product on this image is naturally defined by hPdiff S ; Pdiff S0 iKdiff  ðPdiff S ÞðS0 Þ

½12

The space Kdiff obtained in this manner is separable. The images jsi = Pdiff jSi of the spin network states are called s-knot states. They span Kdiff . They are determined only by the diffeomorphism equivalence class s of the spin network S. Namely, by an abstract (non-imbedded) knotted graph, colored with spins and intertwiners. These colored knots are called s-knots or abstract spin networks. The s-knot states have a straightforward physical interpretation as quantum excitations of space, discussed below.

Operators and Quanta of Space The state space defined above carries a quantum representation of classical observables of general relativity. The classical quantity U[A, ], a function of the field variable A, acts naturally as a multiplicative operator on K. Thus, K provides a Schro¨dinger functional representation [A] of quantum gravity, which diagonalizes the (holonomy of the) gravitational connection. The two constraints [1] and [2] generate SU(2) gauge and diffeomorphism transformations on A. The corresponding transformations on the Schro¨dinger functional states [A] are given by the unitary representations mentioned above. The quantum implementation of the two constraint equations [1] and [2], following Dirac’s theory of constrained quantum systems, is the requirement of invariance under these transformations. The space Kdiff is the solution to these requirement. The triad field operator E can be defined only if suitably smeared. Since E is a 2-form, its geometrically natural smearing is with a 2D surface. (The 1-form field A is smeared over a line in U[A, ].) Given a finite 2D surface S :  = (1 , 2 ) 7! xa () 2 , the smeared field Z Z @xa @xb E½S ¼ E ¼ d2  abc 1 2 Ec ðxðÞÞ ½13 @ @ S is quantized by the functional derivative operator Z 8G @xa @xb  E½S  i h 3 d2  abc 1 2 ½14 c @ @ Ac ðxðÞÞ This operator is well defined on K and the quantum operators E[S] and U[A, ] define a linear representation of the Poisson algebra of the corresponding classical quantities. Thus, they define a quantization

341

of the kinematics of general relativity. Notice that in a general covariant quantum field theory field operators can be well defined even if smeared on low-dimensional regions, while in conventional quantum field theory, these operators need to be smeared over 3D or 4D regions. A simple calculation shows that if S and  intersect once, Ev ½SU½A;  ¼ ih

8G U½A; 1 vU½A; 2  c3

½15

where v 2 su(2), we have written Ev = tr[vE], 1, 2 are the two paths into which  is partitioned by the surface, and the sign is determined by the relative orientation of S and . More generally, E[S]U[A, ] is a sum of one such term per intersection between S and . Composite operators can be constructed in terms of these operators. In particular, using standard formulas in classical general relativity, the area of the surface S can be written as a Riemann sum X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A½S ¼ lim tr½EðS n ÞEðS n Þ ½16 N!1

n

where S n , n = 1, . . . , N, is a Riemann partition of the surface. A straightforward calculation based on eqn [15] shows that, if S cuts n links of a spin network carrying spins ( j1 . . . jn ) = j, then the spin network state jSi is an eigenstate of A[S] with eigenvalue Aj ¼

8 hG X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ji ðji þ 1Þ c3 i¼1;n

½17

where ji = 1=2, 1, 3=2, 2, . . . These are therefore discrete eigenvalues of the area. All eigenvalues of the area operator A[S] are real and discrete and A[S] is a self-adjoint operator. Similar results are obtained for the volume operator. This gets a discrete contribution for each node of a spin network. These spectral properties of the area and volume operators determine the physical interpretation of the spin network states: the nodes of the spin network represent quanta of space with quantized volume; the nodes are connected by links representing quanta of surface with quantized area. The graph  determines the adjacency relations between the individual quanta of space; the intertwiners in are volume quantum numbers; the spins j are area quantum numbers. The interpretation carries over to the s-nodes, which represent the same quantum excitations of space, up to its manifold coordinatization, which is physically irrelevant because of the gauge invariance under

342 Loop Quantum Gravity

Figure 1 The graph of an s-knot, namely an abstract spinfoam, and the set of quanta of space it represents. Each node n of the graph defines a quantum of space. The associated intertwiner in is the corresponding volume quantum number. Two quanta of space are adjacent if the corresponding nodes are linked. A link  cuts the elementary surface separating the two quanta and its spin j is the area quantum number of this surface.

diffeomorphisms of . An s-knot state jsi with N nodes represents a quantum excitation of space with N quanta of space adjacent to one another according to the connectivity of  (see Figure 1). Notice that the quantum states jsi do not represent quantum excitations living in the physical space: they represent quantum excitations of the physical space. For instance, the state j0i defined by the empty graph does not represent an ‘‘empty’’ physical space, but the absence of any physical space. A generic quantum state of the physical space is represented by a normalizable linear superposition of these discrete quantized spacetimes (see Knot Invariants and Quantum Gravity). In a nongeneral covariant context, the kinematical quantization predictions of quantum theory (such as the quantization of the angular momentum) are obtained from the spectral properties of operators that represent measurements at a given time. In the general covariant Hamiltonian formalism, the corresponding kinematical quantization predictions are given by spectral properties of ‘‘partial observables’’ operators, which in general are not gauge invariant in the sense of Dirac. Area and volume are partial observables of this kind. Their spectra are therefore interpreted as physical predictions of LQG (up to an overall numerical factor, called the Immirzi parameter, which is obtained in certain variants of the theory).

Dynamics The dynamics of the theory is obtained in terms of a ‘‘Hamiltonian constraint’’ operator C that quantizes the constraint [3]. Different variants of the operator C, and of its Lorentzian version, have been constructed. The operator is defined via a suitable regularization procedure. The description of these constructions exceeds the scope of this article, and

we limit ourself here to mentioning the main result and a few general comments. The main result of the LQG dynamics is that C turns out to be well defined and ultraviolet-finite when restricted to Kdiff . Finiteness holds also when standard matter couplings, such as Yang–Mills fields and fermions, are added. The reason for this finiteness can be understood as a consequence of the discrete nature of space implied by the spectral properties of the geometric operators described above. The limit in which the ultraviolet cutoff, introduced to regulate C, is removed turns out to be trivial on the diffeomorphism-invariant states in Kdiff . This is because this limit probes the shortdistance regime, but there is no physical (gaugeinvariant) short distance, in a theory in which geometry turns out to be quantized at the Plank scale. Since the physical states in Kdiff define a physical geometry only at scales larger than the Planck scale hGc3 , the ‘‘short-distance’’ modes in the coordinate manifold  turn out to be pure gauge. This interplay between quantum field-theoretical and generalrelativistic physics is the distinctive character of LQG. Finally, we sketch the formal structure that dynamics can take in the general covariant Hamiltonian formalism of LQG. The operator C defines a linear operator P  (C), usually (improperly) denoted the ‘‘projector,’’ which sends states in Kdiff into the kernel of C, formed by the generalized Kdiff vectors that solve the Wheeler–De Witt equation C = 0 (see Wheeler–De Witt Theory). Matrix elements of P are interpreted as transition amplitudes between quantum states of space. Physical predictions for processes that take place in a finite spacetime region R can be obtained, in principle, as follows. One considers a state ji representing the result of the measurement of partial observables of the 3D boundary of a spacetime region R. ji codes the nonrelativistic notions of initial, boundary and final conditions. Then h0jPji can be interpreted as a relative probability amplitude associated to this result. A formal expansion of this amplitude in powers of C generates a spinfoam sum (see Spin Foams) that can be understood as the ‘‘quantum gravity sum over histories’’ in R. A systematic technique for computing physical transition amplitudes from the backgroundindependent and nonperturbative formalism of LQG has not yet been developed. See also: BF Theories; Black Hole Mechanics; Canonical General Relativity; Knot Invariants and Quantum Gravity; Knot Theory and Physics; Quantum Cosmology; Quantum Dynamics in Loop Quantum Gravity; Quantum Geometry and its Applications; Spin Foams; Wheeler–De Witt Theory.

Lorentzian Geometry

Further Reading Ashtekar A and Lewandowski J (2004) Background independent quantum gravity: a status report. Classical and Quantum Gravity 21: R53. Rovelli C (1998) Loop quantum gravity. Living Reviews in Relativity (electronic journal). (http://www.livingreviews.org/ Articles/Volume1/1998-1rovelli). Rovelli C (2004) Quantum Gravity. Cambridge: Cambridge University Press.

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Rovelli C and Smolin L (1990) Loop space representation for quantum general relativity. Nuclear Physics B 331: 80. Rovelli C and Smolin L (1995) Discreteness of area and volume in quantum gravity. Nuclear Physics B 442: 593. Rovelli C and Smolin L (1995) Discreteness of area and volume in quantum gravity – erratum. Nuclear Physics B 456: 734. Smolin L (2002) Three Roads to Quantum Gravity. New York: Perseus Books Group. Thiemann T Modern Canonical Quantum General Relativity. Cambridge: Cambridge University Press (to appear).

Lorentzian Geometry P E Ehrlich, University of Florida, Gainesville, FL, USA S B Kim, Chonnam National University, Gwangju, South Korea ª 2006 Elsevier Ltd. All rights reserved.

Introduction Einstein’s (1916) use of differential geometry as an essential tool in his theory of general relativity has long been a motivation for the study of Lorentzian geometry. More recently, the influential monographs of R Penrose (1972) and of S Hawking and G Ellis (1973), the latter still cited by some as the Bible of general relativity, so fascinated differential geometers that Lorentzian geometry took its place alongside of global Riemannian geometry as a worldwide research area. Let M be a smooth n-dimensional manifold, n 2, with a countable basis. A Lorentz metric g = < , > on M is a symmetric nondegenerate (0, 2) tensor field on M of index (, þ , . . . , þ). The existence of such a tensor field implies that M admits a (non-oriented) line field; hence, some compact manifolds like S2 do not admit such metrics. A nonzero tangent vector v in TM is then timelike (resp., nonspacelike, null, spacelike) according to whether g(v, v) < 0 (resp.,

0, = 0, > 0). A Lorentzian manifold (M, g) is a pair consisting of a smooth manifold together with a choice of Lorentz metric. In this article, we use the convention that a spacetime (M, g) is a Lorentzian manifold together with a choice of time orientation, that is, a continuous timelike vector field X on M. Then a tangent vector v based at p may be consistently defined to be future (resp., past) directed if g(X(p), v) < 0 (resp., > 0). (Some authors also require that (M, g) be space oriented.) If a Lorentzian manifold happens not to be time orientable, then a 2-fold covering manifold with the induced pullback metric will be time orientable. Also basic are the

notations p  q (resp., p q) if there is a futuredirected timelike (resp., nonspacelike) curve from p to q and the corresponding chronological (resp., causal) future of p given by Iþ (p) = {q 2 M; p  q} and Jþ (p) = {q 2 M; p q}. For a Riemannian manifold (N, g0 ), the Riemannian distance function d0 : N N ! ½0; þ1Þ

½1

given by d0 (p, q) = inf {L(c); c : [0, 1] ! N is a piecewise smooth curve with c(0) = p and c(1) = q}. A fundamental result in global Riemannian geometry is the celebrated Hopf–Rinow theorem. Hopf–Rinow Theorem For any Riemannian manifold (N, g0 ), the following conditions are equivalent: (i) metric completeness: (N, d0 ) is a complete metric space; (ii) geodesic completeness: for any v in TN, the geodesic cv (t) in N with initial condition c0v (0) = v is defined for all values of an affine parameter t; (iii) for some point p in N, the exponential map expp is defined on all of Tp N; (iv) finite compactness: every subset K of N that is d0 bounded has compact closure. Moreover, if any one of (i)–(iv) holds, then (N, g0 ) also satisfies (v) minimal geodesic connectedness: given any p, q in N, there exists a smooth geodesic segment c : [0, 1] ! N with c(0) = p, c(1) = q and L(c) = d0 (p, q). A Riemannian metric for a smooth manifold is then said to be complete if it satisfies any of the above properties (i) through (iv). The Heine–Borel property of basic topology implies (via (iv)) that all Riemannian metrics for a compact manifold are automatically complete and many of the examples studied in basic Riemannian geometry are complete.

344 Lorentzian Geometry

Also, if Riem(N) denotes the space of all Riemannian metrics for a smooth manifold N, both geodesic completeness (property (ii) above) and geodesic incompleteness (the failure of property (ii) to hold for all geodesics) are C0 stable properties on Riem(N), that is, given a complete (resp., incomplete) metric g for N, there exists an open neighborhood U(g) of g in Riem(N) in the Whitney C0 fine topology such that all Riemannian metrics h in U(g) are complete (resp., incomplete). For spacetimes (M, g), however, many basic examples furnished by general relativity fail to be geodesically complete and compactness of the underlying smooth manifold M does not imply that the given Lorentz metric g (let alone all Lorentz metrics for M) are complete. Also, the stability of geodesic completeness and incompleteness is more complicated than in the Riemannian case, necessitating concepts like pseudoconvex geodesic systems and disprisonment as studied by Beem and Parker. To summarize, for spacetimes and their associated Lorentzian distance functions, no naive analogs for the Hopf–Rinow theorem are valid. Under additional hypotheses, geodesic completeness may be guaranteed. Marsden noted that a compact spacetime with a homogenous Lorentz metric is geodesically complete. Then Carriere showed that a compact spacetime whose curvature tensor vanishes is geodesically complete. Later Kamishima (assuming constant curvature) and then Romero and Sanchez more generally showed that a compact Lorentzian manifold which admits a timelike Killing field is geodesically complete. At any point p in a given spacetime, emanating from p are three families of geodesics: timelike, spacelike, and null. It was hoped in the 1960s that possibly continuity arguments could be obtained for different types of geodesic completeness. However, a series of examples showed by the mid-1970s that timelike geodesic completeness, null geodesic completeness, and spacelike geodesic completeness are logically inequivalent. (Here, a given geodesic is said to be complete if it may be extended to be defined for all values of an affine parameter.) Nomizu and Ozeki for Riemannian manifolds showed that any given Riemannian metric g0 for the smooth manifold N could be made geodesically complete by making a conformal change of metric g0 , where  : N ! (0, þ1) is a smooth function. Especially in general relativity, such conformal changes are natural because the causal character of tangent vectors and curves (and hence of the basic causality conditions) are preserved. For spacetimes while generally nonspacelike geodesic completeness could not be produced by conformal changes, for some

subclasses of spacetimes, such as the strongly causal ones, it was possible with a global conformal change. For a large class of spacetimes, the warped or multiwarped products (originally inspired by several cosmological models in general relativity and a basic construction from Riemannian geometry), explicit integral criterion involving the warping functions have been given for timelike or null geodesic completeness. Several early examples of this type of result are discussed in Beem et al. (1996, pp. 111–112).

Lorentz Distance and the Nonspacelike Cutlocus For an arbitrary, not necessarily complete, Riemannian manifold (N, g0 ), the Riemannian distance function given in eqn [1] is continuous, the metric topology induced by d0 coincides with the given manifold topology, and d0 (p, q) is finite for all p, q in N. Now, for an arbitrary spacetime (M, g), and p, q in M, if there is no future-directed nonspacelike curve from p to q, set d(p, q) = 0; if there is such a curve, let dðp; qÞ ¼ supfLðcÞ; c : ½0; 1 ! ðM; gÞ is a piecewise smooth futuredirected nonspacelike curve with cð0Þ ¼ p and cð1Þ ¼ qg

½2

(Unlike the Riemannian case, [2] does not bound d(p, q) from above by L(c) for any selected curve c and hence the Lorentz distance may assume the value þ1.) This then defines what some authors term the ‘‘Lorentzian distance function’’ d ¼ dðgÞ : M  M ! ½0; þ1

½3

and other authors term ‘‘proper time.’’ It is linked to the causal structure of the given spacetime since dðp; qÞ > 0 iff q is in Iþ ðpÞ

½4

and in place of the triangle inequality for the Riemannian distance function, a reverse triangle inequality holds: if p  r  q; then dðp; qÞ  dðp; rÞ þ dðr; qÞ

½5

Also in the context of eqn [2], a future-directed nonspacelike curve c : [0, 1] ! M from c(0) = p to c(1) = q is defined to be maximal if L(c) = d(p, q). Corresponding to the Riemannian theory, a maximal nonspacelike curve turns out to be a smooth null or timelike geodesic segment.

Lorentzian Geometry

As mentioned earlier, geodesic completeness is generally not a natural requirement to place on a spacetime. But what emerges from [4] in place of Riemannian completeness is an interplay between the causal properties of the given spacetime and the continuity (and other properties) of the Lorentzian distance function (cf. Beem et al. (1996, chapter 4)). At the extreme of totally vicious spacetimes, the Lorentz distance is always þ1. Less drastically, if (M, g) contains a closed timelike curve passing through p, then d(p, q) = þ1 for all q in Jþ (p). Also, certain cosmological models contain pairs of points at infinite distance. In general, Lorentzian distance is only lower semicontinuous. Adding upper semicontinuity forces a distinguishing spacetime to be causally continuous. A spacetime is chronological iff d(p, p) = 0 for all p in M. At the other extreme from totally vicious spacetimes are globally hyperbolic spacetimes, which share many properties somewhat analogous to complete Riemannian manifolds. The Lorentzian distance function of a globally hyperbolic spacetime is both continuous and finite valued. (Indeed, a strongly causal spacetime is globally hyperbolic iff all Lorentz metrics g 0 in the conformal class C(M, g) also have finite-valued distance functions d(g 0 ).) Second, corresponding to property (v) of the Hopf–Rinow Theorem, these spacetimes all satisfy maximal nonspacelike geodesic connectability: given any p, q in M with p  q, there exists a future nonspacelike geodesic segment c : [0, 1] ! M with c(0) = p, c(1) = q and L(c) = d(p, q). A basic concept from the calculus of variations is that of a pair of conjugate points along a geodesic segment c : [0, a] ! (M, g). A smooth vector field J(t) along c is said to be a ‘‘Jacobi field’’ if J satisfies the Jacobi differential equation J00 þ RðJ; c0 Þc0 ¼ 0

½6

where R denotes the curvature tensor. Then c(t), c(s) are said to be conjugate points along c if there exists a nonzero Jacobi field J along c with J(t) = J(s) = 0. Much of the basic comparison techniques in global Riemannian geometry involving lengths of geodesics in manifolds satisfying curvature inequalities, such as the ‘‘Rauch comparison theorems,’’ the ‘‘Toponogov triangle comparison theorem,’’ and volume comparison theorems, were first obtained through Jacobi field techniques (cf. Petersen (1998) for a contemporary account). Later, Riccati equation techniques became more popular (cf. Karcher (1989)). For spacetimes, especially in the globally hyperbolic case, analogous results have been obtained for nonspacelike geodesic

345

segments, with a key breakthrough in 1979 being Harris’s version of the ‘‘Toponogov triangle comparison theorem’’ for timelike geodesic triangles in globally hyperbolic spacetimes. The Raychaudhuri equation used earlier in general relativity corresponds for spacetimes to this passage in the Riemannian setting from the Jacobi equation to the Riccati equation. The basic conjugate point theory and the Morse index theory for an arbitrary timelike or null geodesic segment in a general spacetime are reasonably close to the earlier Riemannian theory, if vector fields of the form J(t) = f (t) 0 (t) are accounted for in the case of a null geodesic segment  : [0, 1] ! (M, g). But spacelike geodesics and conjugate points are more problematic, as was first established using symplectic techniques by Helfer in 1994. More recently, progress has been made in applying important ideas of Gromov (1999) for Riemannian manifolds to the spacetime context (cf. Noldus (2004) for an example). Inspired by fundamental concepts in global Riemannian geometry, Beem and Ehrlich in 1979 introduced the concept of nonspacelike cut point, again most tractable for globally hyperbolic spacetimes. Let  : [0, a) ! (M, g) be a futureinextendible, future-directed nonspacelike geodesic in an arbitrary spacetime. Define t0 ¼ supft 2 ½0; aÞ; dðð0Þ; ðtÞÞ ¼ Lðj½0; t Þg ½7 (If there is a closed timelike curve through (0), then d((0), (0)) = þ1 and t0 will not exist. If  is a nonspacelike geodesic ray and hence d((0), (t)) = L(j[0, t] ) for all t, then t0 = a.) However, if 0 < t0 < a, then (t0 ) is said to be the future nonspacelike cut point of p = (0) along . For general spacetimes, it may be shown that: 1. for 0 < s < t < t0 , that j[s, t] is the unique maximal nonspacelike geodesic in all of (M, g) between (s) and (t); 2. j[0, t] is maximal for all t with 0  t  t0 ; and 3. for all t with t0 < t < a, there is a longer nonspacelike curve in (M, g) than j[0, t] between (0) and (t). A nonspacelike cut point is a subtler concept than a nonspacelike conjugate point since the existence of a cut point is not necessarily captured by the behavior of families of future nonspacelike curves (or geodesics) close to the given geodesic segment , the basic viewpoint of the calculus of variations. But since calculus of variations arguments shows that past a nonspacelike conjugate point, longer ‘‘neighboring curves’’ join (0) to (t), the future cut point of p = (0) along  comes no later than the first

346 Lorentzian Geometry

future conjugate point to p along  in either the timelike or null geodesic case. In a startling result which contradicted erroneous arguments in all the standard textbooks, Margerin in 1993 gave examples to show that even for compact Riemannian manifolds, the first conjugate locus of a point (i.e., the set of all first conjugate points along all geodesics issuing from a given point) need not be closed, even though elementary arguments correctly show that the cut locus of any point (i.e., the set of all cut points along all geodesics issuing from the given point) is always closed. The timelike first conjugate locus of a point in a spacetime will generally not be closed, but because a nonspacelike geodesic in a globally hyperbolic spacetime must escape from any compact subset in finite affine parameter, the future (or past) first nonspacelike conjugate locus of any point in such a spacetime is a closed subset. In a result analogous to the Riemannian characterization, nonspacelike cut points in globally hyperbolic spacetimes may be characterized as follows: let q = (t0 ) be the future cut point of p = (0) along the timelike (resp., null) geodesic segment  from p to q. Then either one of both of the following conditions hold: (1) q is the first future conjugate point to p along , or (2) there exist at least two maximal timelike (resp., null) geodesic segments from p to q. Now given p in an arbitrary spacetime (M, g), the future timelike (resp., null) cut locus of p is defined to be the set of all timelike (resp., null) cut points along all future timelike (resp., null) geodesics issuing from p and the future nonspacelike cut locus of p is defined as the union of the future timelike and null cut loci. Employing alternatives (1) and (2) in the preceeding paragraph, it may be shown for globally hyperbolic spacetimes that the null and nonspacelike cut loci are closed subsets of M. The null cut locus has a privileged status by virtue of a phenomena not encountered for Riemannian manifolds. Under a conformal change of back-ground spacetime metric, null geodesics remain null pregeodesics (i.e., may be reparametrized to be null geodesics in the deformed Lorentz metric) while such deformations fail to preserve timelike or spacelike geodesics, or to preserve geodesics in the Riemannian case. Even though null conjugate points along a null geodesic will not remain invariant under conformal change of spacetime metric, it is remarkable that elementary arguments involving the spacetime distance function show that global conformal diffeomorphisms do preserve null cut points and hence the null cut locus of any point.

Geodesic Incompleteness and the Lorentzian Splitting Theorem In global Riemannian geometry, an important concept is that of a geodesic ray. In a complete Riemannian manifold (N, g0 ), a unit geodesic c : [0, þ1) ! (N, g0 ) is said to be a (geodesic) ray if d0 (c(0), c(t)) = t for all t  0. By the triangle inequality, c(t) is minimal between every pair of its points. By making a limit construction, it may be shown that for each p in N, there exists a geodesic ray c(t) with c(0) = p. An allied concept is that of a (geodesic) line c : R ! (N, g0 ); here d0 (c(t), c(s)) = jt  sj for all t, s is required, that is, c is minimal between every pair of its points. The existence of a line is much stronger than the existence of a ray. If (N, g0 ) has positive Ricci curvature everywhere, then (N, g0 ) contains no lines despite the fact that it contains a ray issuing from each point. A helpful tool in this setting is the compactness of sets of tangent vectors of the form fw 2 Tp N; g0 ðw; wÞ ¼ 1g

½8

for any p in N; hence, any infinite sequence of tangent vectors based at p automatically has a convergent subsequence. For spacetimes, geodesic completeness cannot generally be assumed. Yet a future nonspacelike geodesic ray  : [0, b) ! (M, g) may be defined to be a future-directed, future-inextendible nonspacelike geodesic with d((0), (t)) = L(j[0, t] ) for all t in [0, b). The reverse triangle inequality implies that  is maximal between any pair of its points. Similarly, a nonspacelike geodesic line  : (a, b) ! (M, g) is a past- and future-inextendible nonspacelike geodesic with d((t), (s)) = L(j[t, s] ) for all s, t. Hence,  is maximal between any pair of its points. If nonspacelike geodesic completeness is assumed, a = 1 and b = þ1 above. Constructions here are more delicate than in the Riemannian case because the sets fv 2 Tp M; gðv; vÞ ¼ 1g

½9

of unit timelike tangent vectors, while closed in the tangent space, are noncompact. Despite this technicality, using the limit curve machinery of general relativity in place of the compactness in [8], it has been shown that a strongly causal spacetime admits a past and future nonspacelike geodesic ray issuing from every point (cf. Beem et al. (1996, chapter 8)). (If the spacetime is not nonspacelike geodesically complete, these rays will not necessarily be past or future complete.) As in the Riemannian case, the existence of a complete line is a stronger geometric condition. For that reason, in 1977 Beem and Ehrlich introduced the concept of a spacetime causally disconnected by a compact set K and

Lorentzian Geometry

showed that a strongly causal spacetime which is causally disconnected by a compact set contains a nonspacelike geodesic line which intersects the compact set. (Again, unless the spacetime is nonspacelike geodesically complete, this line need not be future or past complete.) A pattern common to many results in global Riemannian geometry especially since the 1950s is the following: the existence of a complete Riemannian metric on a smooth manifold which also satisfies a global curvature inequality implies a topological or geometric conclusion. A celebrated early example from the 1950s and 1960s, obtained by separate results of Rauch, Berger, and Klingenberg, is the topological sphere theorem. Topological Sphere Theorem Suppose (N, g0 ) is a complete, simply connected Riemannian n-manifold whose sectional curvatures satisfy 1=4 < K  1. Then N is homeomorphic to Sn . By contrast, for spacetimes, the assumption of geodesic completeness is generally unwarranted. Here is an example of one of the celebrated singularity theorems of general relativity, published in 1970 as originally stated: Hawking–Penrose Singularity Theorem No spacetime (M, g) of dimension n  3 can satisfy all of the following three requirements together: (i) (M, g) contains no closed timelike curves; (ii) Every inextendible nonspacelike geodesic in (M, g) contains a pair of conjugate points; and (iii) There exists a future- or past-trapped set S in (M, g). This theorem may be reinterpreted more akin to the Riemannian pattern above as follows: suppose (M, g) is a chronological spacetime of dimensions n  3 which satisfies the timelike convergence condition (Ric(v, v)  0 for all timelike tangent vectors) and the generic condition (every inextendible nonspacelike geodesic contains a point which has some appropriate nonzero sectional curvature). If (M, g) contains a future- or past-trapped set, then (M, g) is nonspacelike geodesically incomplete. Hence, this result models the pattern: global curvature inequalities (reflecting the physical assumptions that gravity is assumed to be attractive and every inextendible nonspacelike geodesic experiences tidal acceleration) and a further physical or geometric assumption (the first and third conditions) implies the existence of an incomplete timelike or null geodesic. An influential concept in global Riemannian geometry formulated during the 1960s and 1970s

347

is that of curvature rigidity, which first became widely known through the introduction to the text Cheeger and Ebin (1975). The above statement of the ‘‘sphere theorem’’ contains one hypothesis that the sectional curvature is strictly greater than 1/4. In curvature rigidity, the hypothesis of strict inequality is relaxed to include the possibility of equality as well, and then one tries to show that either the old conclusion is still valid, or if it fails, it fails in an isometric (hence ‘‘rigid’’) manner. Thus in the example of the sphere theorem, if the sectional curvature is now allowed to satisfy 1=4  K  1, then either the given Riemannian manifold remains homeomorphic to the n-sphere, or if not, it is isometric to a Riemannian symmetric space of rank 1. Already in an article in 1970, Geroch had expressed the opinion that most spacetimes should be nonspacelike geodesically incomplete and also that a spacetime should fail to be nonspacelike geodesically incomplete only under special circumstances. Apparently by the early 1980s, S T Yau had formulated the idea that timelike geodesic incompleteness of spacetimes ought to display a curvature rigidity. In the paragraph following the statement of the Hawking–Penrose singularity theorem, there are two curvature conditions mentioned – the timelike convergence condition and the generic condition. Now the timelike convergence condition already allows for the case of equality (i.e., zero timelike Ricci curvature) in its formulation; hence, curvature rigidity here would imply dropping the generic condition that each inextendible nonspacelike geodesic contains a point of nonzero sectional curvatures as a hypothesis. This notion seems first to have been published by Yau’s Ph.D. student R Bartnik in 1988 as follows: Conjecture  3 which

Let (M, g) be a spacetime of dimension

(i) contains a compact Cauchy surface and (ii) satisfies the timelike convergence condition Ric(v, v)  0 for all timelike v. Then either (M, g) is timelike geodesically incomplete, or (M, g) splits isometrically as a product (jR  V, dt2 þ h) where (H, h) is a compact Riemannian manifold. This conjecture has been proven in many cases with the following proof scheme. From the physical or geometric assumptions made, produce an inextendible nonspacelike geodesic line. Further, prove that the line happens to be timelike rather than null. Then if the spacetime were timelike geodesically complete, it would contain a complete

348 Lorentzian Geometry

timelike line. But then the desired splitting may be obtained using the Lorentzian splitting theorem. Lorentzian Splitting Theorem Let (M, g) be a spacetime of dimension 3 which satisfies each of the following conditions: (i) (M, g) is either globally hyperbolic or timelike geodesically complete; (ii) (M, g) satisfies the timelike convergence condition; and (iii) (M, g) contains a complete timelike line. Then (M, g) splits isometrically as a product (R  V, dt2 þ h) where (H, h) is a complete Riemannian manifold. This result, which corresponds to obtaining the spacetime analog of a celebrated splitting theorem of Cheeger and Gromoll for lines in complete Riemannian manifolds of non-negative Ricci curvature, published in 1971, was posed as a problem by S T Yau in a problem list stemming from the conference Special Year in Differential Geometry held at the Institute for Advanced Study in Princeton during the 1979–80 academic year. Early progress was made using maximal hypersurface methods by Gerhardt in 1983, Bartnik in 1984, and Galloway in 1984. Then in 1985, Beem, Ehrlich, Markvorsen, and Galloway introduced the methodology of employing the Busemann function of the complete timelike line, motivated by techniques from Riemannian geometry, and succeeded in obtaining a splitting under the hypothesis of global hyperbolicity and everywhere nonpositive timelike sectional curvatures. In separate publications, Eschenburg and Galloway extended the result to the desired curvature hypothesis of nonnegative timelike Ricci curvatures. Finally, Newman in 1990 achieved the originally desired goal of obtaining the splitting under the assumption of timelike geodesic completeness, rather than global hyperbolicity. This is a more delicate setting, since timelike geodesic completeness does not imply maximal nonspacelike geodesic connectability, a fairly basic geometric tool in many standard constructions. But the idea emerged with Newman’s solution that the existence of a timelike geodesic line or segment in a nonglobally hyperbolic spacetime implies an adequate level of control in a tubular neighborhood of the given line to enable the proof to work. Galloway and Horta in 1996 published a much simplified working out of these concepts. A fuller exposition of these developments may be found in Beem et al. (1996, chapter 14). In addition, in 2000, Galloway published a version of the splitting theorem for a null maximal geodesic line.

Two-Dimensional Spacetimes Two-dimensional spacetimes, sometimes termed Lorentz surfaces, are especially tractable because given (M, g) with dim M = 2, then (M, g) is also a spacetime. Hence, it may be shown that any Lorentzian 2-manifold (M, g) homeomorphic to R2 may be made geodesically complete (not just nonspacelike geodesically complete) by a conformal change of metric. Also, any simply connected twodimensional Lorentzian manifold is strongly causal. In Weinstein (1996), an extensive study is made of Lorentz surfaces generally and particularly, of a conformal boundary for such surfaces first given by Kulkarni in 1985. One of the prettiest classical results linking the geometry and topology of a Riemannian surface is the Gauss–Bonnet theorem. Let (N, g0 ) be a Riemannian manifold of dimension 2 and let P be a polygonal subregion with piecewise smooth bounding curves ci , 1  i  k. Let K denote the Gauss curvature of (N, g0 ) and  the geodesic curvature of the smooth curves ci (which vanishes if ci happens to be a geodesic). If i denote the corresponding interior angles between the successive boundary curves ci and ciþ1 , then the Gauss–Bonnet formula over P is Z Z Z X K dA þ  ds þ ð  i Þ ¼ 2 ½10 P

@P

i

By considering a triangulation of N itself and summing up the corresponding terms in [10], it follows for a compact oriented Riemannian manifold (N, g0 ) of dimension 2 that Z Z K dA ¼ 2ðNÞ ½11 N

where (N) denotes the Euler characteristic. Also lurking in the background here is a formula for computing the angle between unit tangent vectors v, w as cos  ¼ g0 ðv; wÞ

½12

In the spacetime setting, different versions of a Gauss–Bonnet formula for subregions of a twodimensional spacetime (M, g) corresponding to [10] have been given in 1974 by Helzer and in 1984 by Birman and Nomizu. First, the angle computation is a bit trickier for spacetimes than in the Riemannian case; eqn [12] has to be replaced by techniques which use the hyperbolic functions cosh u and sinh u to define the angle u (sometimes called the ‘‘hyperbolic angle’’) between two unit vectors and

Lyapunov Exponents and Strange Attractors

then to allow for null vectors. Birman and Nomizu obtained an analog of [10] assuming that the boundary curves for P are successive smooth unit timelike curves: Z Z Z X K dA þ  ds  i ¼ 0 P

@P

i

Helzer in his formulation allows the different boundary curves to be either unit timelike, unit spacelike or null separately. Since the only compact, orientable smooth surface which admits a spacetime metric is the 2-torus, which has zero Euler characteristic, the Riemannian formula [11] above translates into the uniform constraint on the Gauss curvature of the spacetime: Z Z K dA ¼ 0 M

See also: General Relativity: Overview; Geometric Analysis and General Relativity; Pseudo-Riemannian Nilpotent Lie Groups; Spacetime Topology, Causal Structure and Singularities.

Further Reading Beem J, Ehrlich P, and Easley K (1996) Global Lorentzian Geometry. In: Marcel Dekker Pure and Applied Mathematics, vol. 202, 2nd edn. New York: Dekker.

349

Cheeger J and Ebin D (1975) Comparison Theorems in Riemannian Geometry, North Holland Mathematical Library. Amsterdam: North-Holland. Einstein A (1916) Die Grundlage der allgemeinen Relativita¨tstheorie. Annalen der Physik 49: 769–822. Gromov M (1999) Metric Structures for Riemannian and NonRiemannian Spaces, Birkhauser Progress in Mathematics, vol. 152. Boston: Birkhauser. Hawking S and Ellis G (1973) The Large Scale Structure of Spacetime. Cambridge: Cambridge University Press. Karcher H (1989) Riemannian comparison constructions. In: Chern SS (ed.) Global Differential Geometry, Mathematical Association of America Studies in Mathematics, vol. 27, pp. 170–222. Washington, DC: MAA. Kriele M (1999) Spacetime: Foundations of General Relativity and Differential Geometry, Springer Lecture Notes in Physics (Monographs), vol. 59. Heidelberg: Springer. Noldus J (2004) The limit space of a Cauchy sequence of globally hyperbolic spacetimes. Classical Quantum Gravity 21: 851–874. O’Neill B (1983) Semi-Riemannian Geometry with Applications to Relativity, Academic Press Pure and Applied Mathematics. New York: Academic Press. Penrose R (1972) Techniques of Differential Topology in Relativity. Society for Industrial and Applied Mathematics, Regional Conference Series in Applied Mathematics, vol. 7. Philadelphia: SIAM. Petersen P (1998) Riemannian Geometry, Springer Verlag Graduate Texts in Mathematics, vol. 171. New York: Springer. Sachs R and Wu H (1977) General Relativity for Mathematicians, Springer Verlag Graduate Texts in Mathematics, vol. 48. New York: Springer. Weinstein T (1996) An Introduction to Lorentz Surfaces, de Gruyter Expositions in Mathematics, vol. 22. Berlin: de Gruyter.

Lyapunov Exponents and Strange Attractors 1 <    < k1 < k , and there exists a filtration F1 <    < Fk1 < Fk = Rd into vector subspaces, such that

M Viana, IMPA, Rio de Janeiro, Brazil ª 2006 Elsevier Ltd. All rights reserved.

ðvÞ ¼ i

Lyapunov Exponents The Lyapunov exponents of a sequence {An , n  1} of square matrices of dimension d  1 are the values of 1 ðvÞ ¼ lim sup log kAn  vk n!1 n

½1

over all nonzero vectors v 2 Rd . For completeness, set (0) = 1. It is easy to see that (cv) = (v) and (v þ v0 )  max{(v), (v0 )} for any nonzero scalar c and any vectors v, v0 . It follows that, given any constant a, the set of vectors satisfying (v)  a is a vector subspace. Consequently, there are at most d Lyapunov exponents, henceforth denoted by

for all v 2 Fi nFi1

and every i = 1, . . . , k (write F0 = {0}). In particular, the largest exponent is 1 k ¼ lim sup log kAn k n!1 n

½2

One calls dim Fi  dim Fi1 the multiplicity of each Lyapunov exponent i . There are corresponding notions for continuous families of matrices At , t 2 (0, 1), taking the limit as t goes to 1 in the relations [1] and [2]. The theories for the two types of families, discrete and continuous, are analogous and so at each point in what follows we refer to either one or the other.

350 Lyapunov Exponents and Strange Attractors

for every n  1. Assume the function logþ kA(x)kx is -integrable:

Lyapunov Stability Consider the linear differential equation _ vðtÞ ¼ BðtÞ  vðtÞ

½3

where B(t) is a bounded function with values in the space of d  d matrices, defined for all t 2 R. The theory of differential equations ensures that there exists a fundamental matrix At , t 2 R, such that vðtÞ ¼ At  v0 is the unique solution of [3] with initial condition v(0) = v0 . If the Lyapunov exponents of the family At , t > 0, are all negative then the trivial solution v(t)  0 is asymptotically stable, and even exponentially stable. The stability theorem of Lyapunov asserts that, under an additional regularity condition, stability is still valid for nonlinear perturbations wðtÞ ¼ BðtÞ  w þ Fðt; wÞ

logþ kAðxÞkx 2 L1 ðÞ

½5

(we write logþ  = log max {, 1}, for any  > 0). It is clear that the sequence of functions an (x) = log kAn (x)kx satisfies amþn ðxÞ  am ðxÞ þ an ðf m ðxÞÞ for every m, n, and x. It follows from J Kingman’s subadditive ergodic theorem that the limit lim

1

n!1 n

an ðxÞ

exists for -almost all x. In view of [2], this means that the largest Lyapunov exponent k (x) of the sequence An (x), n  1 is a limit, and not just a lim sup, at almost every point.

½4

1þc

with kF(t, w)k  const.kwk , c > 0. That is, the trivial solution w(t)  0 is still exponentially asymptotically stable. The regularity condition means, essentially, that the limit in [1] does exist, even if one replaces vectors v by elements v1 ^    ^ vl of any lth exterior power of Rd , 1  l  d. By definition, the norm of an l-vector v1 ^    ^ vl is the volume of the parallelepiped determined by the vectors v1 , . . . , vk . This condition is usually tricky to check in specific situations. However, the multiplicative ergodic theorem of V I Oseledets asserts that, for very general matrix-valued stationary random processes, regularity is an almost sure property. This result sets the foundation for the modern theory of Lyapunov exponents. We are going to discuss the precise statement of the theorem in the slightly broader setting of linear cocycles, or vector bundle morphisms.

Linear Cocycles Let  be a probability measure on some space M and f : M ! M be a measurable transformation that preserves . Let  : E ! M be a finite-dimensional vector bundle, endowed with a Riemannian metric k  kx on each fiber E x = 1 (x). Let A : E ! E be a linear cocycle over f. What we mean by this is that

Multiplicative Ergodic Theorem The Oseledets theorem states that the same holds for all Lyapunov exponents. Namely, for -almost every x 2 M there exists k = k(x) 2 {1, . . . , d}, a filtration Fx1 <    < Fxk1 < Fxk ¼ E x and numbers 1 (x) <    < k (x) such that 1 log kAn ðxÞkx ¼ i ðxÞ n!1 n lim

½6

for all v 2 Fxi nFxi1 and i 2 {1, . . . , k}. The Lyapunov exponents i (x), and their number k(x), are measurable functions of x and they are constant on orbits of the transformation f. In particular, if the measure  is ergodic then k and the i are constant on a full -measure set of points. The subspaces Fxi also depend measurably on the point x and are invariant under the linear cocycle: AðxÞ  Fxi ¼ Ffi ðxÞ

and the action A(x) : E x ! E f (x) of A on each fiber is a linear isomorphism. Notice that the action of the nth iterate An is given by

It is in the nature of things that, usually, these objects are not defined everywhere and they depend discontinuously on the base point x. When the transformation f is invertible, one obtains a stronger conclusion, by applying the previous kind of result also to the inverse of the cocycle. Namely, assuming that logþ kA1 k is also in L1 (), one gets that there exists a decomposition

An ðxÞ ¼ Aðf n1 ðxÞÞ    Aðf ðxÞÞ  AðxÞ

E x ¼ E1x    Ekx

 A¼f 

Lyapunov Exponents and Strange Attractors

defined at almost every point and such that A(x)  Eix = Eif (x) and lim

1

n!1 n

log kAn ðxÞkx ¼ i ðxÞ

½7

for all v 2 Eix different from zero and all i 2 {1, . . . , k}. These Oseledets subspaces Eix are related to the subspaces Fxi through Fxj ¼

j M

Eix

i¼1

dim Eix

dim Fxi

Hence, =  dim Fxi1 is the multiplicity of the Lyapunov exponent i (x). The angles between any two Oseledets subspaces decay subexponentially along orbits of f:   1 j lim log angle Eif n ðxÞ ; Ef n ðxÞ ¼ 0 n!1 n for every i 6¼ j and almost every point. These facts imply the regularity condition mentioned previously and, in particular, k X 1 lim log j det An ðxÞj ¼ i ðxÞ dim Eix n!1 n i¼1

½8

Consequently, for cocycles with values in SL(d, R), the sum of all Lyapunov exponents, counted with multiplicity, is identically zero. As we are dealing with almost certain properties, we may generally restrict the vector bundle to some full measure subset over which it is trivial. Then each fiber E x is identified with the space Rd , and we may think of A(x) as a d  d matrix. Then An (x) = A(f n (x)) is a stationary random process relative to (f , ). Thus, in this context it is no serious restriction to view a linear cocycle as a stationary random process with values in the linear group GL(d, R) of invertible d  d matrices. Furthermore, given any such random process An , n  0, one may consider its normalization Bn = An =jdetAn j. The Lyapunov exponents of the two random processes An , n  0, and Bn , n  0, differ by the time average n1 1X log jdetAj ðxÞj n!1 n j¼0

lim

of the determinant. The Birkhoff ergodic theorem ensures that the time average is well defined almost everywhere, as long as the function log j det Aj is in L1 (); this is the case, for instance, if both logþ kA1 k are integrable. This relates the general case to random processes with values in the special linear group SL(d, R) of d  d matrices with determinant 1.

351

The Oseledets theorem was extended by D Ruelle to certain linear cocycles in infinite dimensions. He assumes that the A(x) are compact operators on a Hilbert space H and logþ kAk is in L1 (). The conclusion is the same as in finite dimensions, except that the filtration    < Fxi <    < Fx1 ¼ H may involve infinitely many subspaces, and the Lyapunov exponents may be 1. There is also a version for cocycles over invertible transformations, where one assumes each A(x) to be invertible and the sum of a unitary operator with a compact operator, such that both log kA k are integrable. The conclusion is that there exists an Oseledets decomposition H = E1x    Eix    at almost every point, with finitely or countably many factors.

Random Matrices Relation [8] implies that, for SL(d, R) cocycles, if there is only one Lyapunov exponent (with full multiplicity) then it must be zero. When this happens, the theory contains no information on the behavior of the iterates An (x)  v, apart from the fact that there is no exponential growth nor decay of their norms. Thus, the question naturally arises under which conditions is there more than one Lyapunov exponent or, equivalently, under which conditions is the largest Lyapunov exponent strictly positive. This problem was first addressed by H Furstenberg for products of independent random variables, corresponding to the following class of linear cocycles. Let  be a probability measure on the group G = GL(d, R). Consider M = GN and  =  N (or M = GZ and  =  Z ), and let f : M ! M be the shift map   f ð j Þj ¼ ð jþ1 Þj It is clear that  is invariant and also ergodic for the transformation f. Consider the cocycle A : E ! E defined by E = M  R d and   A ð j Þj  v ¼ 0  v Clearly,   An ð j Þj  v ¼ n1    1 0  v Corresponding to the hypothesis of the multiplicative ergodic theorem, assume that logþ k k (and logþ k 1 k) are -integrable functions of the matrix . Furstenberg’s theorem states that if the closed group G() generated by the support of  is

352 Lyapunov Exponents and Strange Attractors

noncompact and strongly irreducible in Rd then the largest Lyapunov exponent of the cocycle A is strictly positive. Strong irreducibility means that there exists no finite union of subspaces of Rd that is invariant under all elements of the group. Improvements, extensions, and alternative proofs have been obtained by several authors since then. Especially, Y Guivarc’h and A Raugi provided conditions under which there are exactly d distinct Lyapunov exponents or, in other words, the multiplicity of every Lyapunov exponent is equal to 1. A matrix semigroup has the contraction property if there exists a sequence of elements hn and a probability measure on the projective space of R d that gives zero weight to any projective subspace, such that the images (hn ) m of m under the hn converge to a Dirac mass in the projective space. They proved that if the closed semigroup H() generated by the support of the probability  is strongly irreducible and has the contraction property then the largest Lyapunov exponent has multiplicity 1. Applying this to the exterior powers of the cocycle, one obtains sufficient conditions for simplicity of the other Lyapunov exponents as well. This statement has been improved by I Ya Gol’dsheid and G A Margulis, who formulated the ~ hypotheses in terms of the algebraic closure G() of ~ the semigroup H(). They assumed that G() has the contraction property and the connected component ~ of the identity inside G() is irreducible in Rd , meaning that its elements do not have any common invariant subspace. Then the largest Lyapunov exponent is simple.

Schro¨dinger Cocycles The one-dimensional discrete Schro¨dinger equation is the second-order difference equation ðunþ1 þ un1 Þ þ Vn un ¼ Eun

½9

derived from the stationary Schro¨dinger equation in dimension 1 by space discretization. Here the energy E is a constant and Vn = V(f n ()), where the potential V() is a bounded scalar function and f : M ! M is a transformation preserving some probability measure  on M. In what follows, we take  to be ergodic. Equation [9] may be rewritten as a first-order relation,      unþ1 un Vn  E 1 ¼ 1 0 vnþ1 vn

Hence, it may also be interpreted as a linear cocycle A over f, where the vector bundle is E = M  R2 and   VðÞ  E 1 AðÞ ¼ ½10 1 0 takes values in SL(R, 2). By ergodicity, the Lyapunov exponents are essentially independent of the base point . Let (E) denote the largest exponent: by the relation [8], the other one is (E). The Lyapunov exponent (E) is related to the spectral theory of the linear operators L , ðL uÞn ¼ ðunþ1 þ un1 Þ þ Vn un on the space ‘2 (Z) of complex square-integrable sequences un , n 2 Z. These are bounded Hermitian operators and so the spectra are compact subsets of R. Using the assumption that  is ergodic, one can prove that the spectrum spec(L ) is constant almost everywhere. If the transformation f is minimal, the spectrum is even independent of the point . Moreover, for all energies, ðEÞ  const: distðE; specðL ÞÞ In particular, (E) is always positive on the complement of the spectrum. A fundamental problem (Anderson localization) is to decide when the spectrum is pure-point. This is reasonably well understood for a few classes of base dynamics only, for example, the very chaotic systems such as Bernoulli and Markov processes (random potentials) or uniformly hyperbolic maps and flows, or the irrational rotations on the d-dimensional torus (quasiperiodic potentials). In the latter case, the results are more complete when there is only one frequency (d = 1). It was shown by K Ishii and by L Pastur that if (E) is positive for almost all values of E in some Borel set then the absolutely continuous part of the spectrum is essentially disjoint from that set. The converse is also true (due to S Kotani). Thus, checking that (E) is positive is an important step towards proving localization. A very general criterion for positivity of the Lyapunov exponent was obtained by Kotani. Namely, he proved that if the potential is not deterministic then (E) is positive for almost all E. In particular, for nondeterministic potentials the absolutely continuous spectrum is empty, almost surely. In simple terms, the hypothesis means that from the values of the potential for negative n one cannot determine the values for positive n. More formally, one calls the potential deterministic if every Vn , n  0 is almost everywhere a measurable function of {Vn : n  0}. For instance, quasiperiodic potentials are deterministic, whereas Bernoulli potentials are not.

Lyapunov Exponents and Strange Attractors

Subharmonicity Method

Nonuniform Hyperbolicity m

m

Let D be the set of complex vectors (z1 , . . . , xm ) 2 C such that jzj j  1 for all j and let Tm be the subset defined by jzj j = 1 for all j. Let f : Tm ! Tm and A : Tm ! SL(d, R) be continuous maps that admit holomorphic extensions to the interior of Dm with f (0) = 0. Assume that f preserves the natural (Haar) measure  on Tm . Let Z ðA; Þ ¼ ðzÞd Tm

where (z) denotes the largest Lyapunov exponent for the cocycle defined by A over f. It also follows from the subadditive ergodic theorem that Z 1 ðA; Þ ¼ lim log kAn ðzÞkd n Tm M Herman observed that, since the function log kAn (z)k is plurisubharmonic on Dm , one may use the maximum principle to conclude that Z 1 1 log kAn ðzÞkd  log kAn ð0Þk n Tm n Then, taking the limit when n ! 1 one obtains that ðA; Þ  ðAÞ

½11

where (A) denotes the spectral radius of the matrix A(0). Starting from this observation, he developed a very effective method for bounding Lyapunov exponents from below, that received several applications and extensions, in particular, to the theory of Schro¨dinger cocycles with quasiperiodic potentials. The best-known application is the following bound for integrated Lyapunov exponents of two-dimensional cocycles. Let f : M ! M be a continuous transformation on a compact metric space, preserving some probability measure , and A : M ! SL(2, R) be a continuous map. For each fixed , let AR be the cocycle obtained by multiplying A(x), at every point x, by the rotation of angle . Herman proved that Z Z 1 ðAR ; Þd  NðxÞ d 2 M (A Avila and J Bochi later showed that the equality holds) where NðxÞ ¼ log

353

kAðxÞk þ kAðxÞ1 k 2

Apart from the exceptional case when A acts by rotation at every point in the support of , the righthand side of the inequality is positive, and so the Lyapunov exponent of the cocycle AR is positive for many values of .

The prototypical example of a linear cocycle is the derivative of a smooth transformation on a manifold. More precisely, let M be a finite-dimensional manifold and f : M ! M be a diffeomorphism, that is, a bijective smooth map whose derivative Df (x) depends continuously on x and is an isomorphism at every point. Let E = TM be the tangent bundle to the manifold and A = Df be the derivative. If M is compact or, more generally, if the norms of both Df and its inverse are bounded, then the hypothesis in Oseledets theorem is automatically satisfied for any f-invariant probability . Lyapunov exponents yield deep geometric information on the dynamics of the diffeomorphism, especially when they do not vanish. For most results that we mention in the sequel, one needs the derivative Df to be Ho¨lder continuous: kDf ðxÞ  Df ðyÞk  const: dðx; yÞc Let Esx be the sum of the Oseledets subspaces corresponding to negative Lyapunov exponents. Pesin’s stable manifold theorem states that there s exists a family of embedded disks Wloc (x) tangent to s Ex at almost every point and such that the orbit of s every y 2 Wloc (x) is exponentially asymptotic to the orbit of x. This lamination {W s (x)} is invariant, in the sense that f ðW s ðxÞÞ W s ðf ðxÞÞ and has an ‘‘absolute continuity’’ property. There are analogous results for the sum Eux of the Oseledets subspaces corresponding to positive Lyapunov exponents. The entropy of a partition P of M is defined by h ðf ; PÞ ¼ lim

1

n!1 n

H ðP n Þ

where P n is the partition into sets of the form P = P0 \ f 1 (P1 ) \    \ f n (Pn ) with Pj 2 P and X H ðP n Þ ¼ ðPÞ log ðPÞ P2P n

The Kolmogorov–Sinai entropy h (f ) of the system is the supremum of h (f , P) over all partitions P with finite entropy. The Ruelle–Margulis inequality says that h (f ) is bounded above by the average sum of the positive Lyapunov exponents. A major result of the theory, Pesin’s entropy formula, asserts that if the invariant measure  is smooth (e.g., a volume element) then the two invariants coincide: ! Z X k þ h ðf Þ ¼ j d j¼1

354 Lyapunov Exponents and Strange Attractors

A complete characterization of the invariant measures for which the entropy formula is true was given by F Ledrappier and L S Young. The invariant measure  is called hyperbolic if all Lyapunov exponents are nonzero at almost every point. Hyperbolic measures are exact dimensional: the pointwise dimension dðxÞ ¼ lim r!0

log ðBr ðxÞÞ log r

exists at almost every point, where Br (x) is the neighborhood of radius r around x. This fact was proved by L Barreira, Ya Pesin, and J Schmeling. Note that it means that the measure (Br (x)) of neighborhoods scales as rd(x) when the radius r is small. Another remarkable feature of hyperbolic measures, proved by A Katok, is that periodic motions are dense in their supports. More than that, assuming the measure is nonatomic, there exist Smale horseshoes Hn with topological entropy arbitrarily close to the entropy h (f ) of the system. In this context, the topological entropy h(f , Hn ) may be defined as the exponential rate of growth, lim

1

k!1 k

log #fx 2 Hn: f k ðxÞ ¼ xg

of the number of periodic points on Hn .

Generic Systems Given any area-preserving diffeomorphism on any surface M, one may find another whose first derivative is arbitrarily close to the initial one and which has Lyapunov exponents identically zero at almost every point, or else is globally uniformly hyperbolic (Anosov). This surprising fact was discovered by R Man˜e´, and a complete proof was given by J Bochi. Uniform hyperbolicity means that the tangent bundle admits a Df-invariant splitting TM ¼ Es Eu such that the line bundle Es is uniformly contracted and Eu is uniformly expanded by the derivative. It is well known that Anosov diffeomorphisms can only occur if the surface is the torus T2 . In fact, the theorem of Man˜e´–Bochi is stronger: for a residual subset (a countable intersection of open dense sets) of all once-differentiable areapreserving diffeomorphisms on any surface, either the Lyapunov exponents vanish almost everywhere or the diffeomorphism is Anosov. This shows that zero Lyapunov exponents are actually quite common for surface diffeomorphisms that are only oncedifferentiable. Moreover, this theorem has been

extended to diffeomorphisms on manifolds with arbitrary dimension, in a suitable formulation, by J Bochi and M Viana. However, this phenomenon should be specific to systems with low differentiability. Indeed, already for Ho¨lder-continuous linear cocycles over chaotic transformations it is known that vanishing Lyapunov exponents can only occur with infinite codimension. That is, unless the cocycle satisfies an infinite number of independent constraints, there exists some positive exponent. By ‘‘chaotic’’ we mean here that the invariant probability  of the base transformation is assumed to be hyperbolic and to have local product structure: it is locally equivalent to a product of two measures, respectively, along stable and unstable sets. Under additional assumptions, one can even prove that all Lyapunov exponents have multiplicity 1 outside an infinite-codimension subset. This follows from extensions of the Guivarc’h–Raugi criterion for certain linear cocycles over chaotic transformations, obtained by A Avila, C Bonatti, and M Viana.

Strange Attractors This expression was coined by D Ruelle and F Takens in their celebrated study on the nature of fluid turbulence. E Hopf and also L D Landau and E M Lifshitz had suggested that turbulent motion arises from the existence in the phase space of invariant tori carrying quasiperiodic flows with large number of frequencies. Ruelle and Takens observed that dissipative systems such as viscous fluids do not generally have such quasiperiodic tori, and concluded that turbulence must be credited to a different mechanism: the presence of some ‘‘strange’’ attractor. While they did not propose a precise definition, two main features were mentioned: 1. Complex geometry: a strange attractor is not reduced to an equilibrium point or a periodic solution of the system and, generally, should have a fractal structure. 2. Chaotic dynamics: solutions accumulating on the attractor should be sensitive to their initial states. As more examples were found, it became apparent that the above two features do not always come together. This led to two types of definitions in the literature, depending on whether one emphasizes the geometry or the dynamics. We adopt the second point of view, and propose to define the strange attractor as one carrying an invariant ergodic physical measure which has some positive Lyapunov exponent. The notion of physical measure will be

Lyapunov Exponents and Strange Attractors

defined near the end. The condition on the Lyapunov exponent ensures that the dynamics near the attractor is (exponentially) sensitive to the initial states.

Lorenz-Like Attractors The uniformly hyperbolic attractors introduced by S Smale provided an interesting class of examples of strange attractors, both chaotic and fractal. Perhaps more striking, given that they originated from a concrete problem in fluid dynamics, were the strange attractors introduced by E N Lorenz. The Lorenz system of differential equations, x_ ¼ x þ y;

 ¼ 10

y_ ¼ rx  y  xz;

r ¼ 28

z_ ¼ xy  bz;

b ¼ 8=3

½12

was derived from Lord Rayleigh’s model for thermal convection, by Fourier expansion of the stream function and temperature, and truncation of all but three modes. Lorenz observed that its solutions depend sensitively on their initial states. Consequently, predictions based on the numerical integration of the equations may turn out to be very inaccurate, given that the initial data obtained from experimental measurements are never completely precise. This remarkable observation brought the issue of predictability in deterministic systems to a whole new light and motivated intense investigation of this and many other chaotic systems. The dynamical behavior of the eqns [12] was first interpreted through certain geometric models where the presence of strange attractors, both chaotic and fractal, could be proved rigorously. It was much harder to prove that the original eqns [12] themselves have such an attractor. This was achieved just a few years ago, by W Tucker, by means of a computer-assisted rigorous argument. At about the same time, a mathematical theory of Lorenz-like attractors in three-dimensional space was developed by C Morales, M J Pacifico, and E Pujals. In particular, this theory shows that uniformly hyperbolic attractors and Lorenz-like attractors are the only ones which are robust under all small modifications of the vector field.

He´non-Like Attractors Starting from the work of Lorenz, many models of strange attractors have been found and described to some extent, often related to concrete problems.

355

From a mathematical point of view, it is usually hard to give even a rough description of the dynamics in the chaotic regime. However, this was especially successful for the family of strange attractors introduced by M He´non. He considered a very simple nonlinear system, particularly suited for numerical experimentation: the transformation f ðx; yÞ ¼ ð1  ax2 þ by; xÞ

½13

where a and b are constant parameters. In a breakthrough, M Benedicks and L Carleson were able to prove that, for a set of parameter values with positive probability, this transformation has some nonhyperbolic attractor such that the orbits accumulating on it are sensitive to the starting point. The system [13] is also a model for many other situations, including the phenomenon of creation of homoclinic motions as parameters unfold, and the conclusions of Benedicks and Carleson have been extended to such situations, starting from the work of L Mora and M Viana. Moreover, a detailed theory of He´non-like attractors has been developed by M Benedicks, M Viana, D Wang, L S Young, and other authors. It follows from this theory that these attractors carry an invariant ergodic probability measure  which describes the statistical behavior of almost all trajectories f j (x), j  1, that accumulate the attractor: n 1X ’ðf j ðxÞÞ ¼ n!1 n j¼1

lim

Z

’ d

for any continuous function ’. This property implies that, despite the fact that it is supported on a zero-volume set, the measure  is, in some sense, physically observable. For this reason, one calls it a physical measure. In other words, time averages along typical orbits in the domain of attraction coincide with the space averages determined by the probability . Another property with physical relevance is that  is the zero-noise limit of the stationary measures associated to the Markov chains obtained by adding random noise to f. One says that the system (f , ) is stochastically stable. See also: Chaos and Attractors; Dissipative Dynamical Systems of Infinite Dimension; Ergodic Theory; Fractal Dimensions in Dynamics; Generic Properties of Dynamical Systems; Gravitational N-Body Problem (Classical); Homoclinic Phenomena; Hyperbolic Dynamical Systems; Lagrangian Dispersion (Passive Scalar); Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations; Random Dynamical Systems; Synchronization of Chaos.

356 Lyapunov Exponents and Strange Attractors

Further Reading Barreira L and Pesin Ya (2002) Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lecture Series, vol. 23. American Mathematical Society. Bochi J and Viana M (2004) Lyapunov exponents: how frequently are dynamical systems hyperbolic? In: Modern Dynamical Systems and Applications, pp. 271–297. Cambridge: Cambridge University Press. Bonatti C, Dı´az LJ, and Viana M (2004) Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopedia of Mathematical Sciences, vol. 102. Springer.

Eckmann J-P and Ruelle D (1985) Ergodic theory of chaos and strange attractors. Reviews of Modern Physics 57: 617–656. Gol’dsheid IYa and Margulis GA (1989) Lyapunov indices of a product of random matrices. Uspekhi Mat. Nauk. 44: 13–60. Man˜e´ R (1987) Ergodic Theory and Differentiable Dynamics. Springer. Spencer T (1990) Ergodic Schro¨dinger operators Analysis, et cetera 623–637. Academic Press. Viana M (2000) What’s new on Lorenz strange attractors? Math. Intelligencer 22: 6–19.

M Macroscopic Fluctuations and Thermodynamic Functionals G Jona-Lasinio, Universita` di Roma ‘‘La Sapienza,’’ Rome, Italy ª 2006 Elsevier Ltd. All rights reserved.

Introduction There is no theory so far of irreversible processes that is of the same generality as equilibrium statistical mechanics and presumably it may not exist. While in equilibrium the Gibbs distribution provides all the information and no equation of motion has to be solved, the dynamics plays the major role in nonequilibrium. The theory illustrated below refers to stationary states that are not restricted to being close to equilibrium, and for a wide class of models it can be shown to be exact. In this case one begins to see the appearance of some general principles. In equilibrium statistical mechanics, there is a welldefined relationship, established by Boltzmann, between the probability of a state and its entropy. This fact was exploited by Einstein to study thermodynamic fluctuations. When we are out of equilibrium, for example, in a stationary state of a system in contact with two reservoirs, it is not completely clear how to define thermodynamic quantities such as the entropy or the free energy. One possibility is to use fluctuation theory to define their nonequilibrium analogs. In fact in this way, extensive quantities can be obtained, although not necessarily simply additive due to the presence of long-range correlations which seem to be a rather generic feature of nonequilibrium. This possibility has been pursued in recent years leading to a considerable number of interesting results. One can recognize two main lines. 1. Exact calculations in simplified models. This is well exemplified by the work of Derrida et al. (2002). 2. A general treatment of a class of continuous time Markov chains for which the simplified models provide examples. This is the point of view developed by Bertini et al. (2002, 2004). Both approaches have been very effective and of course give the same results when a comparison is possible.

The second approach seems to encompass a wide class of systems and has the advantage of leading to equations which apply to very different situations. This is the point of view we shall adopt in the following. The question whether there are alternative more natural ways of defining nonequilibrium entropies or free energies is, for the moment, open.

Boltzmann–Einstein Formula The Boltzmann–Einstein theory of equilibrium thermodynamic fluctuations, as described for example in the book Physique Statistique by Landau–Lifshitz, states that the probability of a fluctuation from equilibrium in a macroscopic region of fixed volume V is proportional to exp{VS=k}, where S is the variation of entropy density in the region calculated along a reversible transformation creating the fluctuation and k is the Boltzmann constant. This formula was derived by Einstein simply by inverting the Boltzmann relationship between entropy and probability. He considered this relationship as a phenomenological definition of the probability of a state. Einstein theory refers to fluctuations from an equilibrium state, that is from a stationary state of a system isolated or in contact with reservoirs characterized by the same chemical potentials so that there is no flow of heat, electricity, chemical substances, etc., across the system. When in contact with reservoirs, S is the variation of the total entropy (system þ reservoirs) which, for fluctuations of constant volume and temperature, is equal to F=T, where F is the variation of the free energy of the system and T the temperature. In the following, we refer to F=T, our main object of study, as the entropy and use the letter S for it but no confusion should arise. The important question we address is then: what happens if the system is stationary but not in equilibrium, that is, flows of physical quantities are present due to external fields and/or different chemical potentials at the boundaries? To start with it is not always clear whether a closed macroscopic dynamical description is possible. If the system admits such a description of the kind provided by hydrodynamic

358 Macroscopic Fluctuations and Thermodynamic Functionals

equations, a fact which can be rigorously established in simplified models, a reasonable goal is to find an explicit connection between time-independent thermodynamic quantities (e.g., the entropy) and dynamical macroscopic properties (e.g., transport coefficients). As we shall see, the study of large fluctuations provides such a connection. It leads in fact to a dynamical theory of the entropy which is shown to satisfy a Hamilton–Jacobi equation (HJE) in infinitely many variables requiring the transport coefficients as input. Its solution is straightforward in the case of homogeneous equilibrium states and highly nontrivial in stationary nonequilibrium states (SNSs). In the first case we recover a well-known relationship widely used in the physical and physico-chemical literature. There are several one-dimensional models, where the HJE reduces to a nonlinear ordinary differential equation which, even if it cannot be solved explicitly, leads to the important conclusion that the nonequilibrium entropy is a nonlocal functional of the thermodynamic variables. This implies that correlations over macroscopic scales are present. The existence of long-range correlations is probably a generic feature of SNSs and more generally of situations where the dynamics is not time-reversal invariant. As a consequence if we divide a system into two subsystems, the entropy is not necessarily simply additive. The first step toward the definition of a nonequilibrium entropy is the study of fluctuations in macroscopic evolutions described by hydrodynamic equations. In a dynamical setting, a typical question one may ask is the following: what is the most probable trajectory followed by the system in the spontaneous emergence of a fluctuation or in its relaxation to an equilibrium or a stationary state? To answer this question, one first derives a generalized Boltzmann–Einstein formula from which the most probable trajectory can be calculated by solving a variational principle. The entropy is related to the logarithm of the probability of such a trajectory and satisfies the HJE associated to the variational principle. For states near equilibrium, an answer to this type of questions was given by Onsager and Machlup in 1953. The Onsager–Machlup theory gives the following result under the assumption of time reversibility of the microscopic dynamics. In the situation of a linear hydrodynamic equation and small fluctuations, that is, close to equilibrium, the most probable creation and relaxation trajectories of a fluctuation are time reversals of one another. This conclusion holds also in nonlinear hydrodynamic regimes and without the assumption of small fluctuations. This follows from the study of concrete models. In SNSs, on the other hand, time-reversal invariance is broken and the creation and relaxation trajectories of a fluctuation are not time reversals of one another.

In the following we refer to boundary-driven stationary nonequilibrium states, for example, a thermodynamic system in contact with reservoirs characterized by different temperatures and chemical potentials, but there is no difficulty in including an external field acting in the bulk.

Microscopic and Macroscopic Dynamics We consider many-body systems in the limit of infinitely many degrees of freedom. The basic general assumption of the theory is Markovian evolution. Microscopically, we assume that the evolution is described by a Markov process X which represents the state of the system at time . This hypothesis probably is not so restrictive, because the dynamics of Hamiltonian systems interacting with thermostats finally is also reduced to the analysis of a Markov process. Several examples are discussed in the literature. To be more precise, X represents the set of variables necessary to specify the state of the microscopic constituents interacting among themselves and with the reservoirs. The SNS is described by a stationary, that is, invariant with respect to time shifts, probability distribution Pst over the trajectories of X . Macroscopically, the usual interpretation of Markovian evolution is that the time derivatives of thermodynamic variables _ i at a given instant of time depend only on the i ’s and the affinities (thermodynamic forces) @S=@i at the same instant of time. Our next assumption can then be formulated as follows: the system admits a macroscopic description in terms of density fields which are the local thermodynamic variables. For simplicity of notation, we assume that there is only one thermodynamic variable (e.g., , the density). The evolution of the field  = (t, u), where t and u are the macroscopic time and space coordinates (see below), is given by diffusion-type hydrodynamic equations of the form @t  ¼ 12r  ðDðÞrÞ X   @ui Di;j ðÞ@uj  ¼ 12 1i; jd

¼ DðÞ

½1

The interaction with the reservoirs appears as boundary conditions to be imposed on solutions of [1]. We assume that there exists a unique stationary solution  of [1], that is, a profile (u), which satisfies the appropriate boundary conditions and is such that D() = 0. This holds if the diffusion matrix Di, j () in [1] is strictly elliptic, namely there exists a constant c > 0 such that D()  c (in matrix sense). These equations derive from the underlying microscopic dynamics through an appropriate

Macroscopic Fluctuations and Thermodynamic Functionals

scaling limit in which the microscopic time and space coordinates , x are rescaled as follows: t = =N 2 , u = x=N, where N represents the linear size of the system. For lattice systems, N is an integer. The hydrodynamic equation [1] represents a law of large numbers with respect to the probability measure Pst conditioned on an initial state X0 . The initial conditions for [1] are determined by X0 . Of course, many microscopic configurations give rise to the same value of (0, u). In general,  = (t, u) is an appropriate limit of a local observable N (X ) as the number N of degrees of freedom diverges. The hypothesis of Markovian evolution is also the basis of the 1931 Onsager’s theory of irreversible processes near equilibrium. Onsager, however, did not rely on any microscopic model and assumed, near the equilibrium, linear hydrodynamic equations or regression equations as he called them. His equations, ignoring space dependence, were of the form X _ i ¼  Dij j ½2 i

The diffusion matrix D is related to Onsager transport matrix  and the entropy by the relationship D ¼ s

½3 2

where the elements of s are @ S=@i @j . The matrix  is defined by the relationship between flows and affinities X @S ij ½4 _ i ¼  @j j The indices ij here label different thermodynamic variables. The matrix  is symmetric, a property known as Onsager reciprocity. Equations [2] and [3] follow by developing the entropy near an equilibrium state, that is, by taking a quadratic expression as an approximation. The minus sign in eqn [4] is due to our convention in which the entropy has the same sign as the free energy. Equation [3] permits to reconstruct the entropy from the knowledge of the coefficients D and  and has been widely used especially in physical chemistry. In SNSs, eqn [3] is replaced by a Hamilton– Jacobi-type equation for the entropy.

Dynamical Boltzmann–Einstein Formula The basic assumption is that the stationary ensemble Pst admits a principle of large deviations describing the fluctuations of the thermodynamic variables appearing in the hydrodynamic equation. This means the following. The probability that for large

359

N, the evolution of the random variable N deviates from the solution of the hydrodynamic equation and is close to some trajectory (t) ˆ is exponentially small and of the form Pst ðN ðXN 2 t Þ  ^ðtÞ; t 2 ½t1 ; t2 Þ  eN

d

½Sð^ ðt1 ÞÞþJ½t1; t2  ð^ Þ

¼ eN

d

I½t1 ; t2  ð^ Þ

½5

where d is the dimensionality of the system, J() ˆ is a functional which vanishes if (t) ˆ is a solution of [1] and S((t ˆ 1 )) is the entropy cost to produce the initial density profile (t ˆ 1 ). We normalize S so that S( ) = 0. Therefore, J() ˆ represents the extra cost necessary to follow the trajectory (t). Finally, ˆ N (XN2 t )  (t) means closeness in some metric ˆ and  denotes logarithmic equivalence as N ! 1. Equation [5] is the dynamical generalization of the Boltzmann–Einstein formula. Experience with many models justifies this assumption. To understand how [5] leads to a dynamical theory of the entropy, we discuss its properties under time reversal. Let us denote by  the time inversion operator defined by X = X . The probability measure P st describing the evolution of the time-reversed process X  is given by the composition of Pst and 1 , that is,  P st X  ¼  ;  2 ½1 ; 2 Þ ¼ Pst ðX ¼  ;  2 ½2 ; 1 Þ

½6

Let L be the generator of the microscopic dynamics. We remind that L induces the evolution of observables (functions on the state space) according to the equation @ EX0 [f (X )] = EX0 [(Lf )(X )], where EX0 stands for the expectation with respect to Pst conditioned on the initial state X0 . The time-reversed dynamics, that is, the dynamics which inverts the direction of the fluxes through the system, for example, heat flows under this dynamics from lower to higher temperatures, is generated by the adjoint L of L with respect to the invariant measure : E ½ f Lg ¼ E ½ðL f Þg

½7

The measure , which is the same for both processes, is a distribution over the configurations of the system and formally satisfies L = 0. The expectation with respect to  is denoted by E and f, g are observables. We note that the probability Pst , and therefore P st , depends on the invariant measure . The finitedimensional distributions of Pst are in fact given by Pst ðX1 ¼ 1 ; . . . ; Xn ¼ n Þ ¼ ð1 Þ p2 1 ð1 ! 2 Þ    pn n1 ðn1 ! tn Þ

½8

360 Macroscopic Fluctuations and Thermodynamic Functionals

where p (1 ! 2 ) is the transition probability. According to [6] the finite-dimensional distributions of P st are   P st X 1 ¼ 1 ; . . . ; X n ¼ n ¼ ð1 Þ p 2 1 ð1 ! 2 Þ    p n n1 ðn1 ! tn Þ ¼ ðn Þpn n1 ðn ! n1 Þ    p2 1 ð2 ! 1 Þ ½9 In particular, the transition probabilities p (1 ! 2 ) and p (1 ! 2 ) are related by

ð1 Þ p ð1 ! 2 Þ ¼ ð2 Þ p ð2 ! 1 Þ

½10

This relationship reduces to the well-known detailed balance condition if p (1 ! 2 ) = p (1 ! 2 ). We require that also the evolution generated by L admits a hydrodynamic description, that we call the adjoint hydrodynamics, which, however, is not necessarily of the same form as [1]. In fact, we consider models in which the adjoint hydrodynamics is nonlocal in space. In order to avoid confusion, we emphasize that what is usually called an equilibrium state for a reversible dynamics, as distinguished from an SNS, corresponds to the special case L = L, that is, the detailed balance principle holds. In such a case, Pst is invariant under time reversal and the two hydrodynamics coincide. We now derive a first consequence of our assumptions, that is, the relationship between the functionals I and I associated to the dynamics L and L by [5]. From eqn [6], it follows that I½t 1 ; t2  ð^ Þ ¼ I½t2 ;t1  ð^ Þ

½11

with obvious notations. More explicitly, this equation reads Sð^ ðt1 ÞÞ þ J½t 1 ; t2  ð^ Þ ¼ Sð^ ðt2 ÞÞ þ J½t2 ;t1  ð^ Þ

In a SNS the spontaneous emergence of a macroscopic fluctuation takes place most likely following a trajectory which is the time reversal of the relaxation path according to the adjoint hydrodynamics.

This implies that the entropy is related to J by Þ SðÞ ¼ inf J½1; 0 ð^ ^

½14

where the minimum is taken over all trajectories (t) ˆ connecting  to . We note that the reversibility of the microscopic process X , which we call microscopic reversibility, is not needed in order to deduce the Onsager– Machlup result (i.e., that the trajectory which creates the fluctuation is the time reversal of the relaxation trajectory). In fact, Onsager–Machlup result holds if and only if the hydrodynamics coincides with the adjoint hydrodynamics, which we call macroscopic reversibility. Indeed, it is possible to construct microscopic nonreversible models, L 6¼ L , in which the hydrodynamics and the adjoint hydrodynamics coincide. Spontaneous fluctuations, including Onsager– Machlup time-reversal symmetry, have been observed in stochastically perturbed reversible electronic devices. In nonreversible systems, an asymmetry between the emergence and the relaxation of fluctuations has been observed. The above discussion provides the explanation.

½12

where (t ˆ 2 ) are the initial and final points of ˆ 1 ), (t the trajectory and S((t ˆ i )) the entropies associated with the creation of the fluctuations (t ˆ i ) starting from the SNS. The functional J vanishes on the solutions of the adjoint hydrodynamics. To compute J , it is necessary to know the entropy S. We consider now the following physical situation. The system is macroscopically in the stationary state  at t = 1, but at t = 0 we find it in the state . We want to determine the most probable trajectory followed in the spontaneous creation of this fluctuation. According to [5], this trajectory is the one that minimizes J among all trajectories (t) ˆ connecting  to  in the time interval [1, 0]. From [12], recalling that S( ) = 0, we have that J½1; 0 ð^ Þ ¼ SðÞ þ J½0; Þ 1 ð^

() The right-hand side is minimal if J[0,1] ˆ = 0, that is, if ˆ is a solution of the adjoint hydrodynamics. The existence of such a relaxation solution is due to the fact that the stationary solution  is attractive also for the adjoint hydrodynamics. We have therefore the following consequences:

½13

The Hamilton–Jacobi Equation and Its Consequences We assume that the functional J has a density (which plays the role of a Lagrangian), that is, Þ ¼ J½t1 ; t2  ð^

Z

t2

dtLð^ðtÞ; @t ^ðtÞÞ

½15

t1

Let us introduce the Hamiltonian H(, H) as the Legendre transform of L(, @t ), that is, Hð; HÞ ¼ supfh; Hi  Lð; Þg

½16



where h , i denotes integration with respect to the macroscopic space coordinates u.

Macroscopic Fluctuations and Thermodynamic Functionals

Noting that H( , 0) = 0, the Hamilton–Jacobi equation associated to [14] is   S H ; ¼0 ½17  This is an equation for the functional derivative C() = S= , but not all the solutions of the equation H(, C()) = 0 are the derivatives of some functional. Of course, only those which are the derivative of a functional are relevant for us. We now specify the Hamilton–Jacobi equation [17] for boundary-driven lattice gases. For models with purely diffusive hydrodynamics [1], we expect a quadratic large deviation functional of the form Z 1 t2  1 J½t1 ;t2  ð^ Þ ¼ dt r ð@t ^  DðÞÞ; 2 t1 ð^ Þ1 r1 ð@t ^  DðÞÞi

½18

where D() is the right-hand side of the hydrodynamic equation [1], and by r1 f we mean a vector field whose divergence equals f. The form [18], which can be derived for several models, is expected to be very general: the functional J() ˆ measures how much ˆ differs from a solution of the hydrodynamics [1]. The matrix () = () with () has the same role in our more general context, as the Onsager matrix in [4]. This form of J is also typical for diffusion processes described by finite-dimensional Langevin equations (Freidlin–Wentzell theory). In this case, the Lagrangian L is quadratic in @t (t) ˆ and the associated Hamiltonian is given by Hð; HÞ ¼ 12hrH; ðÞrH i þ hH; DðÞi

½19

so that the Hamilton–Jacobi equation [17] takes the form



1 S S S r ; ðÞr ; DðÞ ¼ 0 ½20 þ 2    As is well known in mechanics, the Hamilton–Jacobi equation has many solutions and we must give a criterion to select the correct one. The criterion which the correct solution has to satisfy is that it must be a Lyapunov function with respect to the unique stationary state. It is a simple calculation to show that eqn [3] follows from HJE, if we look for a solution which is a local function of . This is the right choice in equilibrium where correlations over macroscopic distances are not expected if the microscopic forces are short range. Out of equilibrium, it has been shown by direct calculation that for a special model, the symmetric simple exclusion, the entropy is a nonlocal function of the thermodynamic variables, that is, space

361

correlations extend to macroscopic distances. This result can be derived in a simple way from HJE as we will discuss later. Lattice gases which do not conserve the number of particles do not give rise in general to a purely diffusive hydrodynamics but rather to a reaction diffusion equation. In this case, the large deviation functional will not have the quadratic form [18] and also the HJE will not be quadratic. An example in which particles can be created and destroyed is the so-called Kawasaki–Glauber dynamics. In this case, HJE has exponential nonlinearities. Nonequilibrium Fluctuation Dissipation Relation

We now derive a twofold generalization of the celebrated fluctuation dissipation relationship: it is valid in nonequilibrium states and in nonlinear regimes. Such a relationship will hold provided the rate function J of the time-reversed process is of the form [18] with D replaced by D , the adjoint hydrodynamics, @t  ¼ D ðÞ

½21

with the same boundary conditions as [1]. If J has the form Z 1 t2  1 J½t 1 ;t2  ð^ Þ ¼ dt ðr ð@t ^  D ð^ ÞÞ; 2 t1 ÞÞi ð^ Þ1 r1 ð@t ^  D ð^ by taking the variation of eqn [12], we get   S DðÞ þ D ðÞ ¼ r  ðÞr 

½22

½23

This relation can be verified explicitly for the nonequilibrium zero-range process which we discuss later and holds for several other models. It is also easy to check that the linearization of [23] around the stationary profile  yields a fluctuation dissipation relationship which reduces to the usual one in equilibrium. The fluctuation dissipation relation [23] can be used to obtain the adjoint hydrodynamics from D() and S= ; the first is usually known and the second can be calculated from the Hamilton–Jacobi equation. H Theorem

We show that the functional S is decreasing along the solutions of both the hydrodynamic equation [1] and the adjoint hydrodynamics   S @t  ¼ D ðÞ ¼ r  ðÞr  DðÞ ½24 

362 Macroscopic Fluctuations and Thermodynamic Functionals

Let (t) be a solution of [1] or [24]; by using the Hamilton–Jacobi equation [20], we get

d S SððtÞÞ ¼ ððtÞÞ; @t ðtÞ dt 

1 S S r ððtÞÞ; ððtÞÞr ððtÞÞ ¼ 2   0

½25

In particular, we have that (d=dt)S((t)) = 0 if and only if ( S= )((t)) = 0. We remark that the right-hand side of [25] vanishes in the stationary state, that is, there is no internal entropy production due to the evolution. On the other hand, there is a steady entropy production due to the differences in the chemical potentials of the reservoirs. This is not discussed in this article. Decomposition of Hydrodynamics

There is a structural property of hydrodynamics which follows from the HJE. The hydrodynamic equation can be decomposed as the sum of a gradient vector field and a vector field A orthogonal to it in the metric induced by the operator K1 , where Kf = r  (()rf ), namely   1 S DðÞ ¼ r  ðÞr þ AðÞ ½26 2  with K

S 1 ; K AðÞ 



¼

S ; AðÞ 

¼0

Similarly, using the fluctuation dissipation relationship [23] for the adjoint hydrodynamics, we have   1 S D ðÞ ¼ r  ðÞr  AðÞ ½27 2  Since A is orthogonal to S= , it does not contribute to the entropy production. The vector field A is odd under time reversal like a magnetic force. Both terms of the decomposition vanish in the stationary state, that is, when  = . Whereas in equilibrium the hydrodynamics is the gradient flow of the entropy S, the term A() is characteristic of nonequilibrium states. Note that, for small fluctuations   , small differences in the chemical potentials at the boundaries, A() becomes a second-order quantity and Onsager theory is a consistent approximation. Equation [26] is interesting because it separates the dissipative part of the hydrodynamic evolution associated to the thermodynamic force S=  and

provides therefore an important physical information. Notice that the thermodynamic force S=  appears linearly in the hydrodynamic equation even when this is nonlinear in the macroscopic variables. In general, the two terms of the decomposition [26] are nonlocal in space even if D is a local function of . This is the case for the simple exclusion process discussed later. Furthermore while the form of the hydrodynamic equation does not depend explicitly on the chemical potentials, S=  and A do. To understand how the decomposition [26] arises microscopically, let us consider a stochastic lattice gas. Let L ¼ 12ðL þ L Þ þ 12 ðL  L Þ

½28

be its Markov generator, where L is the adjoint of L with respect to the invariant measure, namely the generator of the time-reversed microscopic dynamics. The term L  L behaves like a Liouville operator, that is, it is anti-Hermitian and, in the scaling limit, produces the term A in the hydrodynamic equation. This can be verified explicitly in the boundary-driven zero-range model introduced in the next section. Since the adjoint generator can be written as L = (L þ L )=2  (L  L )=2, the adjoint hydrodynamics must be of the form [27]. In particular, if the microscopic generator is self-adjoint, we get A = 0 and thus D() = D (). On the other hand, it may happen that microscopic nonreversible processes, namely for which L 6¼ L , can produce macroscopic reversible hydrodynamics if L  L does not contribute to the hydrodynamic limit. The decompositions [26] and [27] remind of the electrical conduction in the presence of a magnetic field. Consider the motion of electrons in a conductor: a simple model is given by the effective equation   1 1 p^H  p ½29 p_ ¼ e E þ mc  where p is the momentum, e the electron charge, E the electric field, H the magnetic field, m the mass, c the velocity of the light, and  the relaxation time. The dissipative term p= is orthogonal to the Lorentz force p ^ H. We define time reversal as the transformation p 7! p, H 7! H. The adjoint evolution is given by   1 1 _ ¼ e E þ p^H  p mc 

½30

Macroscopic Fluctuations and Thermodynamic Functionals

where the signs of the dissipation and the electromagnetic force transform in analogy to [26] and [27]. Let us consider in particular the Hall effect where we have conduction along a rectangular plate immersed in a perpendicular magnetic field H with a potential difference across the longer side. The magnetic field determines a potential difference across the other side of the plate. In our setting on the contrary, it is the difference in chemical potentials at the boundaries that introduces in the equations a ‘‘magnetic-like’’ term. There is therefore a kind of equivalence between certain externally applied fields and driving the system at the boundaries. Minimum Dissipation Principle

In 1931 Onsager formulated, within his near equilibrium theory, a variational principle which shows that the hydrodynamic evolution minimizes at each instant of time a quadratic functional of . ˙ He called this the ‘‘minimum dissipation principle.’’ We now show that the decomposition of the previous subsection leads to a natural exact generalization of this principle. We want to construct a functional of the variables  and ˙ such that the Euler equation associated to the vanishing of the first variation under arbitrary changes of ˙ is the hydrodynamic equation [1]. We define the ‘‘dissipation function’’   _ ¼ ð_  AðÞÞ; K1 ð_  AðÞÞ Fð; Þ ½31 and the functional _ _ ¼ SðÞ _ ð; Þ þ Fð; Þ

S ; _ þ hð_  AðÞÞ; ¼  K1 ð_  AðÞÞi

½32

which generalize the corresponding Onsager’s definitions (Onsager 1931a, b). The operator K has been defined in the previous subsection. It is easy to verify that _  ¼ 0

½33

is equivalent to the hydrodynamic equation [1]. Furthermore, a simple calculation gives

1 S S r ; ðÞr ½34 Fj_ ¼ DðÞ ¼ 4   that is, 2F on the hydrodynamic trajectories equals the entropy production rate as in Onsager’s near equilibrium approximation.

363

The dissipation function for the adjoint hydrodynamics is obtained by changing the sign of A in [31]. Entropy and Optimal Control

There is an interesting interpretation of the entropy as a minimal cost to produce a fluctuation by externally acting on the system. The idea is to show that there exists a cost function which on the optimal control trajectory coincides with the entropy difference with respect to the stationary state. We add an external perturbation v to the hydrodynamic equation @t  ¼ 12r  ðDðÞrÞ þ v ¼ DðÞ þ v

½35

We want to choose v so as to drive, with minimal cost, the system from its stationary state  to an arbitrary state . A simple cost function is Z 1 t2 dshvðsÞ; K1 ððsÞÞvðsÞi ½36 2 t1 where (s) is the solution of [35] and we recall that K()f = r  (()rf ). More precisely, given (t1 ) = , we want to drive the system to (t2 ) =  by an external field v which minimizes [36]. This is a standard problem in control theory. Let Z 1 t2 VðÞ ¼ inf dshvðsÞ; K1 ððsÞÞvðsÞi ½37 2 t1 where the infimum is taken with respect to all fields v which drive the system to  in an arbitrary time interval [t1 , t2 ]. The optimal field v can be obtained by solving the Bellman equation which reads

1 V 1 min hv; K ðÞvi  DðÞ þ v; ¼ 0 ½38 v 2  It is easy to express the optimal v in terms of V; we get v¼K

V 

Hence, [38] now becomes



1 V V V ; KðÞ þ DðÞ; ¼0 2   

½39

½40

By identifying the cost functional V() with S(), eqn [40] coincides with the Hamilton–Jacobi equation [20]. By inserting the optimal v [39] in [35] and identifying V with S, we get that the optimal trajectory (t) solves the time-reversed adjoint hydrodynamics, namely @t  ¼ D ðÞ

½41

364 Macroscopic Fluctuations and Thermodynamic Functionals

The trajectory of the spontaneous emergence of a fluctuation coincides therefore with the trajectory of minimal cost for the optimal control. The optimal field v does not depend on the nondissipative part A of the hydrodynamics.

solved exactly and the previous theory can be checked in full detail. Let us introduce the macroscopic coordinates, time t = =N 2 and space u = x=N. To describe the macroscopic dynamics, we introduce the empirical density

Models The general theory will now be illustrated by briefly describing models where it has been successfully applied. We consider examples of different nature in order to emphasize the generality and flexibility of the point of view developed in the previous section. We have chosen three examples in which the theory is used in different ways. The first one, the zero-range process, can be solved in a simple way so that the theory can be verified in detail. In the second one, the symmetric simple exclusion, we derive from the HJE a nonlinear ordinary differential equation first obtained by Derrida, Lebowitz, and Speer through a direct rather complex calculation. This equation implies the nonlocality of the entropy in the SNS of this model. The third model, the Kawasaki– Glauber dynamics, provides the illustration of two aspects. Nonlocality of the entropy, that is, longrange correlations, can appear in isolated equilibrium states if the microscopic dynamics is not time-reversal invariant. This means that long-range correlations as a signature of time-reversal violation are not restricted to SNSs. The second aspect to be underlined is the effectiveness of the HJE in a more complex case: in fact in this model, the number of particles is not conserved which leads to a very complicated structure of the HJE. As a general comment, we emphasize that dynamics microscopically different but leading to the same macroscopic description, in particular the same hydrodynamics and large deviation functional, are indistinguishable for the theory which is purely macroscopic. Zero Range

We consider the so-called zero-range process which models a nonlinear diffusion of a lattice gas. The model is described by a positive integer variable  (x) representing the number of particles at site x and time  of a finite lattice which for simplicity we assume one dimensional. The particles jump with rates g( (x)) to one of the nearestneighbor sites x þ 1, x  1 with probability 1/2. The function g(k) is nondecreasing and g(0) = 0. We assume that our system interacts with two reservoirs of particles in positions N and N with rates pþ and p , respectively. This model can be

N ðt; uÞ ¼

N 1 X

2 ðxÞ ðu  x=NÞ N x¼N N t

½42

where (u  x=N) is the Dirac . One can prove that in the limit N ! 1, the empirical density [42] tends in probability to a continuous function t (u), which satisfies the following hydrodynamic equation: @t  ¼ 12ðÞ ¼ DðÞ

½43

where () can be explicitly defined in terms of the rates g( ). The boundary conditions for [43] are ((t, 1)) = p . The adjoint hydrodynamics is

1 ðÞ ½44 @t  ¼ ðÞ  r ¼ D ðÞ 2 ðuÞ with ðuÞ ¼

pþ  p pþ þ p uþ 2 2

and ¼

pþ  p 2

The boundary conditions for [44] are the same as for [43]. The second term on the right-hand side of [44] is proportional to the difference of the chemical potentials and produces an inversion of the particle flux. The action functionals J() ˆ and J () ˆ for this model have been computed and have the form [18] and [22], respectively, with () = (). The entropy S() can be easily computed directly from the expression of the invariant measure which is of product type and is known explicitly: Z 1  ððuÞÞ SðÞ ¼ du ðuÞ log ðuÞ 1  ZðððuÞÞÞ  log ½45 Zð ðuÞÞ where ZðÞ ¼ 1 þ

1 X

k gð1Þ    gðkÞ k¼1

It is easy to verify that it solves the HJE. Due to the special zero-range character of the interaction in this model, there are no long-range correlations in nonequilibrium states.

Macroscopic Fluctuations and Thermodynamic Functionals Simple Exclusion

The simple exclusion process is a model of a lattice gas with an exclusion principle: a particle can move to a neighboring site, with rate 1/2 for each side, only if this is empty. We consider again a onedimensional case and we denote by x () 2 {0, 1} the number of particles at the site x at (microscopic) time . The system is in contact with particle reservoirs at the boundaries N where a particle is created with rates p if the boundary site is empty and is destroyed 1  p if it is occupied. In contrast to the zero-range model, the invariant measure carries long-range correlations making the entropy nonlocal. The hydrodynamic equation for the simple exclusion process can be derived as for the zero-range process; in fact, it is easier in this case because a simple computation leads directly to a closed equation for the empirical density which is defined as in [42] except that the variable now takes only the values 0 or 1. We find that the limiting density evolves according to the linear heat equation @t ðt; uÞ ¼ 12ðt; uÞ ¼ DðÞ

½46

with boundary conditions ðt; 1Þ ¼

p

¼ 

1 þ p

In this case, the density of particles  takes values in [0,1]. We use the HJE to calculate the entropy. For this model, we have () = (1  ). We show that the solution of the HJE for S() (which is a functional derivative equation) can be reduced to the solution of an ordinary differential equation. The Hamilton–Jacobi equation for the simple exclusion process is



S S S r ; ð1  Þr ;  ¼ 0 ½47 þ    We look for a solution of the form S ðuÞ ¼ log  ðu; Þ ðuÞ 1  ðuÞ

½48

for some functional (u; ) to be determined satisfying the boundary conditions ð 1Þ ¼ log



1  

in the space variable. The first term on the righthand side is the derivative of the equilibrium entropy, that is for boundary conditions  = þ . Inserting [48] into [47], we get (note that   e =(1 þ e ) vanishes at the boundary)

365

 0 ¼  r log



   ; ð1  Þr 1 D E ¼  hr; ri þ ð1  Þ; ðrÞ2  

e ; r ¼ r  1 þ e   

e 1 2     ; ðrÞ 1 þ e 1 þ e * !+  e ðrÞ2 2 ¼   ðrÞ ;  þ 1 þ e 1 þ e

We obtain a nontrivial solution of the Hamilton– Jacobi if we solve the following ordinary differential equation, corresponding to the vanishing of the right side of the scalar product, which relates the functional (u) = (u; ) to : 1 ¼ ðuÞ; 1 þ eðuÞ ½rðuÞ 

ð 1Þ ¼ log 1  

ðuÞ

2

þ

u 2 ð1; 1Þ ½49

It is clear that  is a nonlocal functional of . A computation shows that the derivative of the functional Z SðÞ ¼ du  log  þ ð1  Þ logð1  Þ

r  þð1  Þ  logð1 þ e Þ þ log r  is given by [48] when (u; ) solves [49]. Kawasaki–Glauber Dynamics

The model consists of particles on a lattice evolving according to two basic dynamical processes: 1. a particle can move to a neighboring site if this is empty as in the simple exclusion and 2. a particle can disappear in an occupied site or be created if this is empty, the rate depending on the nearby configuration. The first process is conservative while the second is not. As before the object of our study is the empirical density [42]. It is possible to show that as N goes to infinity, (t, u) is a solution of @t  ¼ 12 þ BðÞ  DðÞ

½50

BðÞ ¼ E  ðcð Þð1  ð0ÞÞÞ

½51

DðÞ ¼ E  ðcð Þ ð0ÞÞ

½52

with

366 Macroscopic Fluctuations and Thermodynamic Functionals

where  is the Bernoulli product distribution with parameter . Typically, B() and D() are polynomials in . For this model we consider equilibrium states so that we can take periodic boundary conditions. An equilibrium state corresponds to a density  which is the solution of the equation B() = D() gives a minimum of the potential R  and V() = [D(0 )  B(0 )]d0 . We admit potentials with several minima. The Hamiltonian associated to the large deviation functional for this model is not quadratic: Z 1 1 H þ ðrHÞ2 ð1  Þ Hð; HÞ ¼ du 2 2  BðÞð1  exp HÞ  DðÞ

 ð1  expðHÞÞ

½53

where H has the role of the conjugate momentum. The Hamilton–Jacobi equation   S ¼0 ½54 H ;  is therefore very complicated but can be solved by successive approximations using as an expansion parameter   , where  is a solution of B() = D() that is a stationary solution of hydrodynamics. For  = , we have S=  = 0. We are looking for an approximate solution of [54] of the form Z Z 1 du dvððuÞ  Þkðu; vÞððvÞ  Þ SðÞ ¼ 2 þ oð  Þ2

½55

The kernel k(u, v) is the inverse of the density correlation function c(u, v). Z cðu; yÞkðy; vÞ dy ¼ ðu  vÞ ½56 By inserting [55] in [54], one can show that k(u, v) satisfies the following equation: 1 2 ð1



 Þu kðu; vÞ  b0 kðu; vÞ 1 2u ðu

 vÞ þ ðd1  b1 Þ ðu  vÞ ¼ 0

½57

where b1 ¼ B0 ðÞj¼ ;

d1 ¼ D0 ðÞj¼

and Þ ¼ Dð Þ ¼ d0 b0 ¼ Bð

½58

If the entropy is a local functional of the density, k(u, v) must be of the form k(u, v) = f ( ) (u  v) which inserted in [57] gives f ð Þ ¼ ½ ð1  Þ1

½59

ð1  Þ1  ðd1  b1 Þ ¼ 0 b0 ½

½60

and

Therefore if b0 , b1 , d1 do not satisfy the last equation, the entropy cannot be a local functional of the density. It can be shown that in this case timereversal invariance is violated and the adjoint hydrodynamics is different from [50]. This calculation supports the conjecture that macroscopic correlations are a generic feature of equilibrium states of nonreversible lattice gases. See also: Interacting Particle Systems and Hydrodynamic Equations; Interacting Stochastic Particle Systems; Nonequilibrium Statistical Mechanics (Stationary): Overview; Quantum Central-Limit Theorems.

Further Reading Bertini L, De Sole A, Gabrielli D, Jona–Lasinio G, and Landim C (2002) Macroscopic fluctuation theory for stationary nonequilibrium states. Journal of Statistical Physics 107: 635–675. Bertini L, De Sole A, Gabrielli D, Jona–Lasinio G, and Landim C (2004) Minimum dissipation principle in stationary nonequilibrium states. Journal of Statistical Physics 116: 831–841. Bertini L, De Sole A, Gabrielli D, Jona-Lasinio G, and Landim C (2005) Nonequilibrium current fluctuations in stochastic lattice gases. To appear in Journal of Statistical Physics, arXIV: cond-mat/0506664. See also references therein. Derrida B, Lebowitz JL, and Speer ER (2002) Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. Journal of Statistical Physics 107: 599–634. Freidlin MI and Wentzell AD (1998) Random Perturbations of Dynamical Systems. New York: Springer. Kipnis C and Landim C (1999) Scaling Limits of Interacting Particle Systems. Berlin: Springer. Landau L and Lifshitz E (1967) Physique Statistique. Moscow: MIR. Lanford OE (1973) In: Lenard A (ed.) Entropy and Equilibrium States in Classical Statistical Mechanics, Lecture Notes in Physics, vol. 20. Berlin: Springer. Luchinsky DG and McClintock PVE (1997) Irreversibility of classical fluctuations studied in analogue electrical circuits. Nature 389: 463–466. Onsager L (1931a) Reciprocal relations in irreversible processes. I. Physical Review 37: 405–426. Onsager L (1931b) Reciprocal relations in irreversible processes. II. Physical Review 38: 2265–2279. Onsager L and Machlup S (1953) Fluctuations and irreversible processes. Physical Review 91: 1505–1512 and 1512–1515. Spohn H (1991) Large Scale Dynamics of Interacting Particles. Berlin: Springer.

Magnetic Resonance Imaging

367

Magnetic Resonance Imaging C L Epstein, University of Pennsylvania, Philadelphia, PA, USA F W Wehrli, University of Pennsylvania, Philadelphia, PA, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction Nuclear magnetic resonance (NMR) is a subtle quantum-mechanical phenomenon that, through magnetic resonance imaging (MRI), has played a major role in the revolution in medical imaging over the last 30 years. Before being conceived for use in imaging, NMR was employed by chemists to do spectroscopy, and it remains a very important technique for determining the structure of complex chemical compounds like proteins. In this article we explain how NMR is used to create an image of a three-dimensional object. Scant attention is paid to both NMR spectroscopy, and the quantum description of NMR. Those seeking a more complete introduction to these subjects should consult the article Nuclear Magnetic Resonance in this Encyclopedia, as well as the monographs of Abragam (1983) or Ernst et al. (1987), for spectroscopy, and that of Callaghan (1993) for imaging. All three books consider the quantum-mechanical description of these phenomena. Comprehensive discussions of MRI can be found in Bernstein et al. (2004) and Haacke et al. (1999), and a historical appreciation of the development of MRI is given in Wehrli (1995).

The Bloch Equation We begin with the Bloch phenomenological equation, which provides a model for the interactions between applied magnetic fields and the nuclear spins in the objects under consideration. This is a macroscopic averaged model that describes the interaction of aggregates of spins, called isochromats, with applied magnetic fields. An isochromat is a collection of ‘‘like’’ spins, which is spatially large on the atomic scale, but very small on the scale of the variations present in the applied magnetic fields. Spins are alike if they belong to the same species and are in the same chemical environment. There may be several different classes of spins, but, in this article, it is assumed that they are noninteracting and so it suffices to consider each separately. Heretofore, we suppose that there is a single class of like spins. The distribution of isochromats for these spins is described macroscopically by the spin density

function, which we denote by (x, y, z). In most medical applications, one is imaging the distribution of spins arising from hydrogen protons in water molecules. The state of the isochromat at spatial location (x, y, z) is given by a 3-vector: Mðx; y; zÞ ¼ ðm1 ðx; y; zÞ; m2 ðx; y; zÞ; m3 ðx; y; zÞÞ which is interpreted as the magnetic moment per unit volume. It is an ensemble mean of the quantum dipoles caused by the spins within the isochromat. In most applications of NMR to imaging, the applied magnetic field is described as the sum of a large, time-independent field, B0 (x, y, z), and smaller timedependent fields, B0 (x, y, z; t). In the presence of a static field, thermal fluctuations cause the nuclear spins to slightly prefer an orientation aligned with the field. Using the Boltzmann distribution, one obtains that the nuclear paramagnetic susceptibility of water protons is given by ¼

 22 h 4kB T

½1

here h is Planck’s constant, kB the Boltzmann’s constant, and T the absolute temperature, (see Levitt (2001)). The constant  is called the gyromagnetic (or magnetogyric) ratio. For a proton,   2  42:5764  106 rad s1 T1

½2

For water molecules at room temperature,   3.6  109 . If the sample is held stationary in the field B0 for a sufficiently long time, then the spins become polarized and a bulk magnetic moment appears; this is called the equilibrium magnetization: M0 ðx; y; zÞ ¼ ðx; y; zÞB0 ðx; y; zÞ

½3

The Bloch equation describes the evolution of M under the influence of the applied field B = B0 þ B0 : dMðx; y; z; tÞ ¼ Mðx; y; z; tÞ  Bðx; y; z; tÞ dt 1 1  M? ðx; y; z; tÞ þ ðM0 ðx; y; zÞ T2 T1  Mk ðx; y; z; tÞÞ

½4

Here  is the vector cross-product, M? (x, y, z; t) the component of M(x, y, z; t) perpendicular to B0 (x, y, z) (called the transverse component), and Mk the component of M parallel to B0 (called the longitudinal component). For hydrogen protons in other molecules, the gyromagnetic ratio is expressed in the form (1  ). The coefficient  is called the

368 Magnetic Resonance Imaging

nuclear shielding; it is typically between 104 and þ104 . The difference in the nuclear shielding causes a shift in the resonance frequency by . The second and third terms in eqn [4] are relaxation terms. They provide a phenomenological model for the averaged interactions of the spins with one another and their environment. The coefficient 1=T1 (x, y, z) is the spin lattice relaxation rate; it describes the rate at which the magnetization returns to equilibrium. The coefficient 1=T2 (x, y, z) is the spin–spin relaxation rate; it describes the rate at which the transverse components of M decay. The physical processes causing these relaxation phenomena are different and so are the rates themselves, with T2 less than T1 . The relaxation rates largely depend on the localized thermal fluctuations of the molecules and provide a useful contrast mechanism in MR imaging. Spin–spin relaxation occurs very rapidly in solids (S   ZZZ i SL ð!; u  sÞ ¼ s ðxÞ exp h    Fð!Þu ðzÞD! dx dz

where the left-hand side involves a time discretization of the second derivative. When F(!) = !(t), Feynman obtains the path space version of Heisenberg commutation relation between position and momentum observables:   !ðtÞ  !ðt  Þ !ðt þ Þ  !ðtÞ !ðtÞ !ðtÞ    SL

½6

This is a shorthand for the time discretization version along broken paths ! interpolating linearly between point xj = !(tj ), tj = j(u  s)=N, j = 0, 1, . . . , N. In [6]  h is Planck’s constant and S = SL denotes the action functional with Lagrangian L of the underlying classical system. For a particle with mass m in a scalar potential V on the real line, Z u  m 2 SL ð!; u  sÞ ¼ !_ ðÞ  Vð!ðÞ d ½7 2 s The ‘‘D!’’ of [6] is used as a Lebesgue measure, although there is no such thing in infinite dimensions. More generally, the construction of measures or integrals on the various path spaces required for general quantum systems is still nowadays a field of investigation. When F = 1 and u (the complex conjugate of u ) reduces to a Dirac mass at z, [6] is the path-integral representation of the solution (x, u) of the initialvalue problem in L2 : @ ¼H @u ðx; sÞ ¼ s ðxÞ

i h

½8

2

where H = (h =2) þ V and when SL is as in [7]. Feynman’s framework is time symmetric on I: when s = x (still for F = 1), [6] provides a path-integral representation of the solution of the final-value  s). problem for (z, According to Feynman, ‘‘it would be possible to use the integration-by-parts formula  

F i

S F ¼ ½9

!ðsÞ h

!ðsÞ as a starting point to define the laws of quantum mechanics’’ (Feynman and Hibbs 1965, p. 173). The functional derivative corresponds to variations of the underlying paths in directions ! and Z

F

!ðsÞ ds

F ¼

!ðsÞ to an L2 analog of [4].

€ >SL ¼  SL < m!

¼i

 h m

½10

½11

and from this the crucial fact that ‘‘quantum mechanical paths are very irregular. However, these irregularities average out over a reasonable length of time to produce a reasonable drift or average velocity’’ (Feynman and Hibbs 1965, p. 177). A probabilistic interpretation (cf. Cruzeiro and Zambrini (1991)) of Feynman’s calculus uses (Bernstein) diffusion processes solving the SDE  1=2 h h dzðtÞ ¼ dWðtÞ þ r log ðzðtÞ; tÞ dt ½12 m m where the drift stems from a positive solution of the Euclidean version of the above final-value problem for , @ ¼ H @t ðx; uÞ ¼ u ðxÞ h

½13

For any regular function f, we can make sense of the ‘‘continuous limit’’ 1 Df ðzðtÞ; tÞ ¼ lim Et ½f ðzðt þ Þ; t þ Þ !0   f ðzðtÞ; tÞÞ

½14

where Et denotes conditional expectation with respect to the past P t and check, indeed, that DzðtÞ ¼

h r log ðzðtÞ; tÞ m

is Feynman’s ‘‘reasonable drift.’’ Using Feynman– Kac formula, one shows that the diffusions [12] have laws which are absolutely continuous with respect to the Wiener measure of parameter  h=m, with Radon–Nikodym density given by   Z ðzðuÞ; uÞ 1 1 ðzÞ ¼ exp  VðzðÞÞ d ðzðsÞ; sÞ h 0

Malliavin Calculus

We can, therefore, use Malliavin calculus on the path space of these diffusions and the associated integration-by-parts formula to make sense of [9] and all its consequences. The probabilistic counterpart of the time symmetry of Feynman’s framework is interesting: Heisenberg’s original argument to deny the existence of quantum trajectories (1927) was that any position can be associated with two velocities. Feynman’s interpretation [11] and the definition [14] suggest that this has to do with a past or future conditioning at time t. Indeed, there is another description of diffusions z(t) with respect to a family of future -fields, using the Euclidean version of the initialvalue problem for , underlying [6]. Another drift built on the model of the drift in [12] results, and Feynman’s commutation relation [11] becomes rigorous (without, of course, the factor i). We refer to Cruzeiro and Zambrini (1991) for a development of this approach using Malliavin calculus. See also: Euclidean Field Theory; Functional Integration in Quantum Physics; Measure on Loop Spaces; Stochastic Differential Equations; Stochastic Hydrodynamics.

Further Reading Airault H and Malliavin P (1988) Inte´gration ge´ometrique sur l’ espace de Wiener. Bulletin des Sciences Mathe´matiques 112(2): 3–52. Bismut JM (1984) Large Deviations and the Malliavin Calculus. Birkha¨user Progress in Mathematics 45.

389

Cipriano F (1999) The two-dimensional Euler equation: a statistical study. Communications in Mathematical Physics 201(1): 139–154. Cruzeiro AB and Malliavin P (1996) Renormalized differential geometry on path space: structural equation, curvature. Journal of Functional Analysis 139: 119–181. Cruzeiro AB and Zambrini JC (1991) Malliavin calculus and Euclidean quantum mechanics. I. Functional Calculus. Journal of Functional Analysis 96: 62–95. Feynman RP and Hibbs AR (1965) Quantum Mechanics and Path Integrals. International Series in Pure and Applied Physics. New York: McGraw-Hill. Ikeda N and Watanabe S (1989) Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, vol. 24. Amsterdam–New York–Oxford: North-Holland. Imkeller P (1998) The smoothness of laws of random flags and Oseledets spaces of linear stochastic differential equations. Potential Analysis 9: 321–349. Malliavin P (1978) Stochastic calculus of variations and hypoelliptic operators. Proceedings of the International Symposium on Stochastic Differential Equations, Kyoto 1976, pp. 195–263. Wiley. Malliavin P (1997) Stochastic Analysis. Springer Verlag Grund. der Math. Wissen. vol. 313. Berlin: Springer. Malliavin MP and Malliavin P (1990) Integration on loop groups. I. Quasi-invariant measures. Journal of Functional Analysis 93: 207–237. Malliavin P and Taniguchi S (1997) Analytic functions, Cauchy formula, and stationary phase on a real abstract Wiener space. Journal of Functional Analysis 143: 470–528. Malliavin P and Thalmaier A (2005) Stochastic Calculus of Variations in Mathematical Finance. Berlin: Springer. Nualart D (1995) The Malliavin Calculus and Related Topics, Springer Verlag Problem and Its Applications. New York– Berlin–Heidelberg: Springer. Stroock DW (1981) The Malliavin Calculus and its application to second order parabolic differential equations I (II). Mathematical Systems Theory 14: 25–65 and 141–171.

Marsden–Weinstein Reduction see Cotangent Bundle Reduction; Poisson Reduction; Symmetry and Symplectic Reduction

Maslov Index see Optical Caustics; Semiclassical Spectra and Closed Orbits; Stationary Phase Approximation

390 Mathai–Quillen Formalism

Mathai–Quillen Formalism S Wu, University of Colorado, Boulder, CO, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction Characteristic classes play an essential role in the study of global properties of vector bundles. Particularly important is the Euler class of real orientable vector bundles. A de Rham representative of the Euler class (for tangent bundles) first appeared in Chern’s generalization of the Gauss– Bonnet theorem to higher dimensions. The representative is the Pfaffian of the curvature, whose cohomology class does not depend on the choice of connections. The Euler class of a vector bundle is also the obstruction to the existence of a nowherevanishing section. In fact, it is the Poincare´ dual of the zero set of any section which intersects the zero section transversely. In the case of tangent bundles, it counts (algebraically) the zeros of a vector field on the manifold. That this is equal to the Euler characteristic number is known as the Hopf theorem. Also significant is the Thom class of a vector bundle: it is the Poincare´ dual of the zero section in the total space. It induces, by a cup product, the Thom isomorphism between the cohomology of the base space and that of the total space with compact vertical support. Thom isomorphism also exists and plays an important role in K-theory. Mathai and Quillen (1986) obtained a representative of the Thom class by a differential form on the total space of a vector bundle. Instead of having a compact support, the form has a nice Gaussian peak near the zero section and exponentially decays along the fiber directions. The pullback of Mathai–Quillen’s Thom form by any section is a representative of the Euler class. By scaling the section, one obtains an interpolation between the Pfaffian of the curvature, which distributes smoothly on the manifold, and the Poincare´ dual of the zero set, which localizes on the latter. This elegant construction proves to be extremely useful in many situations, from the study of Morse theory, analytic torsion in mathematics to the understanding of topological (cohomological) field theories in physics. In this article, we begin with the construction of Mathai–Quillen’s Thom form. We also consider the case with group actions, with a review of equivariant cohomology and then Mathai–Quillen’s construction in this setting. Next, we show that much of the above can be formulated as a ‘‘field theory’ on a

superspace of one fermionic dimension. Finally, we present the interpretation of topological field theories using the Mathai–Quillen formalism.

Mathai–Quillen’s Construction Berezin Integral and Supertrace

Let V be an oriented real vector space of dimension n with a volume element  2 ^n V compatible with the orientation. The ‘‘Berezin R B integral’’ of a form ! 2 ^ V  on V, denoted by !, is the pairing h, !i. Clearly, only the top degree component of ! contributes. For example, if  2 ^2 V  is a 2-form, then 8  ^ðn=2Þ > Z B < ;  ; if n is even ðn=2Þ! e ¼ > : 0; if n is odd If V has a Euclidean metric ( , ), then  is chosen to be of unit norm. If  2 End(V) is skew-symmetric, then (1=2)( ,  ) is a 2-form and, if n is even, the Pfaffian of  is   Z B 1 PfðÞ ¼ exp ð ;  Þ 2 The Berezin integral can be defined on elements in ^ A, where A is any a graded tensor product ^ V   Z2 -graded commutative algebra. For example, if we consider the identity operator x = idV as a V-valued function on V, then dx is a 1-form on V valued in V, and (dx, ) is a 1-form valued in V  . Let {e1 , . . . , en } be an orthonormal basis of V and write x = xi ei , where xi are the coordinate functions on V. We let   Z ð1Þnðnþ1Þ=2 B 1 ðx; xÞ  ðdx; Þ exp  uðxÞ ¼ 2 ð2Þn=2 ^ ^ V  . The result is The integrand is in  (V)    1 1 ðx; xÞ dx1 ^    ^ dxn ½1 uðxÞ ¼ exp  n=2 2 ð2Þ a Gaussian n-form whose (usual) integration on V is 1. Let Cl(V) be the Clifford algebra of V. For any orthonormal basis {ei }, let  i be the corresponding generators of Cl(V) and let  = ei   i 2 V  Cl(V). For any ! 2 ^k V  , we have !ð; . . . ; Þ ¼

1 !i i  i1     ik 2 ClðVÞ k! 1 k

If n is even, the Clifford algebra has a unique Z2 -graded irreducible spinor representation S(V) = Sþ (V)  S (V). For any element a 2 Cl(V), the

Mathai–Quillen Formalism

supertrace is str a = trSþ (V) a  trS (V) a. If  2 End(V) is skew-symmetric, then   pffiffiffiffiffiffiffi 1 1=2 ^ ð; Þ ¼ AðÞ str exp PfðÞ 4 where  ^ AðÞ ¼ det

=2 sinhð=2Þ

is concentrated on the zero locus of the section s. In fact, the Euler class is the Poincare´ dual to the homology class represented by s1 (0). Hence, if n  m and if ! 2 nm (M) is closed, we have Z Z ! ^ er;s ðEÞ ¼ ! ½6 M



More generally, supertrace can be defined on ^ A for any Z2 -graded commutative algebra Cl(V)  þ A = A  A . If  is skew-symmetric and  2 V   A , then  pffiffiffiffiffiffiffi 1=2  pffiffiffiffiffiffiffi  1 1 ð; Þ þ str exp ðÞ 4 2   Z B 1 1=2 ^ ½2 ¼ AðÞ exp ð ;  Þ þ  2

Representatives of the Euler and Thom Classes

Let M be a smooth manifold and let  : E ! M be an oriented real vector bundle of rank r. Suppose E has a Euclidean structure ( , ) and r is a compatible connection. The curvature R 2 2 (M, End (E)) is skew-symmetric, and hence ( , R ) 2 2 (M, ^2 E ). A de Rham representative of the Euler class of E is     Z B 1 1 R ð ; R Þ ¼ Pf exp er ðEÞ ¼ ½3 r=2 2 2 ð2Þ Here, the Berezin integration is fiberwise in E: it is the pairing between the integrand and the unit section  of the trivial line bundle ^r E that is consistent with the orientation of E. The de Rham cohomology class of [3] is independent of the choice of ( , ) or r. Let s be a section of E. Following Berline et al. (1992) and Zhang (2001), we consider sr; s ¼ 12 ðs; sÞ þ ðrs; Þþ 12 ð; R Þ 

½4



a differential form on M valued in ^ E . Mathai– Quillen’s representative of the Euler class is Z ð1Þrðrþ1Þ=2 B r; s er; s ðEÞ ¼ e ½5 ð2Þr=2 One can show that er, s (E) is closed and that as  varies, the cohomology class of er, s (E) does not change. By taking  ! 0, the de Rham class of er, s (E) is equal to that of er (E) when r is even. The form er, s (E) provides a continuous interpolation between [3] and the limit as  ! 1, when the form

391

s1 ð0Þ

when s intersects the zero section transversely. To obtain Mathai–Quillen’s representative of the Thom class, we consider the pullback of E to E itself. The bundle  E ! E has a tautological section x. Applying [5] to this setting, we get  Z ð1Þrðrþ1Þ=2 B 1 r ðEÞ ¼ exp  ðx; xÞ r=2 2 ð2Þ  1 ðrx; Þ  ð ; R Þ ½7 2 where (  ,  ), r, and R are understood to be the pullbacks to  E. This is a closed form on the total space of E. Moreover, its restriction to each fiber is the Gaussian form [1]. The cohomology groups of differential forms with exponential decay along the fibers are isomorphic to those with compact vertical support or the relative cohomology groups H  (E, EnM). Here M is identified with its image under the inclusion i : M ! E by the zero section. Under the above isomorphism, the cohomology class represented by r (E) coincides with the Thom class (E) = i 1 2 H r (E, EnM) defined topologically. For any section s 2 (E), we have er, s (E) = s r (E). Character Form of the Thom Class in K-Theory

Let E = Eþ  E be a Z2 -graded vector bundle over ^ M. The spaces  (M, E), (End(E)) and  (M)  ^ (End(E)) are also Z2 -graded. The action of   T 2 ^ (End(E)) on   s 2  (M, E) is  (M)  ^ T :   s 7! ð1ÞjTj jj ð ^ Þ  ðTsÞ  The supertrace of A 2 (End(E)) is str A = trEþ A  ^ trE A; it extends  (M)-linearly to str :  (M)   (End(E)) !  (M). Let r be a connection on E preserving the grading. r is an odd operator on  (M, E). If L 2 (End(E) ) is odd, then D = r þ L is called a ‘‘superconnection’’ on E; the ‘‘curvature’’ D2 = R þ rL þ L2 2 ( (M)  (End(E)))þ is even. With the superconnection, the Chern character of the virtual vector bundle Eþ E can be represented by  pffiffiffiffiffiffiffi  1 2 þ  D chr; L ðE ; E Þ ¼ str exp ½8 2

392 Mathai–Quillen Formalism

It is a closed form on M and its de Rham cohomology class is independent of the choice of r or L. If L ispinvertible everywhere on M and the ffiffiffiffiffiffi eigenvalues of 1L2 are negative, then [8] is exact: chr;L ðEþ ; E Þ ! ! pffiffiffiffiffiffiffi Z 1 pffiffiffiffiffiffiffi 1 1 2 d ðr þ LÞ L d str exp ¼ 2 2 1 Now let E be an oriented real vector bundle of rank r = 2m over M with a Euclidean structure ( , ). Suppose further that E has a spin structure. The associated spinor bundle S(E) = Sþ (E)  S (E) is a graded complex vector bundle over M. For any section s 2 (E), let c(s) 2 (End(E) ) be the Clifford multiplication on E. Then for any s, s0 2 (E), we have {c(s), c(s0 )} = 2(s, s0 ). Given a connection r on E preserving ( , ), the induced spinor connection rS on S(E) preserves the grading. If R is the curvature of r, that of rS is RS = (1=4) (, R), where  is now a section of E  Cl(E). For any s 2 (E), consider the superconnection  1=2  Ds ¼ rS þ pffiffiffiffiffiffiffi cðsÞ 1 The Chern character form [8] of Sþ (E) S (E) is, using [2],  1=2 m^ R þ  chr; s ðS ðEÞ; S ðEÞÞ ¼ ð1Þ A er; s ðEÞ ½9 2 where er, s (E) is given by [5]. In cohomology groups, [9] reduces to 1=2 ^ chðSþ ðEÞÞ  chðS ðEÞÞ ¼ ð1Þm AðEÞ eðEÞ

If M is noncompact and the norm of s increases rapidly away from s1 (0), then both sides of [9] are differential forms that decay rapidly away from s1 (0) and can represent cohomology classes of such. As before, we take the pullback  E with the tautological section x. Then [9] becomes chr ð Sþ ðEÞ;  S ðEÞÞ  1=2 ^ R ¼ ð1Þm  A r ðEÞ 2

½10

where r (E) is given by [7]. Both sides of [10] are forms on E that decays exponentially in the fiber directions; hence, it descends to an equality in H  (E, EnM). In the relative K-group K(E, EnM), the pair  S (E) with the isomorphism c(x) away form the zero section is, up to a factor of (1)m , the K-theoretic Thom class i! 1 2 K(E, EnM). Therefore, [10] reduces to the well-known formula

1=2 ^ i 1 chði! 1Þ ¼  AðEÞ

in cohomology groups H  (E, EnM). The refinement [10] as an equality of differential forms is due to Mathai and Quillen (1986). In fact, this is how [7] was derived originally.

Equivariant Cohomology and Equivariant Vector Bundles Equivariant Cohomology

Let G be a compact Lie group with Lie algebra g. Fixing a basis {ea } of g, the structure constants are c given by [ea , eb ] = tab ec . Let {#a } and {’a } be the dual  bases of g generating the exterior algebra ^(g ) and the symmetric algebra S(g ), respectively. The Weil ^ S(g ). We define a grading algebra is W(g) = ^ (g )  on W(g) by specifying deg #a = 1, deg ’a = 2. The contraction a and the exterior derivative d are two odd derivations on W(g) defined by a #b ¼ ab ;

a ’b ¼ 0

a b c # # þ ’a ; d#a ¼ 12tbc

a b c d’a ¼ tbc # ’

½11

The Lie derivative is La = { a , d}. These operators satisfy the usual (anti-)commutation relations d2 ¼ 0;

La ¼ f a ; dg;

f a ; b g ¼ 0; ½La ; Lb  ¼

½La ; d ¼ 0

c ½La ; b  ¼ tab c ; c tab Lc

½12

½13

The cohomology of (W(g), d) is trivial. If G acts smoothly on a manifold M on the left, let Va be the vector field generated by the Lie algebra c element ea 2 g. Then, [Va , Vb ] = tab Vc . Denote  a = Va and La = LVa , acting on  (M). In the Weil model of equivariant cohomology, one considers the ^  (M), on which the graded tensor product W(g)  operators ^ 1þ 1 ^ a ~ a ¼ a  ~ ¼ d ^ 1 þ 1 ^d d ~ a ¼ La  ^ 1 þ 1 ^ La L act and satisfy the same relations [12] and [13]. ^  (M) is ‘‘basic’’ if it An element ! 2 W(g)  satisfies a ! = 0, La ! = 0 for all indices a. Let ^  (M))bas be the set of such. G (M) = (W(g)  Elements of G (M) are equivariant differential forms on M. The operator d˜ preserves G (M)  and its cohomology groups HG (M) are the equivariant cohomology groups of M. They are

Mathai–Quillen Formalism

isomorphic to the singular cohomology groups of EG G M with real coefficients. The BRST model of Kalkman (1993) is obtained a ^ by applying an isomorphism  = e#  a of W(g)   (M). The operators become ^1  ~ a 1 ¼ a  ~  ’a  ~ 1 ¼ d ^ a þ #a  ^ La  d ~ a 1 ¼ L ~a  L The subspace of basic forms in the Weil model becomes ðG ðMÞÞ





G

¼ ðSðg Þ   ðMÞÞ

This is precisely the Cartan model of equivariant cohomology, in which the exterior differential is ~ 0 ¼ 1  d  ’a  a d If P is a principal G-bundle over a base space B, we can form an associated bundle P G M ! B. Choose a connection on P and let  = a ea 2 1 (P)  g,  = a ea 2 2 (P)  g be the connection and curvature forms, respectively. The components a , a satisfy the same relations [11]. Replacing #a , ’a by a , a , we have a homomorphism that maps ! 2 W(g)   (M) to !ˆ 2  (P  M). If ! is basic, then so is !, ˆ and the latter descends to a form  on P G M. Furthermore, the operator d~ on G (M) ! descends to d on  (P G M). Thus, we get the Chern–Weil homomorphisms G (M) !  (P G M)  and HG (M) ! H  (P G M). For example, the vector space Rr has an obvious SO(r) action. The Gaussian r-form [1] is invariant under SO(r) and can be extended to an SO(r)-equivariant closed r-form, called the ‘‘universal Thom form.’’ Let E be an orientable real vector bundle E of rank r with a Euclidean structure. E determines a principal SO(r)bundle P; the associated bundle P SO(r) Rr is E itself. By applying the Chern–Weil homomorphism to this setting, we get a closed r-form on E. This is another construction of the Thom form [7] by Mathai and Quillen (1986). Further information of equivariant cohomology can be found there, and in Berline et al. (1992) and Guillemin and Sternberg (1999). Equivariant Vector Bundles

Recall that a connection on a vector bundle E ! M determines, for any k  0, a differential operator r : k ðM; EÞ ! kþ1 ðM; EÞ The curvature R = r2 2 2 (M, End(E)) satisfies the Bianchi identity rR = 0. If the connection preserves a Euclidean structure on E, then R is skew-symmetric.

393

If a Lie group G acts on M and the action can be lifted to E, then G also acts on the spaces (E) and  (M, E). As before, the Lie derivatives La on these spaces are the infinitesimal actions of ea 2 g. We choose a G-invariant connection on E. The ‘‘moment’’ of the connection r under the G-action is a = La  rVa acting on (E). In fact, a is a section of End(E), or  2 (End(E))  g . If a Euclidean structure on E is preserved by both the connection and the G-action, then a is skewsymmetric. On  (M, E), we have La ¼ f a ; rg þ a c a R ¼ ra ; La b ¼ tab c c ½a ; b  ¼ tab c þ Rab

where Rab = R(Va , Vb ) 2 (End(E)). ^  (M, E), On the graded tensor product W(g)  ~ a act and the contraction ~ a and the Lie derivative L satisfy [13]. In the Weil model, equivariant differential forms on M with values in E are the basic ^  (M, E), which form a subspace elements in W(g)   ^  (M, E))bas . The ‘‘equivariant G (M, E) = (W(g)  covariant derivative’’ is ~ ¼ d ^ a ^ 1 þ 1 ^ r þ #a  r

½14

~ a and hence r ~ =L ~ preserves One checks that { a , r}  the basic subspace G (M, E). The equivariant curva~ =r ~ 2 is ture R ~ ¼ R  #a ra þ ’a a þ 1 #a #b Rab R 2

½15

~ = 0. ~R It satisfies the equivariant Bianchi identity r Equivariant characteristic forms are invariant poly~ They are equivariantly closed and nomials of R. their equivariant cohomology classes do not depend on the choice of the G-invariant connection. Hence, they represent the equivariant characteristic classes  of E in HG (M). For the BRST model, we use a similar isomorpha ^  (M, E). The operators ism  = e#  a on W(g)  become ^1  ~ a 1 ¼ a  ~ 1 ¼ r ~  ’a  ^ a þ #a  ^ La  r ~ a 1 ¼ L ~a  L and the basic subspace turns into ðG ðM; EÞÞ ¼ ðSðg Þ   ðM; EÞÞG This is the Cartan model, which can be found in Berline et al. (1992). The equivariant covariant derivative is ~ 0 ¼ 1  r  ’a  a r

394 Mathai–Quillen Formalism

~ 0 = (r ~ 0 )2 = R þ ’a a The equivariant curvature is R and the characteristic forms are defined similarly. Let P ! B be a principal G-bundle with a connection . Following [14], the bundle P  E ! P  M has a connection ^ ¼ d  1 þ 1  r þ a  a r  on the vector bundle It descends to a connection r ~ 7! r  can be considP G E ! P G M. The map r ered as the analog of the Chern–Weil homomorphism for connections. There is also a homomorphism G (M, E) !  (P G M, P G E), which commutes ~ r.  The curvature with the covariant derivatives r, 2  ~  R = r is the image of the equivariant curvature R. Consequently, the equivariant characteristic forms descend to those of P G E ! P G M by the usual Chern–Weil homomorphism. Now let E = Eþ  E be a graded vector bundle over M with a G-action preserving all the structures. We have the G (M)-linear supertrace map str: ^ (End(E)) ! G (M). If r is a G-invariant G (M)  connection on E preserving the grading and if L 2 (End(E) )G is odd and G-invariant, then ~ = r~ þ L is an ‘‘equivariant superconnection.’’ D The equivariant counterpart of [8] is  pffiffiffiffiffiffiffi  1 ~ 2 þ  chr; D 2 G ðMÞ ~ L ðE ; E Þ ¼ str exp 2 representing the equivariant Chern character of  Eþ E in HG (M). Representatives of the Equivariant Euler and Thom Classes

Consider an oriented real vector bundle E ! M of rank r with a Euclidean structure ( , ). Choose a connection r on E preserving ( , ). We assume that a Lie group G acts on M and that the action can be lifted to E preserving all the structures on E. We use the Weil model; the constructions in the Cartan model are similar. For any  2 kG (M, E) and  2 lG (M, E), we obtain (, ^) 2 kþl G (M) by taking the wedge product of forms as well as the pairing in E. The BerezinRintegral of ! 2 G (M, ^ E ) B along the fibers of E is ! = h, !i 2 G (M). Here,  is the unit section of the canonically trivial determinant line bundle ^r E, compatible with the orientation of E. The equivariant Euler form     Z B ˜ 1 1 ˜ Þ ¼ Pf R ð ; R er~ ðEÞ ¼ ½16 exp r=2 2 2 ð2Þ is equivariantly closed. It represents the equivariant  Euler class eG (E) 2 HG (M).

Given a G-invariant section s 2 (E)G , the equivariant counterpart of [4] is 1 1 ~ ~ S r;s ~ ¼ 2 ðs; sÞ þ ðrs; Þ þ 2 ð ; R Þ

½17

and that of Mathai–Quillen’s Euler form [5] is Z ð1Þrðrþ1Þ=2 B S r;s e ~ ½18 er;s ~ ðEÞ ¼ ð2Þr=2 It is also equivariantly closed, and its equivariant cohomology class is eG (E). The equivariant extension of Mathai–Quillen’s Thom form [7] is  Z ð1Þrðrþ1Þ=2 B 1 exp  ðx; xÞ r~ ðEÞ ¼ r=2 2 ð2Þ  ~ Þ ~ Þ  1 ð ; R ðrx; ½19 2 where x is the (G-invariant) tautological section of  E ! E. Finally, G acts on the (graded) spinor bundle S(E). Using the equivariant superconnection  1=2  S ~ ~ cðsÞ Ds ¼ r þ pffiffiffiffiffiffiffi 1 [9] generalizes to ~ R chr;s ~ ðS ðEÞ; S ðEÞÞ ¼ ð1Þ A 2 þ



!1=2

m^

er;s ~ ðEÞ

Now apply the construction to the bundle  E ! E and its tautological section x. The pair  S (E) with an odd bundle map c(x) determines, up to a factor of (1)m , the Thom class i! 1G in the equivariant K-group KG (E, EnM). The equivariant analog of [10] descends to ^ G ðEÞ1=2 i 1G chG ði! 1G Þ ¼  A in equivariant cohomology.

Superspace Formulation Mathai–Quillen Formalism and the Superspace R 0 j 1

Let R0 j 1 be the superspace with one fermionic coordinate but no bosonic coordinates. The translation on R0 j 1 is generated by D = @=@ , which satisfies {D, D} = 0. We consider a sigma model on R 0 j 1 whose target space is an (ordinary) smooth manifold M of dimension n. A pffiffiffiffiffiffimap X : R0 j 1 ! M can be written aspX( ) ffiffiffiffiffiffi = x þ 1 . Here, x = Xj = 0 2 M and =  1DXj = 0 2 Tx M; the latter is fermionic. Under the translation

Mathai–Quillen Formalism

7! þ , x and vary according to the supersymmetry transformations pffiffiffiffiffiffiffi

x ¼ DXj ¼0 ¼ 1 ½20

¼ DðDXÞj ¼0 ¼ 0 Clearly, 2 = 0, which is also a consequence of D2 = 0. For any p-form ! 2 p (M), we have an observable 1 O! ðXÞ ¼ X !ðD; . . . ; DÞj ¼0 p! In local coordinates, !¼

1 !i i ðxÞ dxi1 ^    ^ dxip p! 1 p

¼

pffiffiffiffiffiffiffir Z 1 ð2Þr=2

½d eSMQ ½x;

;

½22

where SMQ ½x; ;  ¼ 12 ðs; sÞ 

pffiffiffiffiffiffiffi 1ð; r sÞ  14 ð; Rð ; ÞÞ

½23

When r is even, [22] is equal to Oe(r, s)(E) (X), where e(r, s)(E) is given by [5]. Furthermore, for any closed form ! on M, the expectation value Z hO! ðXÞi ¼ ½dX½dO! ðXÞ eSMQ ½X; ½24

Equivariant Cohomology and Gauged Sigma Model on R0 j 1 i1



ip

Using C(  ) to denote the set of function(al)s on a space, we can identify C(Map(R0 j 1 , M)) with  (M). Under [20], O! (X) = Od! (X). So, O! (X) is invariant under supersymmetry if and only if ! is closed. The cohomology of is the de Rham cohomology of M. Consider the measure [dX] = [dx][d ]. In local coordinates, [dx] = dx1    dxn is the standard (bosonic) measure and [d ] = d 1    d n is a fermionic measure such that Z ½d ð1Þnðn1Þ=2 1    n ¼ 1 n RFor any ! 2  (M), the superfield integral R [dX]O! (X) is equal to the usual integral M ! if the latter exists. Let E ! M be a real vector bundle of rank r with an inner product ( , ), and let r be a compatible connection whose curvature is R. Consider a theory whose fields are X 2 Map(R0 j 1 , M) and a fermionic section  2 (X E). Let D = (X r)D be the covariant derivative along D in the pullback bundle X E ! R0 j 1 . Then,  = j = 0 2 Ex is fermionic and f = Dj = 0 2 Ex is bosonic. Given a fixed section s 2 (E), we write a superspace action Z pffiffiffiffiffiffiffi SMQ ½X;  ¼ d ð; 12 D þ 1s XÞ 0j1 R pffiffiffiffiffiffiffi ¼ 12 ðf ; f Þ þ 1ðf ; sÞ  ðr s; Þ

þ 14 ð; Rð ; ÞÞ

½de

SMQ ½X;

is equal to [6].

and pffiffiffiffiffiffiffip 1 O! ðx; Þ ¼ !i1 ip ðxÞ p!

Z

395

½21

It is automatically supersymmetric. Performing pffiffiffiffiffiffithe Gaussian integral over f and replacing  by  1, we get

Suppose G is a Lie group and P is a principal G-bundle over R0 j 1 . Since is nilpotent, we can choose a ‘‘trivialization’’ of P such that the connection and curvature are A 2 1 (R 0 j 1 )  g and F 2 2 (R0 j 1 ) g, respectively. (g pffiffiffiffiffiffiis the Lie algebra of G.) In components, c = 1 D A 2 g is fermionic and  = pffiffiffiffiffiffi ( 1=2) 2D F 2 g is bosonic. The space of connections A is the set of pairs (c, ). Under 7! þ , ! pffiffiffiffiffiffiffi 1 ½c; c

c ¼  þ 2 ½25 pffiffiffiffiffiffiffi

 ¼ 1 ½c;  Thus, the algebra C(A) is isomorphic to the Weil algebra W(g) and corresponds to the differential d in [11]. This relation between gauge theory on a fermionic space and the Weil algebra can be found in Blau and Thompson (1997). With a trivialization of P, the group of gauge transformation G can be identified with 0j1 Map(R of the form pffiffiffiffiffi , G). Any group element ispffiffiffiffiffiffi 1 ^g = ge  , with g = ^gj =0 2 G and  = 1 D ^g $ 2 g (fermionic), where $ is the Maurer–Cartan form on G. The action of ^g is A 7! A0 = Ad^g (A  ^g $), or c7! c0 = Adg (c  ) and  7! 0 = Adg . By choosing  = c, we obtained a new trivialization, called the ‘‘Wess–Zumino gauge,’’ in which c0 = 0. The residual gauge redundancy is G, and A=G = g=AdG . The Wess–Zumino gauge is not preserved by the translation on R 0j 1 unless we define 0 by composing with a suitable (infinitesimal) gauge transformation. If so, then 0  = 0. Suppose M is a manifold with a left G-action. As before, let {ea } be a basis of g and let the vector field Va be the infinitesimal action of ea . In the gauged sigma model, we include another field X 2 (P G M). With a trivialization of P, we can identify X with a map

396 Mathai–Quillen Formalism

X : R0 j 1 ! M. The covariant derivative is given by a rX = dX pA ffiffiffiffiffiffiVa , DX = rD X. Let x = Xj = 0 2 M and =  1DXj = 0 2 Tx M. Then the supersymmetric transformations are pffiffiffiffiffiffiffi  

xi ¼ 1 i  ca Vai pffiffiffiffiffiffiffi i

½26

i ¼  a Vai þ 1cj Va;j In the Wess–Zumino pffiffiffiffiffiffi gauge, the transformations simplify to 0 x = 1 , 0 =  a Va . The observables form the G-invariant part of the space C(A  Map(R 0 j 1 , M)). For any ! 2 p (M), we have 1 !ðDX; . . . ; DXÞj ¼0 p! pffiffiffiffiffiffiffip 1 !i1 ip ðxÞ i1    ¼ p!

O! ðX; AÞ ¼

ip

½27

O! (X, A) is gauge covariant: O! (X, A) 7! Og ! (X, A), and the set of gauge-invariant observables is thus identified with (S(g )   (M))G . Moreover, since pffiffiffiffiffiffiffi

O! ðX; AÞ ¼ ðOd! ðX; AÞ  1ca OLa ! ðX; AÞ pffiffiffiffiffiffiffi  1a O a ! ðX; AÞÞ ~ 0 in BRST model.

corresponds to the differential d Let E ! M be an equivariant vector bundle and let r be a G-invariant connection with curvature R and moment . Any s 2 (E)G defines a section of P G E ! P G M, still denoted by s. Consider a theory with superfields X 2 (P G M) and  2 (X (P G E)) (fermionic). Let D be the covariant derivative of the pullback connection. With a trivialization of P, we put  = j = 0 2 Ex (fermionic) and f = Dj = 0 2 Ex (bosonic). The equivariant extension of [21] is Z pffiffiffiffiffiffiffi SMQ ½X; ; A ¼ d ð; 12D þ 1s XÞ R 0j1

Similar to [22], we get, in the Wess–Zumino gauge, pffiffiffiffiffiffiffir Z Z 1 SMQ ½X;;A ½de ½deSMQ ½x; ;; ½28 ¼ ð2Þr=2 where SMQ ½x; ; ;  pffiffiffiffiffiffiffi 1 ¼ ðs; sÞ  1ð; r sÞ 2 pffiffiffiffiffiffiffi 1 1 ð; a a Þ ½29  ð; Rð ; ÞÞ  4 2 When r is even, [28] is equal to O~e(r, s) (X, A), where ~e(r, s) is given by [18].

The Atiyah–Jeffrey Formula

Given the G-action on M, for any x 2 M, there is a linear map Cx : g ! Tx M defined by Cx (ea ) = Va (x). With an invariant inner product ( , ) on g and an invariant Riemannian metric on M, the adjoint of Cx is Cyx : Tx M ! g, that is, Cy 2 1 (M)  g. If G acts on M freely, then Cx is injective and (Cy C)x is invertible for all x 2 M. The projection M !  = M=G is a principal G-bundle. It has a connection M such that the horizontal subspace is the orthogonal compliment of the G-orbits. The connection 1-form is  = (Cy C)1 Cy , whereas the curvature is  = (Cy C)1 dCy on horizontal vectors. Let ! be an equivariant form on M. Suppose G  acts on M freely, then ! descends to a form ! on M. We look for a gauge-invariant, supersymmetric quantity (X, A) such that Z 1 ½dX½dAO! ðX; AÞ ðX; AÞ volðGÞ Z  ! ðXÞ  ¼ ½dXO ½30 Mathematically, corresponds to a closed equivariant form  on M such that Z Z Z 1  ½d !ðÞ ^ ðÞ ¼ ! volðGÞ 2g  M M which is [30] in the Wess–Zumino gauge. In fact,  is distribution valued in the sense of Kumar and Vergne (1993) and can be understood as an equivariant homology cycle, as in Austin and Braam (1995). Let P be a G-bundle over R 0 j 1 with a connection and let Ad P = P G g ! R0 j 1 be the adjoint bundle. Consider a (bosonic) superfield pffiffiffiffiffiffi 2 (AdP). Set  =

j = 0 (bosonic) and  =  1D j = 0 (fermionic). Choosing a trivialization of P,  and  are both in g. Under 7! þ , they transform as pffiffiffiffiffiffiffi

 ¼ 1 ð þ ½c; Þ ½31 pffiffiffiffiffiffiffi

 ¼ ð½;   1½c; Þ The superspace action SCMR ½X; ; A ¼

pffiffiffiffiffiffiffi Z 1

R 0j1

d ð ; Cy DXÞ

is invariant under [25], [26], and [31] and, under the Wess–Zumino gauge, it is SCMR ½x; ; ; ;  pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ¼  1ð; Cy Þ  1ð; dCy ð ; ÞÞ þ ð; Cy CÞ

½32

Mathai–Quillen Formalism

If G acts on M freely, then Z ðX; AÞ ¼ ½d eSCMR ½X; ;A

½33

satisfies [30]. The factor (X, A) in [30] is called ‘‘projection’’ in Cordes et al. (1996). Let E ! M be a G-equivariant vector bundle with a fixed G-invariant connection r, moment , and an invariant section s. Consider the superspace action SAJ ½X; ; ; A ¼ SMQ ½X; ; A þ SCMR ½X; ; A In the Wess–Zumino gauge and after the Gaussian integral over f, it becomes the Atiyah–Jeffrey action SAJ ½x; ; ; ; ;  ¼ SMQ ½x; ; ;  þ SCMR ½x; ; ; ; 

½34

If s intersect the zero section transversely and G acts on s1 (0) freely, then s1 (0)=G is smooth and Z Z  ¼ ½dx½d ½d½d½d½d ! s1 ð0Þ=G

 O! ðx; ; ÞeSAJ ½x;;;;;

½35

for any closed equivariant form ! on M. Equation [35] is the formula of Atiyah and Jeffrey (1990) and of Witten (1988a) in an infinite-dimensional setting. When s1 (0)=G is not smooth, the right-hand side of [35] can be regarded as a definition of the left-hand side. It is often convenient to add to SAJ another term Z 1 ð½ 2 F; ; D Þ S½X; ; A ¼  4 R0j1 D pffiffiffiffiffiffiffi 1 1 ð; ½; Þ þ ð½; ; ½; Þ ½36 ¼ 2 2 Since [36] is -exact and no new field is added, the integral [35] does not change if S is added to SAJ .

Applications to Cohomological Field Theories We now apply the Mathai–Quillen construction formally to a number of cases in which both the rank of the vector bundle and the dimension of the base space are infinite. Thus, the (bosonic and fermionic) integrals in [24] or [35] become path integrals in quantum mechanics or quantum field theory. Supersymmetric Quantum Mechanics

Let (M, g) be a Riemannian manifold and LM = Map(S1 , M), the loop space. At each point u 2 LM, which is a map u : S1 ! M, the tangent space is

397

Tu LM = (u TM). In particular, u_ = du=dt, where t is a parameter on S1 , is a tangent vector at u and u 7! u_ is a vector field on LM. For any Morse function h on M, s(u) = u_ þ (grad h) u is another vector field on LM. Vector fields on LM can be identified as sections of the bundle ev TM ! S1  LM, where ev : S1  LM ! M is the evaluation map. The Levi-Civita connection r on TM pulls back to a connection on ev TM and the covariant derivatives along LM define a natural connection rLM on T(LM). For example, for any tangent vector V 2 Tu LM = (u TM), we u u have rLM V s(u) = rt V þ (rV grad h) u, where r is  the pullback connection on u TM. The Riemann curvature tensor R on M determines that on LM. The (infinite-dimensional) analog of [22] is  Z  Z ½du½d ½d exp  dt L½u; ;  ½37 S1

where

,  2 Tu LM = (u TM) are fermionic and

L½u; ;  ¼ 12 gðu_ þ grad h; u_ þ grad hÞ pffiffiffiffiffiffiffi  1gð; rut þ r grad hÞ  14 gð; Rð ; ÞÞ ½38 pffiffiffiffiffiffi Here and below, factors of 1 and 2 in [22] are absorbed in the path-integral measure. [38] is, up to a total derivative, the Lagrangian of the Euclidean N = 2 supersymmetric quantum mechanics on M. The partition function [37] is equal to Euler characteristic number of LM or M, which can be confirmed by an (exact) stationary-phase calculation. Topological Sigma Model

Let  be a Riemann surface with complex structure " and let (M, !) be a symplectic manifold with a compatible almost-complex structure J. Let E be a vector bundle over Map(, M) so that the fiber over u is E u = (u TM  T  ). For any u 2 Map(, M), du 2 E u and u 7! du is a section of E. The pullback of the Levi-Civita connection on TM, tensored with a connection on T  , defines a connection on E. The vector bundle to which we apply the Mathai– Quillen formalism is the antiholomorphic part E 01 of E.  The fiber over u 2 Map(, M) is E 01 u = ((u TM  01 01  T ) ). The sub-bundle E has a connection r01 via projection from E. E 01 has a natural section s : u 7! @ J u = (1=2)(du þ J du "). Solutions to the equation @J u = 0 are pseudoholomorphic (or J-holomorphic) curves; let M = s1 (0) be the space of such curves. Its (virtual) dimension is dim M ¼ 12 ðÞ dim M þ 2c1 ðu TMÞ

½39

398 Mathai–Quillen Formalism

Along any V 2 Tu Map(, M) = (u TM), the covariant derivative of s = @J is calculated in Wu (1995): 1 u u  r01 V ð@J Þ ¼ 2 ðr V þ J r V "Þ

þ 14 rV J ðdu " þ J duÞ

½40

where ru is the pullback connection on u TM. To write the Mathai–Quillen formalism for the bundle E 01 ! Map(, M), we let 2 (u TM) and  2 ((u TM  T  )01 ) be fermionic fields. Equation [23] becomes the Lagrangian

1 dim M ¼ 4hðgÞkðPÞ 2 dim GððMÞ þ ðMÞÞ

L½u; ;  ¼ 12 kduk2 þ 12ðdu; J du "Þ pffiffiffiffiffiffiffi  1ð; ru þ ðr JÞ du "Þ  18 ð; ðRð ; Þ 12 ðr JÞ2 ÞÞ

where h(g) is the dual Coxeter number of g and ½41

It is precisely the Lagrangian of the topological sigma model of Witten (1988b). Here, the pairing ( , ) is induced by the Riemannian metric !( , J ) on M and a metric on  that is compatible with ". TheR second term in [41], integrated over , is equal to  u ! = h[!], u []i. For any differential form  2 p (M), let O (u, ) be the observable obtained from ev  2 p (  Map(, M)) by identifying  (Map(, M)) with C(Map(R0 j 1 , Map(, M))). If  is closed and  R 2 Hq () is a homology cycle, then W, (u, ) =  O (u, ) is identified with a closed (p  q)-form on Map(, M). For closed i 2 pi (M) and i 2 Hqi ()(1 i r), the expectation values * + r Y Wi ;i i¼1

¼

Z

½du½d ½d

r Y

Wi ;i ðu; Þe

S½u; ;

G acts on E and the bundle is G-equivariant. The trivial connection on E is G-invariant; the moment is given by  2 (ad P) :  2 2þ (M, ad P) 7! [, ]. The bundle E has a natural section s : A 2 A 7! FAþ , the self-dual part of the curvature. Its derivative along V 2 1 (M, ad P) = TA A is LV s = (rA V)þ . The section s is G-invariant, the zero set s1 (0) is the space of anti-self-dual connections, and the quotient M = s1 (0)=G is the instanton moduli space. Its (virtual) dimension is

½42

i¼1

are the Gromov–Witten invariants of (M, !). MoreP over, [42] is nonzero only if ri = 1 (pi  qi ) = dim M. Topological Gauge Theory

Let M be a compact, oriented 4-manifold, G, a compact, semisimple Lie group, and P ! M, a principal G-bundle. Denote by A the space of connections on P and G, the group of gauge transformations. The Lie algebra of G is Lie(G) = (ad P) = 0 (M, ad P). At A 2 A, the tangent space is TA A = 1 (M, ad P). Both spaces have inner products if we choose an invariant inner product ( , ) on the Lie algebra g of G and a Riemannian metric g on M. The infinitesimal action of G on A is C = rA : Lie(G) ! TA A. With a Riemannian metric, any 2-form on M decomposes into self-dual and anti-self-dual parts: 2 (M) = 2þ (M)  2 (M). We consider a trivial vector bundle E ! A whose fiber is 2þ (M, ad P).

kðPÞ ¼ 

1 hp1 ðAdPÞ; ½Mi 2 Z 4hðgÞ

is the instanton number of P. We proceed with the Mathai–Quillen interpretation of Atiyah and Jeffrey (1990). Let 2 1 (M, ad P),  2 2þ (M, ad P),  2 (ad P) be fermionic fields and ,  2 (ad P), bosonic fields. The combination of [34] and [36] is given by the Lagrangian L½A; ; ; ; ;  1 ¼ kFAþ k2 þ ð; ryA rA Þ 2 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi  1ð; rA Þ  1ð; rA Þ pffiffiffiffiffiffiffi  1ð; ½ ; Þ pffiffiffiffiffiffiffi 1 1 ð; ½;  þ ½; Þ k½; k2 þ 2 2

½43

Here, ( , ) is the pairing induced by a Riemannian metric on M and an invariant inner product on g. With an additional topological term proportional to (FA , ^ FA ), [43] is the Lagrangian of topological gauge theory of Witten (1988a). There is a tautological connection on the G-bundle A  P ! A  M. It is invariant under the G-action. Identifying  (A) with C(Map(R0 j 1 , A)) and using the Cartan pffiffiffiffiffiffimodel, the G-equivariant curvature is F = FA þ 1 þ . For any homology cycle  2 Hq (M), Z 1 W ðA; ; Þ ¼ ðF ; ^F Þ ½44 4hðgÞ  corresponds to a closed G-equivariant form on A. For i 2 Hqi (M)(1 i r), the expectation values * + Z r Y 1 ½dA½d ½d½d½d½d Wi ¼ volðGÞ i¼1 

r Y i¼1

Wi ðA; ; ÞeS½A;

;;;;

½45

Mathematical Knot Theory

are, up to a factor of jZ(G)j, Donaldson invariants of Pr M. Moreover, [45] is nonzero only if i = 1 (4  qi ) = dim M. Other cohomological field theories can also be understood or constructed by the Mathai–Quillen formalism. Of such we mention only the topological field theories of abelian and nonabelian monopoles in Labastida and Marin˜o (1995), which are related to the Seiberg–Witten invariants. See also: Characteristic Classes; Donaldson–Witten Theory; Equivariant Cohomology and the Cartan Model; K-Theory; Topological Quantum Field Theory: Overview; Topological Sigma Models.

Further Reading Atiyah MF and Jeffrey LC (1990) Topological Lagrangians and cohomology. Journal of Geometry and Physics 7: 119–138. Austin DM and Braam PJ (1995) Equivariant homology. Mathematical Proceedings of the Cambridge Philosophical Society 118: 125–139. Berline N, Getzler E, and Vergne M (1992) Heat Kernels and Dirac Operators. Berlin: Springer. Blau M and Thompson G (1997) Aspects of NT  2 topological gauge theories and D-branes. Nuclear Physics B 492: 545–590.

399

Cordes S, Moore M, and Ramgoolam S (1996) Lectures on 2D Yang–Mills Theory, Equivariant Cohomology and Topological Field Theories (Les Houches, 1994), pp. 505–682. Amsterdam: North-Holland. Guillemin VW and Sternberg S (1999) Supersymmetry and Equivariant de Rham Theory. Berlin: Springer. Kalkman J (1993) BRST model for equivariant cohomology and representatives for the equivariant Thom class. Communications in Mathematical Physics 153: 447–463. Kumar S and Vergne M (1993) Equivariant cohomology with generalized coefficients. Aste´risque 215: 109–204. Labastida JMF and Marin˜o M (1995a) A topological Lagrangian for monopoles on four-manifolds. Physics Letters B 351: 146–152. Labastida JMF and Marin˜o M (1995b) Non-abelian monopoles on four-manifolds. Nuclear Physics B 448: 373–395. Mathai V and Quillen D (1986) Superconnections, Thom classes, and equivariant differential forms. Topology 25: 85–100. Witten E (1988a) Topological quantum field theory. Communications in Mathematical Physics 117: 353–386. Witten E (1988b) Topological sigma models. Communications in Mathematical Physics 118: 411–449. Wu S (1995) On the Mathai–Quillen formalism of topological sigma models. Journal of Geometry and Physics 17: 299–309. Zhang W (2001) Lectures on Chern–Weil Theory and Witten Deformations. Nankai Tracts in Mathematics, vol. 4. Singapore: World Scientific.

Mathematical Knot Theory L Boi, EHESS, Paris, France ª 2006 Elsevier Ltd. All rights reserved.

Fundamental Concepts of the Topological Theory of Knots and Links The first known discovery relating to knots as mathematical objects was made by Gauss around 1833 in a note that refers to the knotting together of closed curves. This investigation originated in his work on electromagnetic theory that led him to compute inductance in a system of two linked circular wires. In this note he had given an analytic formula for the linking number of a pair of knotted curves. This number is a combinatorial topological invariant (it is an integer number). Moreover, one can now show that this number is invariant under Reidemeister moves (discussed in a later section). The linking coefficient can be generalized for the case of p- and q-dimensional manifolds in Rpþqþ1 . The formula for the parametrized curves 1 (t) and 2 (t) with radius vectors r1 (t), r2 (t) is given by the following formula: Z Z 1 ðr1  r2 ; dr1 ; dr2 Þ3 lkð1 ; 2 Þ ¼ ½1 4 1 2 jr1  r2 j

The linking coefficient allows us to distinguish some two component links. Another approach to the link coefficient is that involving Seifert surfaces. (On this subject, see the section ‘‘Isotopies, Reidemeister moves, torus knots, and the linking number.’’) A systematic study of knots in R3 , however, was only begun in the second half of the nineteenth century by Tait and his followers. They were motivated by Kelvin’s theory of atoms modeled on knotted vortex tubes of ether. It was expected that physical and chemical properties of various atoms could be expressed in terms of properties of knots such as the knot invariants. Even though Kelvin’s theory did not work, the theory of knots grew as a subfield of combinatorial and algebraic topology. Recently, new invariants of knots have been discovered and they have led to the solution of long-standing problems in knot theory. Surprising connections between the theory of knots and statistical mechanics, quantum groups, and quantum field theory are emerging. Moreover, knot theory has been shown to be intimately connected with many problems in physics, chemistry, and biology. Tait classified the knots in terms of the crossing number of a regular projection. A regular projection of a knot on a plane is an orthogonal projection of

400 Mathematical Knot Theory

the knot such that, at any crossing in the projection, exactly two strands intersect transversely. He made a number of observations about some general properties of knots which have come to be known as the ‘‘Tait conjectures.’’ In its simplest form, the classification problem for knots can be stated as follows. Given a projection of a knot, is it possible to decide in finitely many steps if it is equivalent to an unknot. This question was answered affirmatively by W Haken in 1961. (For details, see Burde and Zieschang (1985)).

General Notions and Definitions Let M be a closed orientable 3-manifold. A smooth embedding of S1 in M is called a knot in M. A link in M is a finite collection of disjoint knots. The number of disjoint knots in a link is called the number of components of the link. Thus, a knot can be considered as a link with one component. Two links L, L0 in M are said to be equivalent if there exists a smooth orientation-preserving automorphism f : M ! M such that f (L) = L0 . For links with two or more components, we require f to preserve a fixed given ordering of the components. Such a function f is called an ambient isotopy and L and L0 are called ambient isotopic. Here, we shall take M to be S3 ffi R3 [ {1} and simply write ‘‘a link’’ instead of ‘‘a link in S3 .’’ The diagrams of links are drawn as links in R3 . A link diagram of L is a plane projection with crossings marked as over or under. The simplest combinatorial invariant of a knot K is the crossing number c(K). It is defined as the minimum number of crossings in any projection of the knot K. The classification of knots up to crossing number 17 is now known. The crossing numbers of some special families of knots are known; however, the question of finding the crossing number of an arbitrary knot is still unanswered. Another combinatorial invariant of a knot K that is easy to define is the unknotting number u(K). It is defined as the minimum number of crossing changes in any projection of the knot K which makes it into a projection of the unknot. Upper and lower bounds for u(K) are known for any knot K. An explicit formula for u(K) for a family of knots called torus knots, conjectured by Milnor nearly 40 years ago, has been proved recently by a number of different methods. The 3-manifold S3 nK is called the knot complement of K. The fundamental group 1 (S3 nK) of the knot complement is an invariant of the knot K. It is called the fundamental group of the knot and is denoted by 1 (K). Equivalent knots have homeomorphic complements and conversely. However,

this result does not extend to links. (For details and a proof, see Manturov (2004), chapter 4).

The Fundamental Group of Knots and Its Role in Topology For a better understanding of the above considerations, we need to introduce briefly the important concept of fundamental group in topology. The fundamental group plays an essential role in topology; it is involved in the entire technical apparatus of the subject, and likewise in all applications of topological methods. In fact, for low-dimensional manifolds (i.e., of dimension 2 or 3) the fundamental group underlies all nontrivial topological facts. Classical knot theory is concerned with the space S3 nK = M, an open 3-manifold. There is a natural embedding of the torus T 2 in M, namely as the boundary of small tubular neighborhood of the knot K. Similarly, for a link we obtain a disjoint union of 2-tori in M. The principal topological invariant of a knot K is the fundamental group 1 (M) of the complement M of K, with distinguished subgroup the natural image of 1 (T 2 ), T 2 M2 , with the obvious standard basis. The classical theorem of Papakyriakopoulos of the 1950s asserts that a knot is equivalent to the trivial one if and only if 1 (M) is abelian. It was known by Haken in the early 1960s that there is an algorithm for deciding whether or not any knot is equivalent to the trivial knot. However, while it appears to have been established (by Waldhausen and others in the 1960s and 1970s) that two knots are topologically equivalent if and only if the corresponding fundamental groups with labeled abelian subgroups are isomorphic, the existence of an appropriate algorithm for deciding such equivalence remains an open question. The complexity of the knot group 1 (M) has led to the search for more effectively computable invariants to distinguish knots and links. (On this subject, see the section ‘‘Polynomial invariants of knots and links.’’) Starting with the oriented diagram of the knot or link K on the plane, one calculates in the standard manner (see Crowell and Fox (1963) and Neuwirth (1965)) a presentation of the group 1 (M) of the knot (M = S3 nK), obtaining one generator for the edge of the diagram of a trefoil knot and a pair of relations for each crossing. Since one relation of each such pair simply equates the pair of generators corresponding to the edges forming the upper branch of the crossing, the presentation reduces immediately to the standard one involving the same number of generators and relations. The 2-complex

Mathematical Knot Theory

L with exactly one 0-cell, and with 1-cells labeled by generators and 2-cells labeled by the relations, is then a deformation retract of M. Lifting to the universal cover we obtain a boundary operator on a complex of free Z[1 ]-modules, which takes the form of a square matrix with entries from this group ring, and it is this matrix that is related to some differentiation as follows. Denoting the generators by ai and relators by rj , one defines the operator @ai by @ai ðaj Þ ¼ ij @ai ðbcÞ ¼ @ai ðbÞ þ b@ai ðcÞ the matrix in question then has entries qij given by qij ¼ @ai ðrj Þ Mapping each generator ai to t, we obtain a complex of modules over the ring of integer Laurent polynomials, with boundary operator the corresponding square matrix now with Laurent polynomials as entries. The determinant of this matrix turns out to be zero, and the highest common factor of its cofactors, after multiplication by a suitable power of t, turns out to be just the Alexander polynomials A(t). Let us say a bit more using a little different notation on this question. Let Aq (K) and Jq (K) be the Alexander polynomial and the Jones polynomial, respectively. One of the earliest problems in knot theory was: to what extent does the topological type X of the complementary space X = S3 nK and/or the isomorphism class G of its fundamental group G(K) = 1 (X, x0 ) suffice to classify knots? The trefoil knot is the simplest example of nontrivial knot, so it seems remarkable that, not long after the discovery of the fundamental group of a topological space, Max Dehn (1914) succeeded in proving that the trefoil knot and its mirror image had isomorphic groups, but their knot types were distinct. Dehn’s (ingenious) proof was the beginning of a long story, with many contributions which reduced repeatedly the number of distinct knot types that could have homeomorphic complements and/or isomorphic groups, until it was finally proved, quite recently, that (1) X determines K and (2) if K is prime, then G determines K up to unoriented equivalence. Thus, there are at most four distinct oriented prime knot types which have the same knot group. The knot group G is finitely presented; however, it is infinite, torsion-free, and (if K is not the unknot) nonabelian. Its isomorphism class is in general not easily understood via a direct attack on the problem. In such circumstances, the obvious thing to do is to pass to the abelianized group, but unfortunately G=[G, G] ffi H1 (X; Z) is infinite cyclic for all knots,

401

so it is of no use in distinguishing knots. Passing to the covering space X that belongs to [G, G], we note that there is a natural action of the cyclic group G=[G, G] on  X via covering translations. The action makes the homology group H1 ( X; Z) into a Z[q, q1 ]-module, where q is the generator of G=[G, G]. This module turns out to be finitely generated. It is the famous Alexander module. While the ring Z[q, q1 ] is not a principal ideal domain (PID), relevant aspects of the theory of modules over a PID apply to H1 ( X; Z). In particular, it splits as a direct sum of cyclic module, the first nontrivial one being Z[q, q1 ]=Aq (K). Thus, Aq (K) is the generator of the ‘‘order ideal,’’ and the smallest nontrivial torsion coefficient in the module H1 ( X). In particular, Aq (K) is very clearly an invariant of the knot group. We remark that when a knot is replaced by its mirror image (i.e., the orientation on S3 is reversed), the Alexander and Jones polynomials Aq (K) and Jq (K) go over to Aq1 (K) and Jq1 (K), respectively. As noted earlier, Aq (K) is invariant under such a change, but from the simplest example, the trefoil knot, we see that Jq (K) is not. Now recall that G does not change under changes in the orientation of S3 . This simple argument shows that Jq (K) cannot be a group invariant! Thus, it seems interesting indeed to ask about the underlying topology behind the Jones polynomial.

Isotopies, Reidemeister Moves, Torus Knots, and the Linking Number Because each knot is a smooth embedding of S1 in R3 , it can be arbitrarily closely approximated by an embedding of a closed broken line in R3 . Here we mean a good approximation such that after a very small smoothing (in the neighborhood of all vertices) we obtain a knot from the same isotopy class. However, generally this might not be the case. Definition 1 An embedding of a disjoint union of n closed broken lines in R3 is called a polygonal n-component link. A polygonal knot is a polygonal one-component link. Definition 2 A link is called tame if it is isotopic to a polygonal link and wild otherwise. All C1 -smooth knots are tame. In the sequel, all knots are taken to be smooth, hence, tame. Definition 3 Two polygonal links are isotopic if one of them can be transformed to the other by means of an iterated sequence of elementary isotopies and reverse transformations. The

402 Mathematical Knot Theory

elementary isotopy, generally, is assumed to be a replacement of an edge with two edges provided that the triangle has no intersection points with other edges of the link. It can be proved that the isotopy of smooth links corresponds to that of polygonal links; the proof is technically complicated. Like smooth links, polygonal links admit planar diagrams with overcrossings and undercrossings, having such a diagram one can restore the link up to isotopy. Definition 4 By a planar isotopy of a smooth-link planar diagram we mean a diffeomorphism of the plane onto itself not changing the combinatorial structure of the diagram. Obviously, planar isotopy is an isotopy, that is, it does not change the link isotopy type in R3 . Theorem 1 (Reidemeister) Two diagrams D1 and D2 of smooth links generate isotopic links if and only if D1 can be transformed into D2 by using a finite sequence of planar isotopy and the three Reidemeister moves W1 , W2 , W3 . Theorem 2 Suppose that D and D0 are regular diagrams of two knots (or links) K and K0 , respectively. Then K  K0 , D  D0 . We may conclude from the above theorems that the problem of equivalence of knots, in essence, is just a problem of the equivalence of regular diagrams. Therefore, a knot (or link) invariant may be thought of as a quantity that remains unchanged when we apply any one of the Reidemeister moves to a regular diagram. Knots and links embedded in R3 can be considered as curves (families of curves) in 2-surfaces, where the latter surfaces are standardly embedded in R3 . In this section we shall briefly show that all knots and links can be obtained in this manner. Consider a handle surface Sg standardly embedded in R3 and a curve (knot) K in it. We can now ask the following question: which knot isotopy classes can appear for a fixed g? First, let us note that for g = 0 there exists only one knot embeddable in S2 , namely the unknot. The case g = 1 (torus, torus knots) gives us some interesting information. Consider the torus as a Cartesian product S1  S1 with coordinates , ’ 2 [0, 2], where 2 is identified with 0. In two dimensions, the torus can be illustrated as a square with opposite sides identified. Let us embed this torus standardly in R3 ; more precisely, ð; ’Þ ! ððR þ r cos ’Þ cos ; ðR þ r cos ’Þ sin ; r sin ’Þ

½2

Here R is the outer radius of the torus, r the small radius (r < R),  the longitude, and ’ the meridian. For the classification of torus knots we shall need the classification of isotopy classes of nonintersecting curves in T 2 : obviously, two curves isotopic in T 2 are isotopic in R3 . Without loss of generality, we can assume the considered closed curve to pass through the point (0, 0) = (2, 2). It can intersect the edges of the square several times. In addition, assume all these intersections to be transverse. Let us calculate separately the algebraic number of intersections with horizontal edges and those with vertical edges. Here, passing through the right edge or through the upper edge is said to be positive; that through the left or the lower edge is negative. Thus, for each curve of such type we obtain a pair of integer numbers. So, each torus knot passes p times the longitude of the torus, and q times its meridian, where GCD(p, q) = 1. It is easy to see that for any coprime p and q such a curve exists: one can just take the geodesic line {q  p’ = 0 (mod 2)}. Let us denote the torus knot by T(p, q). So, in order to classify torus knots, one should consider pairs of coprime numbers p, q and see which of them can be isotopic in the ambient space R3 . The simplest case is when either p or q equals 1. The next simplest example of a pair of coprime numbers is p = 3, q = 2 (or p = 2, q = 3). In each of these cases we obtain the trefoil knot. Let us state the following important result. Theorem 3 For any coprime integers p and q, the tori (p, q) and (q, p) are isotopic. Proof For a proof of this theorem, see Rolfsen (1990). Note that the (p, q) torus knot in one full torus is just the (q, p) torus knot in the other one. Thus, mapping one full torus to the other one, we obtain an isotopy of (p, q) and (q, p) torus knots. This homotopy of full tori can be expressed as a continuous process in S3 . Indeed, torus knots of type (p, q) can be represented by a series of planar diagrams. Moreover, it is possible to demonstrate a way of coding a knot (link) as a (p-strand) braid closure. Analogously to the case of torus knots, one can define torus links which are links embedded into the torus standardly embedded in R3 . We know the construction of torus knots. So, in order to draw a torus link, one should take a torus knot K T (one can assume that it is represented by a straight linear curve defined by the equation q  p’ = 0 (mod 2) and add to the torus T some closed nonintersecting simple curves; each curve should be nonintersecting and should not intersect K. Thus, these curves should be embedded in T nK, that is, in the open

Mathematical Knot Theory

cylinder. Each curve on the cylinder is either contractible or passes the longitude of the cylinder once. So, each curve in T nK is either contractible inside T nK, or ‘‘parallel’’ to K inside T, that is, isotopic to the curve given by the equation q  p’ = " (mod 2) inside T nK. Thus, the following theorem holds. Theorem 4 Each torus knot is isotopic to the disconnected sum of a trivial link and a link that is represented by a set of parallel torus knots of the same type (p, q). As we already know, a link invariant is a function defined on links that is invariant under isotopies. We shall represent links by using their planar diagrams. According to the Reidemeister theorem, in order to prove the invariance of some function on links, it is sufficient to check this invariance under the three Reidemeister moves. First, let us consider the simplest integer-valued invariant of two-component links. Let L be a link consisting of two oriented components A and B and let L0 be the planar diagram of L. Consider those crossings of the diagram L0 where the component A goes over the component B. There are two possible types of such crossings with respect to the orientation. For each positive crossing we assign the number (þ1), for each negative crossing we assign the number (1). Let us summarize these numbers along all crossings where the component A goes over the component B. Thus, we obtain some integer number and, in fact, this number is invariant under Reidemeister moves. The so-obtained link invariant is called linking coefficient.

Polynomial Invariants of Knots and Links By changing a link diagram at one crossing we can obtain three diagrams corresponding to links Lþ , L , and L0 which are identical except for this crossing. In the 1920s, Alexander gave an algorithm for computing a polynomial invariant K (t) (a Laurent polynomial in t) of a knot K, called the Alexander polynomial, by using its projection on a plane. He also gave its topological interpretation as an annihilator of a certain cohomology module associated to the knot K. In the 1960s, Conway defined his polynomial invariant and gave its relation to the Alexander polynomial. This polynomial is called the Alexander–Conway polynomial. The Alexander–Conway polynomial of an oriented link L is denoted by rL (z) or simply by r(z) when L is fixed. We denote the corresponding polynomials of Lþ , L , and L0 by rþ , r , and r0 , respectively.

403

The Alexander–Conway polynomial is uniquely determined by the following axioms. Axiom 1 Let L and L0 be two oriented links which are ambient isotopic. Then rL0 ðzÞ ¼ rL ðzÞ

½3

Axiom 2 Let S0 be the standard unknotted circle embedded in S3 . It is usually referred to as the unknot and is denoted by O. Then rO ðzÞ ¼ 1

½4

Axiom 3 The polynomial satisfies the following skein relation: rþ ðzÞ  r ðzÞ ¼ zr0 ðzÞ

½5

We note that the original Alexander polynomial L is related to the Alexander–Conway polynomial of an oriented link L by the relation L ðtÞ ¼ rL ðt1=2  t1=2 Þ

½6

In the 1980s, Jones discovered his polynomial invariant VL (t), called the Jones polynomial, while studying von Neumann algebras and gave its interpretation in terms of statistical mechanics. A state model for the Jones polynomial was then given by Kauffman (1987) using his bracket polynomial. These new polynomial invariants have led to the proofs of most of the Tait conjectures. The Jones polynomial VK (t) of K is a Laurent polynomial in t, which is uniquely determined by a simple set of properties similar to the axioms for the Alexander–Conway polynomials. More generally, the Jones polynomial can be defined for any oriented link L as a Laurent polynomial in t1=2 , so that reversing the orientation of all components of L leaves VL unchanged. In particular, VK does not depend on the orientation of the knot K. For a fixed link, we denote the Jones polynomial simply by V. Recall that there are three standard ways to change a link diagram at a crossing point. The Jones polynomial is characterized by the following properties: 1. Let L and L0 be two oriented links which are ambient isotopic. Then VL0 ðtÞ ¼ VL ðtÞ

½7

2. Let O denote the unknot. Then VO ðtÞ ¼ 1

½8

3. The polynomial satisfies the following skein relation: t1 Vþ  tV ¼ ðt1=2  t1=2 ÞV0

½9

404 Mathematical Knot Theory

An important property of the Jones polynomial that is not shared by the Alexander–Conway polynomial is its ability to distinguish between a knot and its mirror image. More precisely, we have the following result. Let Km be the mirror image of the knot K. Then VKm ðtÞ ¼ VK ðt  1Þ

½10

Since the Jones polynomial is not symmetric in t and t1 , it follows that in general VKm ðtÞ 6¼ VK ðtÞ

½11

We note that a knot is called amphicheiral (achiral in biochemistry) if it is equivalent to its mirror image. We shall use the simpler biochemistry term. So, a knot that is not equivalent to its mirror image is called chiral. The condition expressed by [11] is sufficient but not necessary for chirality of a knot. The Jones polynomial did not resolve the following conjecture by Tait concerning chirality: if the crossing number of a knot is odd, then it is chiral. However, it has been demonstrated recently that a 15-crossing knot provides a counterexample to the chirality conjecture.

New Invariants and Their Applications in Mathematical Physics There was an interval of nearly 60 years between the discovery of the Alexander polynomial and the Jones polynomial. Since then a number of polynomials and other invariants of knots and links have been found. A particularly interesting one is the twovariable polynomial generalizing V, called the HOMFLY polynomial (name formed from the initials of authors of the article (Freyd et al. 1985) and denoted by P. The HOMFLY polynomial P(, z) satisfies the following skein relation: 1 Pþ  P ¼ zP0

½12

Both the Jones polynomial V and the Alexander– Conway polynomial rL are special cases of the HOMFLY polynomial. The precise relations are given by the following theorem. Theorem 5 Let L be an oriented link. Then the polynomials PL , VL , and rL satisfy the following relations: VL ðtÞ ¼ PL ðt; t1=2  t1=2 Þ and rL ðzÞ ¼ PL ð1; zÞ

½13

After defining his polynomial invariant, Jones also established the relation of some knot invariants with statistical mechanical models. Since then this has become a very active area of research. By

constructing a typical statistical mechanics model – the star–triangle relations of the Yang–Baxter equations are an example of such model – one obtains a state model for the Alexander or the Jones polynomial of a knot, by associating to the knot a statistical system, whose partition function X EK ðsÞ!ðsÞ ½14 ZK :¼ gives the corresponding polynomial. (For details, see Jones (1989)). In the function above, ! = F(X, S) ! R is a weight function and the sum is taken over all states s 2 F(X, S). The energy Ek of the system (X, S) is a functional, Ek : FðX; SÞ ! R; k 2 K

½15

where the subscript k 2 K indicates the dependence of energy on the set K of auxiliary parameters, such as temperature, pressure, etc. However, these statistical models did not provide a geometrical or topological interpretation of the polynomial invariant. Such an interpretation was provided by Witten (1989) by applying ideas from quantum field theory to the Chern–Simons Lagrangian. In fact, Witten’s model allows us to consider the knot and link invariants in any compact 3-manifold M.

Vassiliev Invariants and the Space of All Knots: New Generalizations of Knot Theory An entirely new collection of knot invariants, which arose out of techniques pioneered by Arnold in singularity theory, has been introduced by V A Vassiliev in the 1990s. The knot invariants, like the Alexander polynomial, associate a knot with some sort of mathematical quantity. A Vassiliev invariant, on the other hand, is an invariant that satisfies a set of conditions. In this sense, all the invariants introduced above – the Jones polynomial, the HOMFLY and the Kauffman polynomial, the Conway polynomial, and the Alexander polynomial – can all be shown to be Vassiliev invariants. However, not all the knot invariants are Vassiliev invariants, for instance, the signature of a knot is not a Vassiliev invariant. The new Vassiliev invariants have a solid basis in a very interesting new topology, where one studies not a single knot, but a space of all knots. Vassiliev’s knot invariants are rational numbers. They lie in vector space Vi of dimension di , i = 1, 2, 3, . . . , with invariants in Vi having ‘‘order’’ i. These invariants are built from different families of crossing changes.

Mathematical Knot Theory

Considering that Vassiliev’s invariants require introducing an important conceptual change, shifting our attention from the knot K, which is the image of S1 under an embedding  : S1 ! S3 , to the embedding  itself. A knot type K thus becomes an equivalence class {} of embeddings of S1 into S3 . The space of all such equivalence classes of embeddings is disconnected, with a component for each smooth knot type. In this way, one passes from embeddings to smooth maps, thereby admitting maps which have various types of singularities. Let  M be the space of all smooth maps from S1 to S3 . This space is connected and contains all knot types. Our space will remain connected and will contain all knot types if we place two mild restrictions on our maps. Let M denote the collection of all  2  M such that (S1 ) passes through a fixed point  and is tangent to a fixed direction at . The space M has some interesting properties, the main one being that it can be approximated by certain affine spaces, and these affine spaces contain representatives of all knot types. The walls between distinct chambers in M constitute the discriminant , that is,  = { 2 M j } has a multiple point or a place where its derivative vanishes or other singularities. The space M   is our space of all knots. The additive properties of the Alexander and Jones polynomials have a very attractive interpretation in terms of Vassiliev invariants. By a result of Bar-Natan, all coefficients of the Alexander polynomial are Vassiliev invariants (see Bar-Natan (1995)). The same can be said of the Jones polynomial, as proved by a theorem of Birman and Lin (1993). There is an attractive formula due to Kontsevich expressing all Vassiliev invariants analytically in terms of multiple integrals, assuming that the knot or link diagram comes with some generic Morse function (e.g., the projection of the planar diagram on the y-axis). Moreover, from the work of Kontsevich it follows that it is possible to give a purely combinatorial characterization of all Vassiliev invariants (other than the one mentioned above) by associating to an oriented knot K in R3 (given via coordinates z = z(t)(= x(t) þ iy(t)), t) a chord diagram, which is just a circle with 2k distinct points labeled Pj , Qj , j = 1, 2, . . . , k, marked on it, and by imposing certain relations on the free abelian group freely generated by all chord diagrams. Theorem 6 Let VK (t) be the Jones polynomial of a knot K. Let VK (q) be the infinite series obtained q 2 from P1 VKn(t) by substituting e (= 1 þ q þ q =2! þ = q =n!) for t. So we may write n=0 VK ðqÞ ¼ b0 þ b1q þ b2 q2 þ

405

Then Jm (K) = bm is a Vassiliev invariant induced by the Jones polynomial of order (at most) m. The structure and significance of the HOMFLY and Kauffman polynomials can be interpreted in the language of Vassiliev invariants, which are invariants of finite type. The notion of finite type is of extraordinary significance in studying these invariants. One reason for this is the following basic lemma: Lemma 7 If a graph G (an embedded 4-valent graph) has exactly k nodes, then the value of a Vassiliev invariant vk of type k on G, vk (G), is independent of the embedding of G. Let us show briefly this important result. Suppose V is any invariant of oriented links taking values in some abelian group. This V can be extended to be an invariant of singular links in the following way (Kauffman 2001): a singular link is an immersion of simple closed curves in S3 with finitely many transverse double-points. These self-intersections are required to remain transverse in any isotopy demonstrating the equivalence of such singular links. If the definition of V has been extended over singular links with n  1 double points, define it on a singular link L with n singularities by VðL¥ Þ ¼ VðLþ Þ  VðL Þ where V(L ), V(Lþ ), and V(L ) are identical except near a point where they form a node. Note that V(Lþ ) and V(L ) each has n  1 double points. Then V is called a Vassiliev invariant of order n, or an invariant of finite type n, if V(L) = 0 for every L with n þ 1 or more singularities. Recall the Alexander–Conway polynomial invariant, rL (z) 2 Z[z], of oriented links defined by runknot (z) = 1 and rLþ ðzÞ  rL ðzÞ ¼ zrL0 ðzÞ Extend this over singular links by the above method. Then if L is a link with r singularities, rL (z) = zrL0 (z), where L0 is a link with r  1 singularities. Thus, by induction on r, if L has r singularities then rL (z) has a factor of z0 . This implies at once that the coefficient of zn in the Conway polynomial of a link is a Vassiliev invariant of order n. Now suppose one considers the HOMFLY polynomial and makes the substitution (l, m) = (itN=2 , i(t1=2  t1=2 )). The characterizing skein relation becomes tN=2 PðLþ Þ  tN=2 PðL Þ ¼ ðt1=2  t1=2 ÞPðL0 Þ Note that this becomes the Jones polynomial when N = 2. Now make the further substitution t = exp x. Here exp x should be thought of as the classical power series expansion. Of course, exp (x=2) and

406 Mathematical Knot Theory

exp(x=2) have power series expansions; the power series can be multiplied and added to give another power series. Thus, P(L) has a power series expansion in powers of x. It follows immediately that P(Lþ )P(L ) = xS(x) for some power series S(x). Hence, the proof used for the Conway polynomial shows at once that the coefficient of xn in the power series expansion of P(L) is a Vassiliev invariant of order n. All present studies of Vassiliev invariants clearly indicate a major role of these invariants in the future developments of knot theory and topological quantum field theories. Many questions in knot theory remain open, nevertheless, in future it will, very likely be one of the most fruitful and beautiful subjects of research in mathematics and in mathematical physics. Knot theory also attracts attention from the fact that it is revealing new astounding and profound links between geometry, algebra, and topology. See also: Finite-Type Invariants; The Jones Polynomial; Knot Invariants and Quantum Gravity; Knot Theory and Physics; Kontsevich Integral; String Topology: Homotopy and Geometric Perspectives; Topological Knot Theory and Macroscopic Physics; Topological Quantum Field Theory: Overview.

Further Reading Alexander JW (1923) Topological invariants of knots and links. Transactions of the American Mathematical Society 20: 257–306. Atiyah M (1990) The Geometry and Physics of Knots. Cambridge: Cambridge University Press.

Birman JS and Lin X-S (1993) Knot polynomials and Vassiliev’s invariants. Inventiones Mathematicae 111: 225–270. Burde G and Zieschang H (1985) Knots. Studies in Mathematics, vol. 5. Berlin: Walter de Gruyter. Crowell RH and Fox RH (1963) Introduction to Knot Theory. Toronto: Ginn & Company. De La Harpe P, Kervaire M, and Weber Cl (1986) On the Jones Polynomial. L’Enseignement Mathe´matique 32: 271–335. Dehn M (1914) Die beiden Kleeblattschlingen. Mathematische Annalen 75: 402–413. Frayd R et al. (1985) A new polynomial invariant of knots and links. In: Freyd P, Yetter D, Hoste J, Lickorish WBR, Millett K, and Ocneau A (eds.) Bulletin of the American Mathematical Society (NS) 12: 239–246. Jones VFR (1985) A polynomial invariant for knots via von Neuman algebras. Bulletin of the American Mathematical Society 12: 103–111. Kauffman LH (2001) Knots and Physics. Singapore: World Scientific. Kawauchi A (1996) A survey of Knot Theory. Boston: Birkha¨user. Lickorish WBR and Millet K (1987) A polynomial invariant for knots and links. Topology 26: 107–141. Manturov V (2004) Knot Theory. Boca Raton, FL: Chapman and Hall/CRC. Murasugi K (1996) Knot Theory and Its Applications. Boston: Birkha¨user. Neuwirth LP (1965) Knot Groups. Ann. Math. Studies, vol. 56. Princeton: Princeton University Press. Reidemeister K (1932) Knotentheorie. Berlin: Springer. Rolfsen D (1990) Knots and Links. Math. Lecture Series. Berkeley: Publish or Perish. Vassiliev VA (1990) Cohomology of knot spaces. In: Arnold VI (ed.) Theory of Singularities and Its Applications, Advances in Soviet Mathematics, vol. 1, pp. 23–70. Providence, RI: American Mathematical Society. Witten E (1989) Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121: 351–399.

Matrix Product States see Finitely Correlated States

Mean Curvature Flow see Geometric Flows and the Penrose Inequality

Mean Field Spin Glasses and Neural Networks

407

Mean Field Spin Glasses and Neural Networks A Bovier, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany ª 2006 Elsevier Ltd. All rights reserved.

Introduction and Models Rarely has a paper with a simple title as ‘‘A solvable model of a spin glass’’ had such a tremendous impact on both physics and mathematics as the seminal paper of 1972 by Sherrington and Kirkpatrick, which introduced what is now known as the Sherrington–Kirkpatrick (SK) mean-field spin glass model. As solvable as it might have appeared to the authors, it was soon found that the heuristic solution, based on the so-called replica method, was physically unacceptable. The reason was a tacit assumption, now known as replica symmetry, that proved unfounded. Several years later, Giorgio Parisi provided an ingenious way out through his continuous replica symmetry-breaking scheme, that presented a solution that, through its complexity and intrinsic beauty, both stunned and fascinated the community. Unraveling the mysteries involved in this solution has presented a challenge and driving force for the last three decades of mathematical statistical mechanics, while the use of the method in theoretical physics opened the path to solving a wide variety of problems not only in the theory of disordered magnets, but also in neural networks and combinatorial optimization. In this article the focus is on the mathematical results obtained in the study of this and a number of related models.

Mean-Field Models Mean-field models have played an important role in statistical mechanics by providing simple, solvable models in which some of the complex phenomena, such as phase transitions, could be studied and understood. For example, the Curie–Weiss model of a ferromagnet describes N spin variables i (taking values 1) in interaction. The simplifying assumption compared to more realistic models, such as the Ising model, is to ignore the spatial structure of the model and allow all spins to interact with each other with equal strength. This yields to a Hamiltonian function of the form HN ðÞ ¼ 

N N X J X i j þ h i N i;j¼1 i¼1

½1

where J is a coupling constant and h a magnetic field. This from of the interaction implies that the

Hamiltonian is in fact just a function of the P empirical magnetization mN () = N 1 i = 1 i , and this allows one to use methods from the theory of large deviations to analyze rather easily the corresponding Gibbs measures ;N ðÞ 

eHN ðÞ Z;N

½2

The SK Model This model was a straightforward attempt to introduce a mean-field version of models with randomly interacting spins. The interest in such models arose from the discovery of certain alloys of ferromagnets and conductors (e.g., AuFe and CuMn) that had been found to exhibit very unusual magnetic properties. Ruderman and co-workers had proposed that in these models the magnetic ions with magnetic moments Si and Sj located at the points xi and xj would interact via an exchange interaction of the form cosðkf ðxi  xj ÞÞ jxi  xj j3

Si  Sj

Since the positions of the magnetic ions in the alloy are random, the signs of their interaction would be oscillatory. Anderson proposed a simplified model, in the spirit of the Ising model, where spins taking values 1 located on a regular lattice would interact via nearest-neighbor couplings Jij modeled as i.i.d. random variables uniformly distributed on an interval [ J, J]. In the spirit of the Curie–Weiss model, Sherrington and Kirkpatrick then proposed the mean-field model where any two spins would interact via i.i.d. Gaussian random variables Jij of mean zero and variance one. The SK Hamiltonian is thus given by N X X J SK H N ðÞ   pffiffiffiffiffi Jij i j þ h i N 1i 0 and  > 0, we have our usual optical materials. Only one of " or  lesser than zero with the other positive would imply a medium which cannot support any propagating modes. This is a consequence of Maxwell’s equations: k  k ¼ "ð!Þð!Þ

!2 c20

½19

which implies that only evanescently decaying waves with an imaginary component of k are possible. Common examples are ordinary metals with " < 0 and  > 0. Now consider a medium with both " < 0 and  < 0, or a negative refractive index medium. The Maxwell’s equations for a plane time-harmonic wave exp[i (k  r  !t)] are: ! k  E ¼ ð!ÞH ½20 c ! k  H ¼  "ð!ÞE ½21 c The ‘‘left-handedness’’ of the triad (E, H, k) is clear from these equations for "(!), (!) < 0. A real refractive index means that waves propagate with the direction of energy flow given by the Poynting vector, S¼EH

½22

opposite to the direction of the wave vector. Since the group velocity is in the direction of the energy flow, we conclude that in these left-handed materials (LHMs) the group velocity and the phase velocity are oppositely directed. The phase accumulated in pffiffiffiffiffiffi propagating a distance x is  =  "!=c0 x. Thus, pffiffiffiffiffiffi the refractive index can be taken to be n =  ", that is, a negative quantity. Mathematically, it is more reasonable to ask for the sign of the squareroot to determine the wave vector given by eqn [19]. It can be shown by arguments of analytic continuity in the complex plane that the negative sign has to be

kr

kt

θr

θt

k||

θr θt k||

ki

ki

487

kt VAC

(a)

RHM

VAC LHM

(b)

Figure 4 Illustration of Snell’s law at an interface between two media with (a) positive refractive index (VAC/RHM) and (b) negative refractive index (VAC/LHM). The arrows indicate the wave vectors and the energy flow is opposite to the wave vector in the negative index medium.

chosen for propagating waves when Re(") < 0 and Re() < 0 (Ramakrishna 2005). The negative refractive index has real effects on the behavior of radiation even in basic processes such as refraction. Consider an interface between vacuum and a negative refractive index medium with n < 0 shown in Figure 4. Continuity conditions on the electromagnetic fields at the interface require for a plane wave incident from the vacuum side at an oblique angle that the parallel wave vector kk is conserved for the transmitted and reflected wave. This is the origin of Snell’s law: sinð i Þ ¼ sinð r Þ ¼ n sinð t Þ

½23

where i , r and t are the angles of incidence, reflection, and transmission, respectively. The flow of energy across the interface determines the direction of the group velocity in the material medium as being away from the interface. Therefore, the component of the phase velocity vector normal to the interface must change sign as we pass from vacuum into the material medium. We are then forced to conclude that the ray is bent toward the same side of the surface normal as the incident wave. This picture is consistent with Snell’s law with the interpretation that n < 0 ) t < 0. Figure 4 illustrates this point which has been experimentally verified by several groups (Shelby et al. 2001, Parazzoli et al. 2003, Eleftheriades et al. 2002). As a direct consequence of this, it is seen that a flat slab of negative refractive medium can act as a lens as shown in Figure 5. Provided that the slab is of sufficient thickness, the refracted rays from a point source come to a focus inside the slab and upon exiting the slab the rays are redirected again such that they come to a focus on the opposite side of the slab (Veselago 1968). Veselago also predicted a negative Doppler shift in such media and an obtuse angle cone for Cerenkov radiation.

488 Negative Refraction and Subdiffraction Imaging

components in the image plane as they decay exponentially in amplitude as one moves away from the source. Hence the resolution, , provided by a conventional lens is limited to those components with

Object plane

Image plane

k2x þ k2y < !2 =c2 )   n = –1 d/ 2

d

d/ 2

Figure 5 Steady-state passage of rays (representing the energy flow) of light from vacuum through a slab made of a LHM with n = 1. The slab acts as a lens mapping a point on the image plane to a point on the object plane.

Perfect Lens: Subwavelength Imaging

;kx ;ky

   exp i kx x þ ky y þ kz z  !t

½24

where E ðkx ; ky Þ ¼

Z

E ðx; y; 0Þ eiðkx xþky yÞ dx dy

½25

x;y

In the above expression, the source is assumed to be monochromatic of frequency !, k2x þ k2y þ k2z = !2 =c20 , c0 is the speed of light in free space, and z is the optical axis. A conventional lens acts by applying a phase correction to each of the propagating components so that they reassemble to a focus at a point beyond the lens. For these components kz is real, thus a phase change is all that is required to form an image containing these components. The higher spatial details in an object, however, are described by the nonpropagating near-field components with an imaginary kz where k2x þ k2y > !2 =c2 . A conventional lens cannot restore these

½26

Now consider the slab of medium with " = 1 and  = 1 and of thickness ds . It can be shown (Pendry 2000) that the transmission and reflection coefficients are lim ~t ¼ exp½ikz ds 

½27

lim ~r ¼ 0

½28

"!1 !1

"!1 !1

A wave analysis of the Veselago lens revealed an extremely novel aspect: it did not suffer from the diffraction limit and the image resolution could be infinite (Pendry 2000), if the negative index material were perfectly nondispersive and nonabsorbing. Before we analyze this, let us first briefly review the problem of imaging and the diffraction limit. Any object is visible because it emits or scatters light. The problem of imaging is then concerned with reproducing the electromagnetic field distribution on a 2D object plane in the 2D image plane. If E(x, y, 0) be the electric field on the object (z = 0) plane, the fields in free space can be decomposed into the Fourier components kx and ky , and polarization defined by : X   Eðx; y; z; tÞ ¼ E kx ; ky

2c ¼ !

respectively, where kz is the component of the wave vector normal to the interface. Thus, the slab reverses the phase advance for the propagating waves as revealed by the ray picture. Analytic continuation to imaginary wave vectors kz = i z implies that the transmittance ~t ! exp(þ z d), that is, the slab also increases the amplitude of the evanescent waves in transmission at exactly the same rate as the rate of the decay in free space outside. Thus, each wave, propagating or evanescent, arrives at the image plane with its phase or amplitude restored exactly to the values at the object plane so as to perfectly reconstruct the image. The lens is also perfectly impedance matched and has zero reflection. These incredible properties have led the phenomenon to be called ‘‘perfect lensing.’’ Note that there is no energy flux associated with purely evanescent waves, and hence the amplification obtained in the steady state corresponds to local field enhancements which would imply the presence of localized resonances. In fact, the entire mechanism of the focusing of the near-field components is due to surface modes that reside on the surfaces of these negative index materials (Ramakrishna 2005). " = 1 and  = 1 are precisely the conditions for these surface modes of electric and magnetic nature, respectively. These surface plasmon resonances which are excited resonantly by the evanescent modes and the secret to the perfect lens is that all the surface modes are completely degenerate. Although the conditions for realizing a perfect lens are easy to specify, in practice these are very difficult to meet. The requirement of negative values for " and  implies that these quantities must disperse necessarily with frequency and be dissipative. Thus, the perfectlens condition can only be met approximately at a single frequency. Any deviation from the ideal

Negative Refraction and Subdiffraction Imaging

conditions can then result in the excitation of slab polariton resonances which can swamp the image. The effects of absorption, which are always present, can also seriously degrade the lens performance by damping out the surface plasmon resonances (Ramakrishna 2005). Consider the transmission for the P-polarized radiation through a negative index slab: ~tðkx Þ ¼

4ðkz1 ="þ Þðkz2 =" Þ eikz2 ds D

½29

where D ¼ ðkz1 ="þ þ kz2 =" Þ2  ðkz1 ="þ  kz2 =" Þ2 e2ikz2 ds Under the perfect-lens conditions, the first term in the denominator goes to zero for evanescent waves and the exponential in the second term decays faster than the exponential in the numerator. However, if there was a mismatch in the conditions, ("þ = 1 and " = 1 þ , say) then the first term in the denominator no longer vanishes. In the large wave vector limit (kx  !=c0 ), the two terms in the denominator become approximately equal when   1   ½30 kx ¼  ln   ds 2 thus yielding a criterion for the largest wave vector for which there is effective amplification. The dependence through the logarithm on the deviations (whether real or imaginary) from the resonant conditions underlines the fact that the perfect lens effect is indeed very sensitive. In practice, the periodicity, d, of the strucuture of the metamaterials comprising the negative index slab itself imposes an upper wave vector cutoff kc = 2=d. The material will become spatially dispersive for wave vectors k ! kc , and for k > kc the very description as a homogeneous material will break down. An important simplification of the perfect-lens conditions results when we consider a situation in which all length scales in the problem are much less than the wavelength of the light (the quasistatic approximation). Under these conditions, the electric and magnetic fields effectively decouple. If we consider the case of P-polarized fields, it can be shown (Pendry 2000) that in the quasistatic limit only the value of the permittivity is important, and there are essentially no conditions on the value of the permeability. This brings metals such as silver into the picture as the permittivity of silver becomes equal to 1 in the optical region of the spectrum and with relatively small losses (Pendry 2000). To overcome the losses, a series of refinements of the simple thin-slab picture have been proposed including dividing the lens into a series of layers and using

489

optical amplification to act against the deleterious effects of absorption (Ramakrishna 2005). The Generalized Perfect-Lens Theorem

The negative refractive slab can be considered as ‘‘optical antimatter’’ in the sense that it cancels out the effects on radiation of the traversal through an equal amount of positive refractive index medium. This cancelation is applicable to the phase changes for the propagating modes and the amplitude changes to the evanescent modes. In fact, the focussing action can happen for more general situations where the requirement of homogeneity of the slab material can be relaxed. Now consider the more general situation where the dielectric permittivity and the magnetic permeability are arbitrary functions of the spatial coordinates: "þ ¼ "ðx; yÞ; " ¼ "ðx; yÞ;

þ ¼ ðx; yÞ

½31

 ¼ ðx; yÞ

½32

corresponding to the Figure 6. We will consider the imaging axis to be the z-axis. Thus, we see that the system is antisymmetric with respect to the z = d plane. It turns out (Pendry and Ramakrishna 2003) that such a system also transfers the image of a source placed at the z = 0 to the z = 2d plane in the same exact sense that it includes both the propagating and evanescent components. In general, the rays in spatially varying media will not be straight lines as shown in Figure 6, but the effect of propagating through the positive medium is nullified by the negative medium. Thus, to an observer on the righthand side, it would appear as if the region between z = 0 and z = 2d did not exist. We will call such media with the same sense of transverse spatial variation but with opposite signs as optical complementary media, and the effect of any such pairs of complementary media on radiation is null.

d

d

d

d

Figure 6 A pair of complementary optical media nullify the effect of each other for the passage of light. Spatially varying positive and negative refractive indices are schematically depicted by the white or shaded regions.

490 Negative Refraction and Subdiffraction Imaging

The most general conditions on the permittivity and permeability tensors for such complementary behavior are: 0 1 "xx "xy "xz B C "~þ ¼ @ "yx "yy "yz A "zx "zy "zz 0 1 ½33 xx xy xz B C ~þ ¼ @ yx yy yz A zx zy zz and 0

1

"xx

"xy

þ"xz

B "~ ¼ @ "yx

"yy

C þ"yz A

þ"zx

þ"zy

"zz

xx

xy

þxz

B ~ ¼ @ yx

yy

C þyz A

þzx

þzy

zz

0

1

½34

and a perfect focus results whenever the two slabs of positive and negative media have such a behavior (see Pendry and Ramakrishna (2003) and Ramakrishna (2005) for the proof). This theorem clearly shows that the dependence along the x- and y-directions transverse to the imaging axis z is completely irrelevant as long as the two slabs are optically complementary. As an extension, it can be shown that any system of optically complementary media will also have a perfect focus as long as the system has a plane of antisymmetry normal to the optical axis. The above effects have also been numerically verified for several such spatially varying complementary media (Pendry and Ramakrishna 2003).

where Q2i ¼



 2  2 @x 2 @y @z þ þ @qi @qi @qi

½37

Note that a distortion of space results in the change of " and  tensors in general. Thus, in many cases, the transformed geometry would involve spatially varying (inhomogeneous) and anisotropic medium parameters. The change in geometry can also make it possible for us to realize lenses with curved surfaces. The original slab lens maps every point on the object plane to another point on the image plane. But the size of the image is identical to that of the source. This is due to the invariance in the transverse direction and the transverse wave vector (kx , ky ) is preserved. In general, to change the size of the images, the translational symmetry would have to be broken and curved surfaces will necessarily be needed. The focussing action for the evanescent waves is crucially dependent on the near degeneracy of the surface plasmons in the case of the slab, and curved surfaces, in general, have a completely different dispersion for the surface plasmons. Thus, one should expect that inhomogeneous materials will be required for such curved lenses of negative refractive index. It can be shown (Ramakrishna 2005) that mapping the slab lens into cylindrical coordinates x ¼ r0 e‘=‘0 cos ;

y ¼ r0 e‘=‘0 sin ;

z¼Z

½38

where ‘0 is some scale factor(= 1) generates a cylindrical annulus of inner and outer radii a1 and a2 , respectively, with the material parameters given by "r ¼ r ¼ 1 " ¼  ¼ 1

Perfect Lens in Other Geometries

The above generalized perfect-lens theorem along with a method of coordinate transformations can enable us to now generate a variety of superlenses in different geometries. In general, if we can find a geometric transformation that maps a given configuration into the geometry for the generalized slab lens, then we would have generated one more arrangement that will exhibit the property of transferring images of sources in a perfect sense. If we define the new coordinates q1 (x, y, z), q2 (x, y, z), and q3 (x, y, z) (assumed orthogonal), then in the new frame, the material parameters and fields are given by (Ward and Pendry 1996) "~i ¼ "i

Q1 Q2 Q3 ; Q2i ~ i ¼ Qi Ei ; E

~i ¼ i

Q1 Q2 Q3 Q2i

~ i ¼ Qi H i H

½35 ½36

½39 2

"z ¼ z ¼ 1=r

for the annular region. The positive material outside the annular region should vary as "r ¼ r ¼ þ1 " ¼  ¼ þ1

½40

"z ¼ z ¼ þ1=r2 where r = r0 exp(‘=‘0 ). This system transfers images in and out of the cylindrical annulus and the image of a source inside at r = a0 will be formed on the surface a3 = a0 (a2 =a1 )2 . Thus, there will be a magnification of the image by the factor  2 a2 M¼ ½41 a1

Negative Refraction and Subdiffraction Imaging

Note that these cylindrical lenses are also shortsighted in the same manner as the slab lens. They can only focus sources from inside to the outside only when a21 =a2 < r < a1 , and the other way around from outside to the inner world when the source is located in a2 < r < a22 =a1 . Similarly the transformation into spherical coordinates (r = r0 e‘=‘0 , , ) can be used to generate a spherical perfect lens wherein a spherical shell of negative refractive material with "(r)  1=r and (r)  1=r with arbitrary dependence along and  (which could be constant too!) have the property of perfectly transferring images of sources in and out of the shell (Pendry and Ramakrishna 2003). This spherical lens also has exactly the same magnification factor given by eqn [41]. In fact, the solutions in these two cases of a cylinder and sphere can also be obtained by a more conventional electromagnetic calculation in terms of the scattering modes (Ramakrishna 2005). One can obtain even more esoteric configurations such as one or two intersecting corners of negative refracting materials that behave as perfect lenses (Pendry and Ramakrishna 2003). Other Approaches to Negative Refraction

There is also an approach to negative refractive materials based on loaded transmission lines (Eleftheriades et al. 2002), which has been implemented at radio- and microwave frequencies using lumped circuit elements. These show all the hallmarks of a negative refractive material within an effective medium approach. Effects which can be interpreted as negative refraction have been observed in certain periodic photonic crystals (PCs) (Luo et al. 2003). An incident propagating plane wave from vacuum appears to undergo negative refraction inside the PC, and a slab of the PC can even work as a Veselago lens. The negative refraction in this case is a result of the curvature of the equifrequency surface and is present in spite of the right-handed nature of the propagation. In these instances, an effective permittivity and permeability cannot be easily ascribed to the crystal as the long wavelength condition is not met. It is difficult to homogenize the PC in the sense of meta-materials, and the energy transport in these PCs is very sensitive to the periodicity and the structural arrangements. Thus, it would be an over-simplification to characterize these

491

effects in PC as merely due to an effective refractive index.

Further Reading Eleftheriades GV, Iyer AK, and Kremer PC (2002) Planar negative refractive index media using periodically L–C loaded transmission lines. IEEE Transactions on Microwave Theory and Techniques 50: 2702–2712. Garland JC and Tanner DB (eds.) (1978) Electrical Transport and Optical Properties of Inhomogeneous Media. New York: American Institute of Physics. Landau LD, Lifschitz EM, and Pitaevskii LP (1984) Electrodynamics of Continuous Media, 2nd edn. Oxford: Pergamon. Luo C, Johnson SG, Joannopoulos JD, and Pendry JB (2003) Subwavelength imaging in photonic crystals. Physical Review B 68: 045115. Marques R and Smith DR (2004) Comment on ‘‘Electrodynamics of Metallic Photonic Crystals and the Problem of Left-Handed Materials’’. Physical Review Letters 92: 059401. O’Brien S and Pendry JB (2002a) Photonic band-gap effects and magnetic activity in dielectric composites. Journal of Physics: Condensed Matter 14: 4035–4044. O’Brien S and Pendry JB (2002b) Magnetic activity at infrared frequencies in structured metallic photonic crystals. Journal of Physics: Condensed Matter 14: 6383–6394. Parazzoli CG, Greegor RB, Li K, Koltenbah BEC, and Tanelian M (2003) Experimental verification and simulation of negative index of refraction using Snell’s law. Physical Review Letters 90: 107401. Pendry JB (2000) Negative refraction makes a perfect lens. Physical Review Letters 85: 3966–3969. Pendry JB (2004) Contemporary Physics 45: 191–202. Pendry JB, Holden AJ, Robbins DJ, and Stewart WJ (1998) Low frequency plasmons in thin-wire structures. Journal of Physics: Condensed Matter 10: 4785–4809. Pendry JB, Holden AJ, Robbins DJ, and Stewart WJ (1999) Magnetism from conductors and enhanced nonlinear phenomena. IEEE Transactions on Microwave Theory and Techniques 47: 2075–2084. Pendry JB and Ramakrishna SA (2003) Focusing light using negative refraction. Journal of Physics: Condensed Matter 15: 6345–6364. Pokrovsky AL and Efros AL (2002) Electrodynamics of metallic photonic crystals and the problem of left-handed materials. Physical Review Letters 89: 093901. Ramakrishna SA (2005) Physics of negative refractive index materials. Reports on Progress in Physics 68: 449–521. Shelby RA, Smith DR, and Schultz S (2001) Experimental verification of a negative index of refraction. Science 292: 77–79. Smith DR, Padilla WJ, Vier DC, Nemat-Nasser SC, and Schultz S (2000) Composite medium with simultaneously negative permeability and permittivity. Physical Review Letters 84: 4184–4187. Veselago VG (1968) The electrodynamics of substances with simultaneously negative values of var " and . Soviet Physics– Uspekhi 10: 509–514. Ward AJ and Pendry JB (1996) Refraction and geometry in Maxwells equations. Journal of Modern Optics 43: 773–793.

492 Newtonian Fluids and Thermohydraulics

Newtonian Fluids and Thermohydraulics G Labrosse and G Kasperski, Universite´ Paris-Sud XI, Orsay, France

jQ

dΦQ

ª 2006 Elsevier Ltd. All rights reserved.

dS

M dΦQ

M

dS

~ flux. Figure 1 Q flux density and Q

Introduction Thermohydraulics is based on the hypothesis of continuous medium. This hypothesis is easily satisfied since, for instance, a one-thousandth of 1 mm3 of a perfect gas at normal temperature and pressure conditions (300 K, 1 atm) contains about 2.5  1013 molecules. Instantaneous balances are made inside a control volume fixed in the system of axes and crossed by the flows. The limit where this volume vanishes leads to the local formulation of the laws governing the flows. The flow is described by velocity ~ v (~ r, t), pressure p(~ r, t), temperature T(~ r, t), and other fields, ~ r being the position vector of a point M, and t the time. The material derivative of q(~ r, t) is   Dq @  þ ð~ v:~ Þ q Dt @t ~ be one of the scalar (vectorial) extensive Let Q (Q) quantities whose balance participates in the flow dynamics. It can be a quantity of matter, heat, impulse, or something else. Let Q be the amount of Q contained in the volume V localized around M, and q(~ r, t) its local representative defined by ð~ r; tÞqð~ r; tÞ ¼ lim

V!0

Q dQ  V dV

½1

where  is the density, similarly defined considering the case where [Q] is taken as the mass m: ð~ r; tÞ ¼

dm dV

½2

Table 1 gives examples of q quantities. The instantaneous local balance of Q reads   @ ðqÞ þ ~  ~ jQ þ q~ v ¼ SQ @t

½3

where SQ stands for any possible local source of Q, and ~ jQ is the Q conduction flux density. Figure 1 Table 1 Some quantities q. T is the absolute temperature, Cp the specific heat at constant pressure, and C the solute mass fraction Mass 1

Impulse

Kinetic energy

Heat

Mass fraction

~ v

2 ~ v 2

Cp T

010

L

L being a fluid container size scale. Thence come the Rayleigh, Prandtl and Lewis numbers,

The curl of [13] yields

V42 gL3 ¼ T T V1 V3 T V3 V2 C Pr ¼ ¼ ; Le ¼ ¼ V1 T V1 T

~ ðp; T; CÞ ^ ~ G þ ðp; T; CÞ~ ^~ G¼0

Ra ¼

which has no reason to be generically satisfied since (p, T, C) and ~ G are totally uncorrelated. The hydrostatic state cannot exist if ~ G does not derive from a scalar potential, as with ~ ~ dW dW ~ ^ ðW ~ ^~ ~ G ¼~ gW rÞ  ^~ r if 6¼ 0 dt dt The Earth’s rotation axis is known to precess with a period of about 26 000 years. This generates a component of 26 000 years timescale in the atmospheric, oceanic, and internal flows. Considering now that

Ra being the experimental control parameter, and Le 1. Table 4 gives Pr orders of magnitude for usual fluids. Let V be the fluid velocity amplitude. The importance of the thermal, solutal, and impulse convections with respect to the corresponding diffusions is, respectively, estimated by the thermal, compositional Pe´clet and Reynolds numbers, PeT ¼

V VL ¼ ; V1 T Re ¼

~ G ¼ ~  the existence of a hydrostatic state only depends on the simultaneous verification of [14] and ~ ðp; T; CÞ ^ ~  ¼0

V3 ¼

V1 ¼ ½14

Water 6.7

PeC ¼

V VL ¼ V2 C

V VL ¼ V3

with Pr ¼

PeT ; Re

Le ¼

PeC ; Re

Ra ¼ ðPeT ReÞjV¼V

4

½15

Iso- surfaces must therefore coincide with isopycnal, isobaric, iso-T, and iso-C surfaces since the p, T, and C sensitivities of  are uncorrelated. The compatibility of this condition with [14] is the key for concluding about the existence of the hydrostatic state. Considering again our planet as an example (forgetting about precession), the isosurfaces are almost ellipsoidal. Such T and C distributions cannot satisfy [14]. Thus, the atmospheric and oceanic dynamics, and thermohydraulics as well, are due to a nonvanishing thermal torque, ~ T ^ ~  . A free surface in hydrostatic state is isothermal ~ and isocompositional, by eqn [10], whatever G.

Dimensionless Local Balances In buoyancy-driven thermohydraulics, we consider four velocity scales – three of molecular origin, and the fourth is the free-fall velocity in the buoyancy,

Capillary thermohydraulics introduces one velocity scale and the Marangoni number, V5 ¼

j j ; ~v

Ma ¼

V5 ¼ PeT V1

with  = (d =dT)T in pure fluid. A small capillary number, Ca = j j= , indicates a weak influence of the dynamics upon the free-surface curvature. Let V1 ,  = 0 V12 ,  = L=V1 , T and C ¼ C0 ð1  C0 ÞST T be the velocity, pressure, time, temperature, and mass fraction scales, with ¼

T  T0 T

and

C¼

C  C0 C

the reduced temperature and mass fraction, respectively. The other quantities, coordinates included, are similarly reduced and noted identically.

496 Newtonian Fluids and Thermohydraulics

Equation [8] does not change and [9], [11] and [12] become, respectively, h i 2 D~ v ¼ ~ P þ Pr Rað þ B CÞ^ez þ ~ ~ v Dt

½16

D ~2 ¼  Dt

½17

2 2 DC ¼ Le ~  C~   Dt

½18

C C T T

~ ~ by 0 , W(t)  = W(t)= ~ In rotating frame, scaling W(t) 0, ~ ~  ^ ðW  ^~ Ra Fr ð þ B CÞW rÞ ! ~  1 dW ~  2W ^ ~ vþ ^~ r  Ek dt must be added inside the square-bracket term of [16]. The Froude and Ekman numbers appear as Fr ¼

g

;

Ek ¼

0 L2

The dimensionless capillarity stress condition [10] reads h i ^t  ~ ^  ~ v þ ð~  ~ vÞt  n     ^t Þ C ½20 ¼ Ma ð~   ^t Þ þ C ð~ with C ¼

½21

0 1 0 1 0 1 ~ v v ~ v   ~ @@  A¼ F þ ~ v~  @  A þ A@  A ½22 @t C C C where F = ( ~ ( P), 0, 0)t , and

is the buoyancy separation ratio and ^ez = ~ g =j~ g j. A B < 0 (> 0) corresponds to opposite (cooperative) thermal and solutal buoyancies. The reduced mass fraction boundary condition on impervious walls is     ~ ^ ¼ ~ ^ C  n   n ½19

20 L

Given a base state S = (~ v, , C), a solution of [8], [16]–[18], how does it behave in presence of an infinitesimal disturbance ( ~ v, , C)? Applying [8], [16]–[18] to (~ v þ ~ v,  þ , C þ C) and discarding the quadratic terms in perturbation provide the disturbance temporal evolution, ~  : ð ~ vÞ ¼ 0

where B ¼ 

Linear Stability

@ =@C C @ =@T T

the capillarity separation ratio, and    @ T   Ma ¼  @T ~v V1  These equations show that, in the Boussinesq framework, the flow physics does not depend on p0 , T0 , and C0 , except through the material properties which enter the numbers.

0

BPr @ 0 A¼ 0

Ra Pr ^ez B1 2 ~ 

1 Ra Pr B ^ez A 0 BLe

½23

2 v~ ) þ a~  . The perturbations ( ~ v, , with Ba = (~ C) have the (~ v, , C) boundary conditions, but homogeneous. On a free surface, the perturbation capillary stress condition is h i ^t  ~ ^   ~ v þ ð~   ~ vÞt  n     ^t Þ C ½24 ¼ Ma ð~   ^t Þ  þ C ð~

Recasting [21]–[23] provides 0 1 0 1 ~ v ~ v @@  A ¼ LðSÞ@  A @t C C whose solution is 0 1 0 1 ~ vðtÞ ~ vðt ¼ 0Þ @ ðtÞ A ¼ eLðSÞt @ ðt ¼ 0Þ A CðtÞ Cðt ¼ 0Þ

½25

½26

Direct System

L(S) is made of ~  acting on the initial perturbation. Conclusions about S stability depend on the sign of

max , the real part of the leading eigenvalue of L found with all the possible perturbations. There is stability if max < 0. At max = 0, the marginal stability, the bifurcation threshold is located at Ra (Pr, Le, B , C , X) = Rac , Rac -being the critical value of the control parameter, X containing all the other parameters of the problem (container aspect ratios, etc.). The nonlinear-stability analysis in the vicinity of Rac supplies  in max / (Ra  Rac ) , which is characteristic of the bifurcation.

Newtonian Fluids and Thermohydraulics

equations describing the temporal evolution of amplitudes, Ai , i = 1, 2, . . . , I, characterizing the perturbation eigenmodes,

1 0.75

dAi ¼ i Ai þ Ni ðAj Þ for i; j ¼ 1; 2; . . . ; I dt

0.5 0.25 N

0 –0.25 –0.5 –0.75 –1

497

0

0.25

0.5 r

0.75

1

Figure 2 Leading axisymmetric thermal adjoint eigenvector (Courtesy of O Bouizi and C Delcarte).

Adjoint System

The leading left eigenmode complex conjugate supplies the response field of the base state to the most destabilizing punctual disturbances. The S state and L eigenspace analytical determinations are often impossible. One must resort to specifically designed numerical tools. A numerical adjoint eigenvector is presented in Figure 2 for a (Ma = 106, Pr = 102 ) side-heated cylindrical liquid bridge, with a free surface on the right and the axis on the left.

Nonlinear Stability When max > 0, the associated disturbance exponentially grows with time, until nonlinearities become essential. The flow progressively evolves from S towards a new state, S 0 , which is a solution of [8], [16]–[18]. How can one proceed analytically to know how the nonlinearities control the bifurcation? A large number of S ! S 0 bifurcations exist, with either both S, S 0 , steady or unsteady but with different flow structure, or one is steady and the other is not. Bifurcations can also be reversible or hysteretic, with respect to Ra. The symmetries of S play an important role and non-Boussinesq effects change the thresholds and the nature of bifurcation. Landau’s works have opened up the way to the theory of nonlinear hydrodynamic stability. The ruling equations are reduced, using an appropriate expansion method, to a set of ordinary differential

½27

where N accounts for the nonlinear action of the I modes on Ai , and the i ’s are the temporal growth rates coming from the linear theory. The stability of the steady solutions, dAi =dt = 0, is determined by local analysis. With one destabilizing mode, the simplest model is dA=dt = A  AjAj, with  > 0, constant, specific of the bifurcation. Symmetry considerations (some of them directly originate from the Boussinesq framework) may impose  = 0, whereby the simplest model becomes dA=dt = A þ A3 , with  another constant. When the flow is weakly confined in one or two space directions, boundary effects can play a subtle dynamical role, allowing, for instance, the existence of multiple solutions, each one made of many interacting modes. A large variety of flow regimes is then observed, as steady/traveling, extended/ localized wave packets, particularly in binary mixtures. Spacetime models, close to [27], such as the Ginzburg–Landau equation, @A @2A ¼ A þ  2 þ jAj2 A @t @x are derived for describing the dynamics of the wave packet envelop (of complex amplitude A).

Hydrostatic State Stability The static-state stability is analytically tractable in unbounded volume. Transverse wave (by [21]) solutions are the potentially destabilizing perturbations, with wave vector ~ k and complex frequency !. The system [22]–[23] gets simplified, and L becomes algebraic upon substituting (i~ k, i!) for (~ , @=@t). Intuitively, the quiescent state loses its stability when ~ (p, T, C)  ~  exceeds a threshold value (positive, by the dissipative effects). This analysis supplies it, together with the data of the oscillatory motions emerging at onset from the rest-state instability. In reality, the fluid is confined to three dimensions, possibly with free surfaces, and wave solutions are no longer usable. The first approach consists in defining a simplified model confined to one dimension. The perturbations must satisfy homogeneous boundary conditions, and/or [24], and they are waves in both other space directions. The resulting problem may be analytically tractable. The stability of many quiescent-state configurations was studied, for fluid layers of infinite or very large

498 Newtonian Fluids and Thermohydraulics

extension, of pure-fluid/mixtures, with/without free surface. Nonetheless, many other configurations are not yet analyzed. Two- and three-dimensional cases must be numerically treated.

q

q

g

Gravitational Buoyancy Convection Among the numberless thermal situations to analyze, research mainly favored the case where the fluid is confined in simple geometries and submitted to two distinct heating directions, ~ T being either aligned or normal to ~ G, that is vertical or horizontal in the gravity field. Each case leads to specific thermohydraulics. The rest-state stability is the first analysis step of the former case, the first to be experimentally studied by Be´nard in 1900, with a horizontal liquid layer. The latter is of more recent interest, with Batchelor’s theoretical work on the parallel convective regimes of pure fluid confined in tall slot. Since then, a large amount of work has been published on those cases, tackling various confinement geometries, and involving high Ra values. This problem became the paradigm of the rich spatiotemporal behaviors arising in nonlinear systems driven away from equilibrium. In binary mixtures the complexity of the dynamics increases considerably. The literature is so far practically devoid of any three-dimensional results in mixtures. Ternary mixtures have so far been only scarcely considered. Steady Parallel-Flow Model

This analytical approach comes from an interesting Batchelor’s remark made about the vorticity but here applied to the velocity of a confined flow. ‘‘A number of flow fields are characterized by values of the magnitude of the’’ velocity ‘‘in the neighborhood of a certain line in the fluid which are much larger than those elsewhere,’’ and (by ~  ~ v = 0) ‘‘this line of necessity’’ is parallel to ~ v and to the container walls. Buoyant forces may contradict this assertion, particularly in Rayleigh–Be´nard configuration with imposed temperatures. There, no parallel solution exists. Nevertheless, steady parallel flows do exist in containers. The thermally active walls (whatever they be – the largest or smallest) are either maintained at constant temperatures, or subjected to a constant heat flux. Figure 3 sketches a cross section (hereafter referred to as the vertical midplane) of such a configuration, with active (uniform heating q) vertical walls. The other sides are adiabatic. No rest state is allowed here. Although intrinsically three dimensional, the steady regime in

eˆz H eˆx

( )

L H Figure 3 Sketch of the cross section of a slender vertical container.

this cavity can be fairly well approximated as two dimensional (in the vertical midplane), and moreover mainly parallel to the active walls, in an Ra range which increases with the aspect ratio, H/L. The influence of the horizontal sides is of limited range compared to the flow extension, H. The parallel flow is then the one-dimensional approximation of what occurs in the major part of the cavity. This configuration is taken with a binary mixture for illustrating an approach applicable with minor variations in other situations. The problem becomes linear. Indeed, ~ v = w(x)^ez by ~  ~ v = 0. Taking T = qL=T as temperature scale, [16]–[18] imply ^ ðx; zÞ ¼ GT z þ ðxÞ;

^ Cðx; zÞ ¼ GC z þ CðxÞ

with GT , GC as constants. The impulse balance is h i d2 w ^ ^ ¼ Ra ðxÞ þ  CðxÞ B dx2 and the ruling equations  d4 w B ð Þ G ¼ Ra G þ þ G T T C w dx4 Le ^ d2  d2 C^ wGT ¼ 2 ; wðGT þ GC Þ ¼ Le 2 dx dx

½28

½29

An internal length scale is predicted, of thickness    1=4 B Ra GT þ ðGT þ GC Þ Le By [28] and [19], the thermal flux condition yields  d3 w  ¼ Ra ð1 þ B Þ dx3 x¼1=2

499

Newtonian Fluids and Thermohydraulics

A last operation allows to determine GT and GC . The overall heat and mass fraction balances are performed in the cavity part (V), which is bounded by an horizontal plane located within the parallelflow region. Since the walls are impervious, the solute is transported only across the lower boundary of (V), through which the net vertical convective supply must be balanced, in steady regime, by vertical diffusion. The heat balance works similarly, since the walls are adiabatic or submitted to equal fluxes. Whence the relations, Z 1=2 ^ wðxÞðxÞ dx ¼ GT

A 1.6 1.2 0.8

–5 ϕ1 –5/2 ϕ1

–5 ϕ1

0.4

2Le

0 –0.4

Le

–1/2 ϕ1

0

–5/2 ϕ1

–0.8 –1.2

1=2

Z

1=2

^ dx ¼ GT þ GC wðxÞCðxÞ

1=2

where 315 ; 218

r¼

–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

r

The steady parallel flow is determined. Its stability can be analyzed as indicated in the section ‘‘Linear stability.’’ Some caution must be taken for the Boussinesq approximations to be valid here, with the temperature and mass fraction increasing constantly (by GT , GC ) along the direction of largest cavity extension. These gradients are at the origin of the ‘‘thermogravitational column’’ separation power, a device designed for the isotope separation. Extremely long columns can provide almost complete separations, with C jCj no longer 1, and then the non-Boussinesq effects occur. As an illustration of aforementioned notions, let us consider the (Pr = 1, Le = 0.1) Rayleigh–Be´nard– Soret (RBS) problem where horizontal solid plates of infinite extension are uniformly heated from above (Ra < 0) or below (Ra > 0). This configuration is simply obtained by rotating the cavity in Figure 3 by =2 with respect to ~ g and to (^ex , ^ez ). The steady parallel-flow model can lead to the right-hand side of an equation like [27] governing the time evolution of A, the parallel-flow amplitude,    dA 1r / A Le2 A4 þ  1 þ A2 dt Le2   r 2 þ 1 ½30 rc

¼

–1.6

Ra ; 720

1 rc ¼ 1 þ B ð1 þ Le1 Þ

Here rc is the critical value or r where the rest state loses its stability towards a steady parallel flow. The roots of dA=dt = 0 are A = A0 = 0, A = Ak (r, Le, B ), for the quiescent, convective states. Figure 4 shows that A0 = 0 and the curves Ak (r) for several

Figure 4 Bifurcation diagram of A0 (r ) and Ak (r ) for various separation ratios B (Le):

B (Le), ’1 = (1 þ Le1 )1 being the rc pole. The solid (dotted) parts correspond to the stable (unstable) steady states, emerging from direct (backward) pitchfork bifurcations of the rest state at rc . Saddle–node bifurcations from unstable to stable steady states are also predicted, on the dashed curve of the equation rffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  c Ak ðrÞ ¼  r  ð1 þ Le2 Þ 2

Fully Nonlinear Problem

Numerical tools are required for solving the system [8], [16]–[18] and analyzing the stability of the flows obtained. The RBS Case Let us illustrate how the rest-state loss of stability occurs in the two-dimensional RBS case, with a (Pr = 1, Le = 0.1, B = 0.2) mixture. The flow lies in the meridian plane of an axisymmetric container with the radius/height ratio equal to 2. No-slip conditions are imposed on impervious walls; the temperature on the bottom plate is higher than on top, and the peripheral wall is adiabatic. At t = 0, the quiescent state is given a small random perturbation. The system evolves (Figure 5) towards a stable periodic solution via a transient regime of exponentially amplified amplitude (eqn [26]). One speaks of a Hopf bifurcation for a steady (here quiescent) state destabilization by oscillatory disturbances. The ‘‘instantaneous’’ frequency (from the time running between two successive identical passes of

500 Newtonian Fluids and Thermohydraulics

2

1

u

0

–1

–2 0

20

40

80

60

100

120

140

t Figure 5 Time evolution of a radial velocity nodal value for Ra = 2600: Reproduced from Millour, Labrosse, and Tric (2003) Physics of Fluids 15(10): 2791–2802, with permission from American Institute of Physics.

10

9.5

9

ωn 8.5

8

7.5

0

20

40

60

80

100

120

140

t Figure 6 Instantaneous angular frequency !n corresponding to Figure 5. Reproduced from Millour, Labrosse, and Tric (2003) Physics of Fluids 15(10): 2791–2802, with permission from American Institute of Physics.

the signal) evolves with time (Figure 6) from its threshold value to its nonlinearly saturated one. Accurate determination the thresholds and identification of the associated bifurcation is possible by fitting the argument  of max (Ra) from the exponential growth of Figure 5, in the Rac vicinity. Figure 7 shows (solid dots) (Ra) measurements, and the solid line (in Figure 8 also) is the linear law given by the two points closest to the vanishing growth rate. The local law announced in the subsection ‘‘Direct system’’ is confirmed, with

an exponent  = 1 for the Hopf bifurcation, and  = 1=2 for saddle–node (Figure 8) and pitchfork bifurcations. The Thermally Driven Cubic Cavity All flows are obviously three dimensional. When do they possess a two-dimensional approximation? How to qualify it? Clearly, the flow that develops in the container of Figure 3 might enjoy (in a given parameter domain, D) the mirror-reflection symmetry property about the vertical midplane. Is there a two-dimensional

Newtonian Fluids and Thermohydraulics

0.015

501

eˆz

0.01 0.005 T0 + ΔT

0 λ

eˆx

– 0.005 –0.01 T0

– 0.015 – 0.02 – 0.025 – 0.03 2574

eˆy Figure 9 Sketch of the thermally driven cubic cavity.

2576

2578

2580

2582

2584

2586

Ra

Figure 7 Temporal growth rate, , of infinitesimal perturbations, in the vicinity of the Hopf bifurcation of the quiescent state. Reproduced from Millour, Labrosse, and Tric (2003) Physics of Fluids 15(10): 2791–2802, with permission from American Institute of Physics.

0.6 0.5 0.4 0.3 λ2

0.2 0.1 0 –0.1 –0.2 2625

2630

2635

2640

2645

Ra

Figure 8 Squared temporal growth rate, 2 , of transient relaxation towards the stationary state close to the saddle– node bifurcation. Reproduced from Millour, Labrosse, and Tric (2003) Physics of Fluids 15(10): 2791–2802, with permission from American Institute of Physics.

where an oscillatory regime appears. The numerical three-dimensional flow is steady until Ra3D, c = 3.2  107 , where it hysteretically bifurcates towards an oscillatory regime breaking the mirror symmetry about the midplane. Let us assess the validity of the two-dimensional approximate solutions. We define dimensionless heat fluxes (Nusselt numbers) which penetrate in one of the active walls,  Z 1 @ 3D  NuðyÞ ¼ dz @x x¼0 0 Three fluxes are interesting to compare: (1) in the Nump = Nu(y = 1=2), (2) globally Nu3D,W = Rmidplane, 1 Nu(y) dy, and (3) the two-dimensional 0 approximation  Z 1 @2D  Nu2D;W ¼ dz @x x¼0 0 Figure 10 shows how they compare themselves, as a function of Ra. Quantitatively, the two-

100 × (Nu2D,W – Nu3D,W)/Nu2D,W 100 × (Nu2D,W – Nump)/Nu2D,W

10

approximation of the flow in this midplane? Is it able to give a correct estimate of the two-dimensional flow stability within D, and to predict the D frontiers, where the mirror-reflection symmetry property ceases to be valid? Only partial answers are available so far, coming from the thermally driven cubic cavity (Figure 9). Filled with a pure fluid, its left and right vertical plates have fixed temperatures, T0 ( = 0 at x = 0) and T0 þ T ( = 1 at x = 1), while the others are adiabatic. Any T 6¼ 0 generates a flow, possibly mirror-symmetric about the vertical (hatched) midplane, and also centrosymmetric about ^ey . The two-dimensional approximation was extensively analyzed, numerically, with air as a fluid. A steady flow is obtained for Ra < Ra2D, c = (1.82  0.01)  108 ,

100 × (Nu3D,W – Nump)/Nu3D,W

5

0

–5

–10 103

104

105

106

107

Ra

Figure 10 Relative 2D–3D Nusselt numbers. Reproduced with permission from Tric E, Labrosse G, and Betrouni M (2000) A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity from accurate numerical solutions. International Journal of Heat and Mass Transfer 43: 4043–4056. ª Elsevier Ltd.

502 Newtonian Fluids and Thermohydraulics

dimensional approximation is not too bad, but not qualitatively, with a nonmonotonic evolution of the discrepancies. These latter become quite negligible when the three-dimensional flow gets unsteady and paradoxically loses the symmetry property on which its two-dimensional approximation is founded.

Thermocapillary Convection Two immiscible liquids, or a liquid and a gas, are separated by a free surface, a region of small thickness (some ten molecular sizes). From a macroscopic viewpoint, it is considered as a singular entity. Its location and geometry are part of the solutions of the governing equations, themselves supposed to satisfy [20] on the free surface. As a first iteration, the free-surface shape can be imposed, fixed, and straight often. Numerous industrial processes involve thermocapillarity wherein thermohydraulics involves complex phenomena, such as phase-change kinetics. A relevant modeling of these situations is a research subject by itself. For thermohydraulics, some academic configurations (Figure 11) have retained the attention of the scientific community. Any thermohydraulic flow transfers heat between hot and cold solid boundaries wherein heat penetrates by conduction. Consequently, the

(a)

(b)

(c)

Figure 11 Open boat ((a) straight and (b) circular) and liquid bridge (c) configurations.

eˆz

∇T

Gas Solid

∂u ∂z

v=0

∂w ∂x

eˆx

Liquid

Figure 12 Thermocapillary origin of vorticity singularity (cold wall configuration).

!

term (  ^t ) of [20] never cancels at the solid boundary/free surface junction, as in Figure 12. A nonzero vorticity is thus generated by thermocapillarity on the free surface until the wall, while flow adherence on the wall gives vorticity values of opposite sign. The problem presents therefore a vorticity singularity at the triple point. This is a deep physical and modeling problem. See also: Bifurcations in Fluid Dynamics; Capillary Surfaces; Compressible Flows: Mathematical Theory; Dynamical Systems and Thermodynamics; Dynamical Systems in Mathematical Physics: An Illustration from Water Waves; Fluid Mechanics: Numerical Methods; Magnetohydrodynamics; Non-Newtonian Fluids; Partial Differential Equations: Some Examples; Stability of Flows; Vortex Dynamics.

Further Reading Batchelor GK (1987) An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press. Bender CM and Orszag SA (1999) Advanced Mathematical Methods for Scientists and Engineers I. New York: Springer. Bird RB, Stewart WE, and Lightfoot EN (1960) Transport Phenomena. New York: Wiley. Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon. Colinet P, Legros JC, and Velarde M (2001) Nonlinear Dynamics of Surface-Tension-Driven Instabilities. Berlin: Wiley. Drazin PG and Reid WH Hydrodynamic Stability, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge: Cambridge University Press. Johns LE and Narayanan R (2002) Interfacial Instability. New York: Springer. Koschmieder EL (1993) Be´nard Cells and Taylor Vortices, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge: Cambridge University Press. Kuhlmann HC (1999) Thermocapillary Convection in Models of Crystal Growth. Springer Tracts in Modern Physics, vol. 152. Berlin: Springer. Labrosse G (2003) Free convection of binary liquid with variable Soret coefficient in thermogravitational column: the steady parallel base states. Physics of Fluids 15(9): 2694–2727. Lu¨cke M, Barten W, Bu¨chel P, et al. (1998) Pattern formation in binary fluid convection and in systems with throughflow. In: Busse FH and Mu¨ller SC (eds.) Evolution of Structures in Dissipative Continuous Systems, Lecture Notes in Physics. New York: Springer. Manneville P (1990) Structures Dissipatives, Chaos and Turbulence. Gif-sur-Yvette (France): Collection Ale´a Saclay. Narayanan R and Schwabe D (eds.) (2003) Interfacial Fluid Dynamics and Transport Processes, Lecture Notes in Physics, vol. 628. Berlin: Springer. Platten JK and Legros JC (1984) Convection in Liquids. Berlin: Springer. Turner JS (1973) Buoyancy Effects in Fluids, Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge: Cambridge University Press.

Newtonian Limit of General Relativity

503

Newtonian Limit of General Relativity J Ehlers, Max Planck Institut fu¨r Gravitationsphysik (Albert-Einstein Institut), Golm, Germany ª 2006 Elsevier Ltd. All rights reserved.

Introduction The general theory of relativity (GRT) unifies special relativity theory (SRT) and Newton’s theory of gravitation (NGT). SRT and NGT describe successfully large domains of physical phenomena; therefore, one would like to understand how they survive as approximations in GRT. In GRT, spacetime is idealized as a four-dimensional Lorentz manifold whose curvature is related to the distribution of energy and momentum. In such a spacetime, the existence of the exponential map implies that the metric near any event (spacetime point) x deviates from a flat metric only by terms given by the curvature there. Thus, if the gravitational tidal field, represented by the curvature tensor, is small near x, one may approximate the GR metric there by a flat Minkowski metric. This explains that SRT is a general local approximation to GRT. Apart from a remark at the end of the subsection ‘‘Local laws’’ the relation GRT ! SRT will not be discussed further. In its traditional formulation, Newton’s theory differs drastically from Einstein’s theory both in its spacetime structure and in its description of gravitation. The main purpose of this report is to show how NGT can nevertheless be understood as a kind of ‘‘limit’’ of GRT. More precisely, the structure of NGT can be viewed as a degenerate version of that of GRT, in parallel to the fact that the Galilei group can be obtained by contracting the Lorentz group. In the next section we state the laws of GRT. We then reformulate these laws with slightly different field variables such that, besides the gravitational constant k, the speed of light appears via  = c2 . The resulting laws remain meaningful if  and/or k are replaced by zero. They turn out to give a common basis for GRT, SRT, and NGT. The possibility of such a framework was indicated independently by Cartan (1923, 1924) and Friedrichs (1927) and extended by several authors; the complete formulation reviewed here was given by Ehlers (1981). The section ‘‘Newton’s theory in spacetime form’’ shows that the laws of NGT and SRT are obtained, with some additional restrictions, from the rescaled laws of GRT by putting, respectively,  = 0 or k = 0. It is emphasized that Newton’s theory proper is a

theory only of isolated systems. Its intrinsic, fourdimensional formulation explains how the distinction between a vectorial gravitational field and inertial forces, as well as the existence of inertial frames, emerge as consequences of asymptotic flatness. These structures are lost in the so-called ‘‘Newtonian’’ cosmology whose dynamics is due to symmetry assumptions, whereas GR cosmology is a proper part of GRT. The penultimate section is concerned with relations between solutions of GRT and NGT, and in the final section some results related to solutions are reported. They illustrate that the limit relation GRT ! NGT may sometimes be inverted to get exact or approximate GR results from NGT. Approximations are related to uniform convergence in , as is indicated at the end of the final section. The limit relations described here may be considered as a model for other theory relations in physics such as quantization or dequantization. Notation Indices will be considered in general as ‘‘abstract’’ ones, characterizing the kind of objects independent of coordinate systems. Greek indices refer to spacetime, Latin ones to 3-space. Fields on spacetime will generally be taken to be smooth.

Basic Concepts and Laws of GRT According to GRT, spacetime is a four-dimensional manifold M endowed with a Lorentzian metric g , here taken to have signature (þ þ þ ). Any kind of matter including nongravitational fields is supposed to determine an energy tensor T  . Metric and matter are interrelated by Einstein’s gravitational field equation   8k 1 ½1 R ¼ 4 T  g T c 2 In this equation, T := T   denotes the trace of the energy tensor, k and c stand for Newton’s constant of gravity and the speed of light, respectively, and the Ricci tensor R is obtained from Riemann’s curvature tensor by contraction R :¼ R  The curvature tensor is constructed from the symmetric, linear connection    determined by the metric. Equation [1] implies the vanishing of the covariant divergence of the energy tensor T  ; ¼ 0

½2

504 Newtonian Limit of General Relativity

the GRT analog of the laws of local conservation of energy and momentum. The energy tensor depends on the kind of matter to be taken into account. In this article, only vacuum fields (T  = 0) and perfect fluids will be considered. For such a fluid, T



2





¼ ð þ c pÞU U þ pg



and to write – presently only as a change of notation – s instead of g . Then the fields t , s ,    , T  , , p, U , called the basic fields below, and constants k > 0,  > 0 satisfy the following laws: t s ¼   

½3a t; ¼ 0;

 and p denote the mass density and the pressure, respectively, and the 4-velocity U is a timelike vector obeying g U U ¼ c2

½3b

If thermodynamical relations are added to specify the kind of fluid – the simplest cases are barotropic equations p = f () – then eqns [1]–[3] admit a well-posed initial value problem for the fields g , U , . Different matter models which could be treated in the context of this report are elastic bodies and ideal gases, but not point particles. Point particles fit into GRT even less than into electrodynamics.

R

s ; ¼ 0

R   ¼ R     1 ¼ 8k t t  t t T  2

To obtain a spacetime formulation of NGT and a limit relation ART ! NGT, we recall that the metric structure of Newton’s spacetime consists of a scalar t, absolute time, which foliates M into instantaneous 3-spaces St , and Euclidean metrics ab (t) on these spaces. If the inverses  ab (t) are pushed forward onto M via the embeddings St ! M, a field s on M results which is assumed to be smooth. By construction, s t; ¼ 0

½4

The pair (t, s ) defines the ‘‘metric,’’ that is, times and distances, in NGT. Such a structure can arise from a Lorentzian metric, for example, the Minkowski metric  , by taking, component-wise, the limits c2  dx dx  dt2  c2 dx2 ! dt2 ;  ! s c!1

c!1

½5

which can be interpreted geometrically as ‘‘opening up the light cones’’ until they degenerate into doubly covered, spacelike hyperplanes, the Newtonian St ’s. The relations [5] suggest to write the GRT laws in terms of the rescaled temporal metric (  c2 ) t :¼ g

½6

½7b ½7c ½7d

T  ; ¼ 0

½7e

T  ¼ ð þ pÞU U þ ps

½7f

t U U ¼ 1

½7g

The Lorentz signature of g can be reexpressed thus: at each event (ffi spacetime point), there exists a ‘‘timelike’’ vector V  , that is, t V  V  > 0

The Cartan–Friedrichs Formalism

½7a

½7h

and V  X = 0 for X 6¼ 0 implies s X X > 0. The indices in eqn [7c] are raised, here and later, by s . Given a set of basic fields on M as listed below eqn [6], the laws [7] remain meaningful for all   0 and k  0. If  = 0, the ‘‘metrics’’ t and s degenerate (and the pair (t , s ) is then called a Galilei metric). Nevertheless, the definition of ‘‘timelike’’ will also be used in that case. Also, X will be said to be ‘‘spacelike’’ if and only if it can be written X = s  with s   > 0. While for  > 0, some of the relations [7] are redundant, this is not so for  = 0. For example, if  = 0, the two eqns [7b] are independent and do not determine the connection    uniquely, in contrast to the case  > 0. The connection will always be assumed to be symmetric. As will be discussed below, these formulas define a framework which serves to relate GRT to NGT and special relativity (SRT). First steps to formulate such a framework have been taken independently by E Cartan and KO Friedrichs. Therefore we call the structure defined by [7] the Cartan–Friedrichs formalism (CFF). We call it a ‘‘formalism’’ and not a ‘‘theory’’ since it is of interest solely as a tool to study relations between theories. Equations [7] remain unchanged if the basic fields and constants are rescaled according to a change of units for time, length, and mass. Here, two sets of basic fields related by such a rescaling will be considered as physically equivalent; they provide the

Newtonian Limit of General Relativity

same relations between observables. Thus,  and k have no physical meanings, but only their signs:  > 0; k > 0 :

GRT

 ¼ 0; k > 0 :

NGT

 > 0; k ¼ 0 :

SRT

½8b

g and w satisfy

Newton’s Theory in Spacetime Form Local Laws

Remarkably, for  = 0 and k > 0 the formulas [7] reproduce almost all the laws on which Newton’s theory of spacetime coupled to Euler’s fluid theory is based. This is summarized in the following: Theorem 1 Let eqn [7] hold on M with  = 0. Then there exists, for any event of M, a neighborhood U with coordinates (xa , t) such that, on U, t coincides with the absolute time, t = t,  t,  , and on the local slices U \ St , s defines Euclidean metrics ab with orthonormal coordinates xa , ab = ab . Vectors are spacelike iff they are tangent to St , otherwise they are timelike. Moreover, the slices are locally geodesic with respect to the connection    , and the induced connection on the slices is the flat connection associated naturally to ab . In addition, in the coordinate chart given by (xa , t), the connection components vanish except 0 a 0 and 0 b a ( = 0 a b ). Therefore, t is an affine parameter on timelike geodesics. Further, U0 = 1, and Ua = va is the 3-velocity of the fluid. If one writes 0 a b ¼: !a b

and uses 3-vector notation with (ga ) = g, (!23 , !31 , !12 ) = w, the timelike geodesics of    are given by € ¼ g þ 2x_  w x

(The last two lines are not sufficient to specify the theories within CFF; in connection with eqn [9] and in Theorem 2 they will be completed.) For discussing limit relations between theories, it is nevertheless useful to represent physical models in different scales. The physical interpretation of t , s in terms of time and distance and that of    through its geodesics as world lines of freely falling test particles, respectively, is the same in the three theories and can be stated in terms of the common framework CFF. For an obvious reason,  may be called causality constant. Note that  and k each occur in only one of the general laws of the theory, apart from the  in [7f]. The laws [7] are invariant under diffeomorphisms of the spacetime manifold. Those diffeomorphisms which map the basic fields of a solution into themselves form the symmetry group of that solution.

0 a 0 ¼: ga ;

505

½8a

Ñ  w ¼ 0;

Ñ  g þ 2w_ ¼ 0

Ñ  w ¼ 0; Ñ  g  2w 2 ¼ 4k

½8c ½8d

and the fluid’s equations of motion are _ þ Ñ  ðvÞ ¼ 0

½8e

ðv_ þ v  Ñv  g  2v  wÞ þ Ñp ¼ 0

½8f

A solution (g, w, , p, v) of eqns [8] on a local chart (xa , t) with t = diag(0, 0, 0, 1) and s = diag(1, 1, 1, 0) provides, via eqn [8a], the general local solution to eqns [7] for  = 0. The proof consists of many, mostly elementary steps which can be gathered from Ku¨nzle (1972) and Ehlers (1981). Given a solution to eqns (7) with  = 0 and k > 0, the coordinates x = (xa , t) referred to in the theorem are determined by the basic fields up to timedependent Euclidean motions, time translations, and time reflections. Such a coordinate system corresponds to a rigid reference frame. As the equation of motion for freely falling particles, eqn [8b], shows, g and w are to be interpreted as the acceleration and rotation fields which determine, relative to a rigid frame, the combined influence of inertia and gravity on particles encoded in the spacetime connection    . (This role of a connection in NGT was recognized by E Cartan.) This interpretation is supported by the (generalized) Euler equation [8f]. As claimed above already, eqns [7] almost reproduce the local laws of the Newton–Euler theory. Indeed, eqns [8] are those of the Newton– Euler theory, provided w depends on time only. Then and only then can the coordinate freedom be used to get nonrotating rigid coordinates with respect to which w = 0. The existence of such coordinates is indispensable for NGT since only with respect to them g is the gradient of a potential U which obeys Poisson’s equation, as shown by eqns [8c] and [8d]. The preceding argument shows that the CFF, specialized to  = 0, has to be restricted by a condition which implies w = w(t) in order to give the local laws of NGT. One such condition is R  ¼ 0

½9

506 Newtonian Limit of General Relativity

as can be verified by computing the curvature tensor via eqn [8a]. Equation [9] for  = 0 expresses that parallel transport of spacelike vectors along arbitrary spacetime curves is integrable, which corresponds to the behavior of free gyroscopes in NGT (in contrast to GRT). Of course, eqn [9] cannot be added to the CFF since it is incompatible with GRT. If, however, the CFF with  > 0, k = 0 is restricted by the condition [9], the spacetime and hydrodynamics of special relativity result.

which expresses covariantly that !a, b ! 0. Since w is harmonic on St (by eqns [8c], [8d]), this in turn implies !a, b = 0; thus, w depends on t only; the asymptotic condition [14] and the local laws imply eqn [9]. We may therefore employ rigid, nonrotating coordinates, w = 0. Then, by eqns [8a], [8c], [8d] the connection coefficients take the form    ¼ t; t; s U;

½15

and Global Laws for Isolated Systems

R   R  ¼ t; t; t; t;

The laws [8] and [9] do not determine the time evolution of the basic fields. Using nonrotating coordinates we put g = ÑU and replace eqns [8c], [8d] by Poisson’s equation U ¼ 4k

½10

In Newtonian dynamics, the potential only serves to compute forces depending instantaneously on the mass distribution. Traditionally, this is achieved by assuming  to have spatially compact support at each time and to solve eqn [10] by Z ðx þ y; tÞ 3 ðx; tÞ ¼ k d y ½11 jyj which implies the fall-off lim ðx; tÞ ¼ 0

jxj ! 1; t¼const:

½12

( will always be used for this solution of eqn [10]). To relate the foregoing isolation assumptions to corresponding assumptions in GRT as far as presently possible, it seems necessary to go back to the laws [7] restricted to  = 0 or the equivalent (3 þ 1) version [8] without the restriction [9]. If some global assumptions are added to eqns [8], eqns [10]–[12] can be deduced from the fourdimensional formulation. One first introduces the following two assumptions: (1) The hypersurfaces St of M (which, for  = 0, are the only spacelike hypersurfaces) are simply connected, complete Euclidean spaces. (2) On each St , the support of  is compact. Using coordinates (xa , t) as in the last subsection, with xa now ranging on R 3 , eqns [8a] imply X ð!a;b Þ2 t ½13 R  R   ¼ 2 a;b

Hence the sum is a 4-scalar, and since t is covariantly constant, it is possible to require R  R   ! 0

at spatial infinity

½14

XX ðU;ab Þ2

½16

a;b a;b

As before, we require R   R  ! 0

½17

and conclude U, ab ! 0. Since the Newtonian potential  of  also has this fall-off and U   is harmonic on St ffi R3 , the following conclusion can be obtained: Lemma 1 The laws [8] and the global conditions (1)–(2), [14], [17] imply: in rigid, nonrotating coordinates, the connection

    t; t; s ; ¼  

½18

is flat ( according to eqn [11] is a scalar, and the -term in eqn [18] is a tensor). In other words,    is asymptotically flat since the -term falls of as jxj2 . Because of this lemma, one can further restrict the coordinates (xa , t) by demanding    = 0. In physical terms this means: by switching to a new, ‘‘unaccelerated’’ frame of reference, one removes from the equations of motion a spatially homogeneous gravitational field which, in contrast to the -term in eqn [16], is not due to matter. The resulting coordinates are defined, up to Galilean transformations, t 0 ¼ t þ c0 0

0

0

0

xa ¼ Da b xb þ ua t þ ca 0

0

0

where ca , ua are constants and D is a constant orthogonal 3  3 matrix. These coordinates are called inertial ones; with respect to them the usual laws of Newtonian mechanics hold; see [8] with w = 0 and U = []. Theorem 2 (Ehlers 1981). The laws [7] of the CFF restricted to  = 0 and augmented by the global and asymptotic conditions (1)–(2), [14], [17], provide a generally covariant, four-dimensional

Newtonian Limit of General Relativity

formulation for the Newtonian theory of space, time, gravitation, and hydrodynamics. The possibility to split the connection  into a flat part which is independent of matter and a tensorial part depending on matter and given by the vector field g = s ,  (with  from eqn [11]), arises only from supplementing the local laws [7] by the global, resp. asymptotic, conditions (1)–(2), [14], [17] stated above. The introduction of inertial coordinates is then convenient, but not necessary. In noninertial, rigid frames of reference,    gives rise to inertial forces. It should be possible to define spatial asymptotic flatness in the CFF, but that has not been done. Remarks on Newtonian Cosmology

In cosmology, the conditions (2) and [17] of the last subsection are not appropriate. Instead one keeps the laws [7] and adds to them eqn [9], so that with respect to nonrotating coordinates the laws [8] with w = 0 and eqn [10] remain valid. Then, there are no longer inertial coordinate systems, and the potential U is not a 4-scalar. For a slightly different approach, see Ru¨ede and Straumann (1997). For the purpose of this article, the term ‘‘cosmological model’’ will be applied to those solutions of the laws [7] and [9] which satisfy  > 0 and which have a symmetry group which acts transitively on the set of world lines representing the motion of the fluid. This strong symmetry assumption determines the time-evolution even in the ‘‘Newtonian’’ case  = 0 in spite of the absence of an evolution equation for the gravitational field g.

Newtonian Limits of Families of GR Solutions The discussion in the sections ‘‘The Cartan– Friedrichs formalism’’ and ‘‘Newton’s theory in spacetime form’’ suggests the following: Definition 1 Let a family F () = (t (), . . .) of basic fields parametrized by , obeying the laws [7] of the CFF, be given for 0   < a. We assume the underlying manifolds M() to be open submanifolds of a fixed manifold M such that M(1 ) M(2 ) if S 1 < 2 and  M() = M. Then we write lim F ðÞ ¼ F ð0Þ

!0

½19

if the fields of F () and their first derivatives converge pointwise to those of F (0).

507

F (0) is then said to be a CF limit of the sequence of (-rescaled) solutions F () of GRT. If the fields of a -family of GR solutions ( > 0) and their first derivatives converge for  ! 0 locally uniformly, then the limit fields satisfy eqns [7]. If F (0) has the additional property [9], the limit is locally Newtonian. On the basis of the section ‘‘The Cartan– Friedrichs formalism’’ one may conjecture that if eqn [19] holds and the F () for  > 0 are spatially asymptotically flat, F (0) will represent an asymptotically flat Newtonian spacetime. Examples such as Example 1 below are in agreement with this conjecture, but a general proof is not known. Example 1 The interior solution for a static, spherically symmetric fluid ball of constant energy density (Schwarzschild 1916) is given by ds2 ¼

dr2 þ r2 ðd#2 þ sin2 # d’2 Þ a2 1  ð3a0  aÞ2 c2 dt2 4

 ¼ const: > 0; U¼

2 @t ; 3a0  a

p ¼ c2 aðrÞ ¼

a  a0 3a0  a

  8 2 2 1=2 kc r  1 3

a0 ¼ aðr0 Þ Inserting into these expressions the parameter  = c2 and treating  and r0 as -independent constants results in a -family with 0   < ((8=3)kr30 )1 . The limit solution represents a Newtonian fluid ball of constant mass density . The Schwarzschild vacuum fields belonging to these fluid balls also have the appropriate Newtonian limits. The resulting complete spacetimes are asymptotically flat. A dimensionless small parameter which could be used instead of  to measure the deviation of the GR solution from its Newtonian limit is the ratio of Schwarzschild radius and the geometric radius: 2kM 8 kr20 ¼ c 2 r0 3 c2 Example 2 A Friedmann–Lemaitre cosmological model of GR containing dust and radiation is given by ds2 ¼ R2 ðtÞ

ab d a d b ð1 

ð1=4ÞðE=c2 Þ

ab

2

a b Þ

where R(t) obeys   _R2  8 k M þ S ¼E 3 R c2 R 2

 c2 dt2

508 Newtonian Limit of General Relativity

M is a mass constant,  = M=R3 is the mass density of ‘‘dust,’’ S is an entropy constant, = S=R4 the energy density and p = (1=3) the pressure of radiation; and E is a constant of dimension (speed)2 . The world lines of the fluid elements are given by a = const. (Lagrangian comoving coordinates). Taking E, M, S constant and  = c2 as a parameter provides a -family of GR models with Newtonian limit. In the limit, t is the Newtonian time, and the spatial metric R2 ab d a d b describes an expanding Euclidian space R3 (if E  0) or an open ball of radius 2R(t) in it (if E > 0). In the coordinates ( a , t) the connection does not have the ‘‘Newtonian’’ components [8a], instead its nonvanishing compoa _ nents are 0 a b = (R=R) b . In local inertial coordinates a a x = R centered on the particle with a = 0 (which could be any particle because of the homogeneity of the model), the spatial metric is dx2 , and the connection components are Newtonian, with U = (2=3)kx2 and U = 4k. In the limit, the radiation no longer influences the expansion; one gets the Newtonian dust models (eqn [9] is satisfied). The connection is, of course, not asymptotically flat. The curvature tensor R   = (4=3)kt s exhibits homogeneity and isotropy. The Gaussian sectional curvature of the 3-space at time t is K = E=R2 . As a dimensionless smallness parameter one can take E=c2 . In the ‘‘open’’ models, with E  0, the coordinates a cover the whole 3-manifold of fluid particles, while in the ‘‘closed’’ case, E < 0, one particle, the antipode of a = 0 on the 3-sphere, is not covered. That particle is missing in the Newtonian limit model. In the Newtonian case the expanding Euclidian space R3 can be replaced by a torus; in the GR cases this is possible only for E = 0. Many examples of GR families with Newtonian limits are known (see, e.g., Ehlers (1997) and references therein). An example of a -family which has an almost Newtonian limit which does not satisfy eqn [9] is provided by NUT spacetimes (see Ehlers 1997), interpreted as due to a gravitomagnetic monopole (Lynden-Bell and Nouri-Zonez 1998).

Applications and Problems Can one construct, for a given Newtonian solution N, a -family of GR solutions which converges to N? Some answers are known and listed below. U Heilig (1995) has shown: given a solution to the Euler–Poisson equations representing a stationary, rigidly rotating, self-gravitating fluid body with its surrounding gravitational field, there exists

a -family of corresponding solutions to the Einstein–Euler system having the given solutions as its limit. The proof is based on the fact that one can reformulate eqns [1], [2] in terms of harmonic coordinates and new dependent gravitational variables instead of g such that the new equations given in Lottermoser (1992) are analytic in  and reduce, for  = 0, to the Euler–Poisson system. In the stationary case these equations are elliptic for   0. Using appropriate function spaces, Heilig shows, via the implicit function theorem, that a solution for  = 0 can be extended to small, positive values of . Since L Lichtenstein has constructed solutions as assumed in the theorem, the existence of GR solutions follows. The gravitational part of the system of equations referred to above is hyperbolic for  > 0, but becomes elliptic for  = 0, whereas the fluid equations remain hyperbolic. In spite of this difficulty Rendall (1994) has shown that -families of timedependent, asymptotically flat solutions to the Einstein–Vlasov system representing gravitating systems of collisionless particles have Poisson– Vlasov limits, and that any Poisson–Vlasov solution can be so obtained. Lottermoser (1992) succeeded in proving the existence of -families of solutions to the Einstein constraint equations which have Newtonian initial data as limits. Nothing seems to be known about solutions evolving from such data. Lottermoser has given an interesting discussion concerning possible extension of his work which apparently has gone unnoticed. Rendall (1992) has defined and analyzed postNewtonian expansions to Einstein’s equations and their solvability, assuming -familiespwhose t , s ffiffiffi are a few times differentiable in =  at = 0. He found that for low orders the equations have asymptotically flat solutions, but that at order 8 divergences occur for general Newtonian seed solutions. Modifications of the method to overcome these difficulties have been considered by Rendall and others; the problem is open. In cosmology, one uses homogeneous background models and studies their perturbations. The latter are frequently based on Newtonian equations. This can perhaps be justified as follows. According to Example 2 the fields of Friedmann– Lemaitre models differ from their Newtonian limits by arbitrarily small amounts uniformly in spacetime regions where the terms involving  are small, that is, pffiffiffiffiffiffi jEj S jxj RðtÞ

RðtÞ; Mc2 c

Noncommutative Geometry and the Standard Model

Additional conditions will be needed to ensure that Newtonian perturbations approximate relativistic ones and that gravitational wave perturbations can be neglected. See also: Cosmology: Mathematical Aspects; Einstein Equations: Exact Solutions; General Relativity: Overview; Gravitational Lensing; Shock Wave Refinement of the Friedman–Robertson–Walker Metric.

Further Reading Cartan E (1923) Sur les varie´te´s a` connexion affine et la the´orie de la relativite´ ge´ne´ralis´ee. Annales Scientifiques de l’Ecole Normale Supe´rieure XL: 325–412. Cartan E (1924) Sur les varie´te´s a` connexion affine et la the´orie de la relativite´ ge´ne´ralis´ee. Annales Scientifiques de l’Ecole Normale Supe´rieure XLI: 1–25. Ehlers J (1981) U¨ber den Newtonschen Grenzwert der Einsteinschen Gravitationstheorie. In: Nitsch J et al. (eds.) Grundlagenprobleme der modernen Physik, pp. 65–84. Mannheim: Bibliografisches Insitut Wissenschaftsverlag. Ehlers J (1997) Examples of Newtonian limits of relativistic spacetimes. Classical and Quantum Gravity 14: A119–A126.

509

Friedrichs KO (1927) Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und des Grenzu¨berganges vom Einsteinschen zum Newtonschen Gesetz. Mathematische Annalen 98: 566–575. Heilig U (1995) On the existence of rotating stars in general relativity. Communications in Mathematical Physics 166: 457–493. Ku¨nzle HP (1972) Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics. Annales de l’Institut Henri Poincare´ XVII: 337–362. Lottermoser M (1992) A convergent post-Newtonian approximation for the constraint equations in general relativity. Annales de l’Institut Henri Poincare´ 57: 279–317. Lynden-Bell D and Nouri-Zonoz M (1998) Classical monopoles: Newton, NUT space, gravomagnetic lensing, and atomic spectra. Reviews of Modern Physics 70: 427–445. Rendall AD (1992) On the definition of post-Newtonian approximations. Proceedings of the Royal Society of London Series A 438: 341–360. Rendall AD (1994) The Newtonian limit for asymptotically flat solutions of the Vlasov–Einstein system. Communications in Mathematical Physics 163: 89–112. Ru¨ede Ch and Straumann N (1997) On Newton–Cartan cosmology. Helvetica Physica Acta 318–335.

Noncommutative Geometry and the Standard Model and g = 3.289  1015 Hz, the famous Rydberg constant. Later quantum mechanics was discovered and allowed to derive the Balmer–Rydberg ansatz and to constrain its parameters:

T Schu¨cker, Universite´ de Marseille, Marseille, France ª 2006 Elsevier Ltd. All rights reserved.

q¼2

Introduction The aim of this contribution is to explain how Connes derives the standard model of electromagnetic, weak, and strong forces from noncommutative geometry. The reader is supposed to be aware of two other derivations in fundamental physics: the derivation of the Balmer–Rydberg formula for the spectrum of the hydrogen atom from quantum mechanics and Einstein’s derivation of gravity from Riemannian geometry. At the end of the nineteenth century, new physics was discovered in atoms, namely their discrete spectra. Balmer and Rydberg succeeded to put order into the fast-growing set of experimental results with the help of a phenomenological ansatz for the frequencies  of the spectral rays of, for example, the hydrogen atom,  ¼ gðnq2  nq1 Þ;

nj 2 N; q 2 Z; g 2 R

½1

The integer variables n1 and n2 reflect the discreteness of the spectrum. On the other hand, the discrete parameter q and the continuous parameter g were fitted by experiment: q = 2

and

g¼

me

e4

4h3 ð4 0 Þ2

½2

in beautiful agreement with the anterior experimental fit.

The Standard Model We propose to introduce the standard model (see Standard Model of Particle Physics) in analogy with the Balmer–Rydberg formula (Table 1).

Table 1 An analogy between atomic and particle physics elements Atomic physics New physics Ansatz Experimental fit Underlying theory

Particle physics

Discrete spectra

Forces mediated by gauge bosons Yang–Mills–Higgs  = g(n2q  n1q ) models q = 2, g = 3.289  1015 Hz Standard model Quantum mechanics

Noncommutative geometry

510 Noncommutative Geometry and the Standard Model The Yang–Mills–Higgs Ansatz

The variables of this Lagrangian ansatz are spin-1 particles A, spin-(1/2) particles decomposed into leftand right-handed components = ( L , R ) and spin0 particles ’. There are four discrete parameters, a compact real Lie group G, the ‘‘gauge group,’’ and three unitary representations on complex Hilbert spaces HL , HR , and HS . The spin-1 particles come in a multiplet living in the complexified of the Lie algebra of G, A 2 Lie(G)C . The left- and right-handed spinors come in multiplets living in the Hilbert spaces, L 2 HL , R 2 HR , respectively. The (Higgs) scalar is another multiplet, ’ 2 HS . The Yang–Mills–Higgs Lagrangian, together with its Feynman diagrams, is spelled out in Table 2. There are several continuous parameters: the gauge coupling g 2 Rþ , the Higgs self-couplings ,  2 R þ , and several Yukawa couplings gY 2 C. Table 2 The Yang–Mills–Higgs Lagrangian and its Feynman diagrams L[A; ; ’] = 12 tr(@ A @  A  @ A @  A )

þg tr(@ A [A ; A ])

þg 2 tr([A ; A ][A ; A ])

þ 6 @

þig (˜ L  ˜ R )(A ) 

Let us choose G = U(1) 3 ei . Its irreducible unitary representations are all one-dimensional, H = C 3 characterized by the charge q 2 Z: (ei ) = eiq . Then with qL = qR and HS = {0}, we get Maxwell’s theory with the photon (or gauge boson or 4-potential) A coupled to the Dirac theory of a massless spinor of electric charge qL whose (relativistic) wave function is pffiffiffiffi . The gauge coupling is given by g = e= 0 . Gauge invariance of the Yang–Mills–Higgs Lagrangian implies, via Noether’s theorem, electric charge conservation in this case (see Symmetries and Conservation Laws). Yang–Mills models are therefore simply nonabelian generalizations of electromagnetism where the abelian gauge group U(1) is replaced by any compact real Lie group. We insist on a compact group because all irreducible unitary representations of compact groups are finite dimensional. Finally, the Higgs scalar is added to give masses to spinors and gauge bosons via spontaneous symmetry breaking (see Symmetry Breaking in Field Theory). We use compact groups and unitary representations as (discrete) parameters. One motivation is Noether’s theorem and conserved quantities. The other comes from Wigner’s theorem: the irreducible unitary representations of the Poincare´ group are classified by mass and spin. Its orthonormal basis vectors are classified by energy–momentum and by the z-component of angular momentum. This theorem leads to the widely accepted definition of a particle as an orthonormal basis vector in a Hilbert space H carrying a unitary representation  of a group G. A precious property of the Yang–Mills–Higgs ansatz is its perturbative renormalizability necessary for fine-structure calculations like the anomalous magnetic moment of the muon. The Experimental Fit

þ12 @ ’ @  ’ þ12 gf(˜S (A )’) @  ’ þ @ ’ ˜ S (A )’g

Physicists have spent some 30 years and some 109 Swiss Francs to distill the fit (Particle Data Group 2004): G ¼ SUð2Þ  Uð1Þ  SUð3Þ=ðZ2  Z3 Þ HL ¼

þ12 g 2 (˜S (A )’) ˜ S (A )’

M  ð2; 16; 3Þ  ð2; 12; 1Þ

½3 ½4

1





þ’ ’’ ’

HR ¼

3  M  ð1; 23; 3Þ  ð1; 13; 3Þ  ð1; 1; 1Þ

½5

1

12 2 ’ ’

þgY

’ þ gY ’

HS ¼ ð2; 12; 1Þ

½6

Here (n2 , y, n3 ) denotes the tensor product of an n2 -dimensional representation of SU(2), ‘‘(weak) isospin,’’ an n3 -dimensional representation of SU(3), ‘‘color,’’ and the one-dimensional representation of

Noncommutative Geometry and the Standard Model

U(1) with ‘‘hyper charge’’ y. For historical reasons, the hypercharge is an integer multiple of 1/6. This is irrelevant: in the abelian case, only the product of the hypercharge with its gauge coupling is measurable, and we do not need multivalued representations, which are characterized by noninteger, rational hypercharges. In the direct sum, we recognize the three generations of fermions, the quarks, ‘‘up, down, charm, strange, top, bottom,’’ are SU(3) triplets, the leptons, ‘‘electron, , ’’ and their neutrinos, are color singlets. The basis of the fermion representation space is       u c t ; ; d s b  L  L  L  

e ; ; e L  L

L u R ; cR ; tR ; eR ;  R ; R dR ; sR ; bR ; The parentheses indicate isospin doublets. The eight gauge bosons associated with su(3) are called gluons. Warning: the U(1) is not the one of electric charge; it is called hypercharge, the electric charge is a linear combination of hypercharge and weak isospin. This mixing is necessary to give electric charges to the W bosons. The W þ and W  are pure isospin states, while the Z0 and the photon are (orthogonal) mixtures of the third isospin generator and hypercharge. As the group G contains three simple factors, there are three gauge couplings,

½7

g3 ¼ 1:218  0:01 The Higgs couplings are usually expressed in terms of the W and Higgs masses: mW ¼ 12g2 v ¼ 80:419  0:056 GeV pffiffiffipffiffiffi m’ ¼ 2 2 v > 98 GeV

½8

½9 pffiffiffi with the vacuum expectation value v := (1=2)= . Because of the high degree of reducibility of the spin(1/2) representations there are 27 complex Yukawa couplings. They constitute the fermionic mass matrix which contains the fermion masses and mixings: me ¼ 0:510998902  0:000000021 MeV mu ¼ 3  2 MeV; md ¼ 6  3 MeV m ¼ 0:105658357  0:000000005 GeV mc ¼ 1:25  0:1 GeV; ms ¼ 0:125  0:05 GeV m ¼ 1:77703  0:00003 GeV mt ¼ 174:3  5:1 GeV;

mb ¼ 4:2  0:2 GeV

For simplicity, we have taken massless neutrinos. Then mixing only occurs for quarks and is given by a unitary matrix, the Cabibbo–Kobayashi–Maskawa matrix 0

CKM

Vud :¼ @ Vcd Vtd

Vus Vcs Vts

1 Vub Vcb A Vtb

½10

whose matrix elements in terms of absolute values are: 0

1 0:9750  0:0008 0:223  0:004 0:004  0:002 @ 0:222  0:003 0:9742  0:0008 0:040  0:003 A 0:009  0:005 0:039  0:004 0:9992  0:0003 ½11

Mathematically, the Cabibbo–Kobayashi–Maskawa matrix comes from a polar decomposition of the mass matrix. The physical meaning of the quark mixings is the following: when a sufficiently energetic W þ decays into a u quark, this u quark  quark with probis produced together with a d 2 ability jVud j , an s quark with probability jVus j2 ,  quark with probability jV j2 . and a b ub The phenomenological success of the standard model is phenomenal: with only a handful of parameters, it reproduces correctly some millions of experimental numbers: cross sections, lifetimes, branching ratios.

Noncommutative Geometry

g2 ¼ 0:6518  0:0003 g1 ¼ 0:3574  0:0001

511

Noncommutative geometry is an analytic geometry generalizing three other geometries that also had important impact on our understanding of forces and time. Let us start by briefly recalling the three forerunners (Table 3). Euclidean geometry underlies Newton’s mechanics as a geometry in the space of positions. Forces are described by vectors living in the same space and the Euclidean scalar product is needed to define work and potential energy. Time is not part of geometry – it is absolute. This point of view is abandoned in special relativity unifying space and time into Minkowskian geometry. This new point of view allows to derive the magnetic

Table 3 Four nested analytic geometries Geometry Euclidean Minkowskian Riemannian Noncommutative

Force R ~  d~ E= F x ~ 0 = 1 c 2 ~ E , 0 ) B, 0 Coriolis $ gravity Gravity ) YMH,  = 13 g22

Time Absolute Universal Proper,

 1040 s

512 Noncommutative Geometry and the Standard Model

field from the electric field as a pseudoforce associated with a Lorentz boost. Although time becomes relative, one can still imagine a grid of synchronized clocks, that is, a universal time. The next generalization is ‘‘Riemannian geometry = curved spacetime.’’ Here gravity can be viewed as the pseudoforce associated with a uniformly accelerated coordinate transformation. At the same time, universal time loses all meaning and we must content ourselves with proper time. With today’s precision in time measurement, this complication of life becomes a bare necessity, for example, the global positioning system (GPS). Our last generalization is ‘‘noncommutative geometry = curved space(time) with an uncertainty principle.’’ As in quantum mechanics, this uncertainty principle is introduced via noncommutativity.

p

•

h/2

x Figure 1 The first example of noncommutative geometry.

the four-component spinor consisting of left- and righthanded, particle and antiparticle wave functions. Unlike the Hamiltonian, the Dirac operator does not lie in A, but it is still an operator on H. In Euclidean spacetime, the Dirac operator is also self-adjoint, 6@  = 6@ .

Quantum Mechanics

Spectral Triples

Consider the classical harmonic oscillator. Its phase space is R 2 with points labeled by position x and momentum p. A classical observable is a differentiable function on phase space such as the total energy p2 =(2m) þ kx2 . Observables can be added and multiplied, and they form the algebra C1 (R2 ), which is associative and commutative. To pass to quantum mechanics, this algebra is rendered noncommutative by means of a noncommutation relation for the generators x and p: [x, p] = i h1. Let us call A the resulting algebra ‘‘of quantum observables.’’ It is still associative, and has an involution  (the adjoint or Hermitian conjugation) and a unit 1. Of course, there is no space anymore of which A is the algebra of functions. Nevertheless, we talk about such a ‘‘quantum phase space’’ as a space that has no points or a space with an uncertainty relation. Indeed, the noncommutation relation implies Heisenberg’s uncertainty relation xp h=2 and tells us that points in phase space lose all meaning; we can only resolve cells in phase space of volume h=2, see Figure 1. To define the uncertainty a for an observable a 2 A, we need a faithful representation of the algebra on a Hilbert space, that is, an injective homomorphism  from A into the algebra of operators on H. For the harmonic oscillator, this Hilbert space is H = L2 (R). Its elements are the wave functions (x), squareintegrable functions on configuration space. Finally, the dynamics is defined by the Hamiltonian, a selfadjoint observable H = H  2 A via Schro¨dinger’s equation (i h@=@t  (H)) (t, x) = 0. Here time is an external parameter; in particular, time is not an observable. This is different in the special-relativistic setting, where Schro¨dinger’s equation is replaced by Dirac’s equation 6@ = 0. Now the wave function is

Noncommutative geometry (Connes 1994, 1995) does to a compact Riemannian spin manifold M what quantum mechanics does to phase space. A noncommutative geometry is defined by the three purely algebraic items (A, H, 6@ ), called a spectral triple. A is a real, associative, and possibly noncommutative involution algebra with unit, faithfully represented on a complex Hilbert space H, and 6@ is a self-adjoint operator on H. As the spectral triple, also the axioms linking its three items are motivated by relativistic quantum mechanics. When A = C1 (M), the functions on a Riemannian spin manifold M, represented on spinors , and 6@ is the gravitational Dirac operator, one has a spectral triple. The converse is also true when A is a suitable commutative algebra (Connes 1996), but the axioms make sense even when A is not commutative. As for quantum phase space, Connes defines a noncommutative geometry by a spectral triple whose algebra is allowed to be noncommutative and he shows how important properties like dimensions, distances, differentiation, integration, general coordinate transformations, and direct products generalize to the noncommutative setting. As a bonus, the algebraic axioms of a spectral triple, commutative or not, include discrete, that is, zero-dimensional spaces that now are naturally equipped with a differential calculus. These spaces have finite-dimensional algebras and Hilbert spaces, meaning that their algebras are just matrix algebras. An ‘‘almost commutative geometry’’ is defined as a direct product of a four-dimensional commutative geometry, ‘‘ordinary spacetime,’’ by a zero-dimensional noncommutative geometry, the ‘‘internal space.’’ If the

Noncommutative Geometry and the Standard Model

latter is also commutative, for example, the ordinary two-point space, then the direct product describes a two-sheeted universe or a Kaluza–Klein space whose fifth dimension is discrete, (Madone 1995). In general, the axioms of spectral triples imply that the Dirac operator of the internal space is precisely the fermionic mass matrix. As a generic example, here is the internal spectral triple underlying the standard model with one generation of quarks and leptons. The algebra A = H  C  M3 (C) 3 (a, b, c) contains quaternions, that is, 2  2 matrices of the form   x  y a¼ ; x; y 2 C  y x complex numbers b and complex 3  3 matrices c. The Hilbert space is 30-dimensional, where we count particles and antiparticles ( c ) separately: H = HL  HR  HcL  HcR = C8  C7  C8  C7 . The representation is block-diagonal, with the four blocks L ðaÞ :¼ 0

a  13

0

0

a

cL ðb; cÞ :¼

0

b13

B B R ðbÞ :¼ B 0 @ 0 

!

0

0

0

c B c R ðb; cÞ :¼ @ 0 0

0 c 0

½12

C C 0C A  b

 3 b1

12  c 0

1

0  2 b1 1 0 C 0A  b



½13

The internal Dirac operator (= fermionic mass matrix) contains two quark masses mu , md and one lepton mass me , and no mixing: 0

0 B M B D¼B @ 0 0 0 B B M¼B @

mu 0

M 0

0 0

0 0

0  M

0 md 0

1 0 0 C C C A M 0 1

 0

 13 

C C C 0 A me

½14

513

These matrices look rather ad hoc; they are not. They define an irreducible spectral triple and, for a given algebra, there is only a finite number of such triples.

The Spectral Action

Chamseddine and Connes (1997) generalize general relativity to noncommutative spacetimes in two strokes, kinematics and dynamics. They explicitly compute this generalization for almost commutative geometries. Kinematics In noncommutative geometry, general coordinate transformations are algebra automorphisms lifted to the Hilbert space of spinors. For almost commutative geometries, these transformations are precisely general coordinate transformations of ordinary spacetime and gauge transformations. Now remember how Einstein uses the equivalence principle to produce ‘‘gravity = curvature’’ starting from the flat metric, which in Connes’ language is the ordinary flat Dirac operator. When applied to an almost commutative geometry (Connes 1996), the equivalence principle produces again a curved metric via the ordinary coordinate transformations on M, while the gauge transformations applied to the fermionic mass matrix produce a new field, the Higgs scalar ’. For the example above, this field is precisely the isospin doublet, color singlet with hypercharge 1=2 of eqn [6]. Gauge transformations also apply to the ordinary Dirac operator, thereby producing the gauge fields A. Dynamics The group of generalized coordinate transformations allowed us to construct the configuration space. In the almost commutative case it consists of Riemannian metrics, gauge fields, and Higgs scalars. We now want a dynamics on this configuration space. Of course, we want this dynamics to be invariant under the group of generalized coordinate transformations. Note that the spectrum of the Dirac operator is invariant under this group and Chamseddine and Connes (1997) define the spectral action as a regularized partition function of these eigenvalues. On almost commutative geometries, the spectral action is equal to the Einstein–Hilbert action plus the Yang–Mills–Higgs ansatz (Figure 2). In other words, almost commutative geometry explains the forces mediated by gauge bosons and Higgs scalars as pseudoforces accompanying the gravitational force in the same way that Minkowskian geometry (i.e., special relativity) explains the magnetic force as a pseudoforce accompanying the electric force.

514 Noncommutative Geometry and the Standard Model

Noncommutative geometry

??

Connes Almost commutative geometry

Connes

Gravity + Yang–Mills–Higgs ansatz + constraints

Einstein Riemannian geometry

Gravity

Figure 2 Deriving the Yang–Mills–Higgs ansatz from gravity.

Yang–Mills–Higgs NCG

Standard model

Left–right symmetry

GUT Supersymmetry

1.4 1.2 1 0.8 0.6 0.4 0.2

g3

g2

(3λ)1/2 mz

109 GeV

Λ

E

Figure 3 Constraints inside the ansatz.

Figure 4 Running coupling constants.

There are constraints on the discrete and continuous parameters in the Yang–Mills–Higgs ansatz deriving from the spectral action Figure 3. In particular, if we consider only irreducible spectral triples and among them only those which produce nondegenerate fermion masses compatible with renormalization, then we only get the standard model with one generation of quarks and leptons, with a massless neutrino and with an arbitrary number of colors, and a few submodels thereof. More than one generation and neutrino masses are possible but imply reducible triples. However, in at least one generation, the neutrino must remain purely left and massless. For the standard model with N generations and Nc colors, we have the constraints g2Nc = g22 = (9=N) on the continuous parameters. If we put N = Nc = 3 and if we believe in the popular ‘‘big desert’’ then these constraints yield a ‘‘unification scale’’  = 1017 GeV at which the uncertainty relation in spacetime should become manifest,  =  h=, and a Higgs mass of m’ = 171.6  5 GeV for mt = 174.3  5.1 GeV (see Figure 4). It is clear that almost commutative geometries only scratch the surface of a gold mine. May we hope that a genuinely noncommutative geometry will solve our present problems with quantum field theory and quantum gravity?

See also: Compact Groups and Their Representations; Dirac Fields in Gravitation and Nonabelian Gauge Theory; Effective Field Theories; General Relativity: Overview; Hopf Algebras and q-Deformation Quantum Groups; Positive Maps on C-Algebras; Quantum Hall Effect; Standard Model of Particle Physics; Symmetries and Conservation Laws; Symmetry Breaking in Field Theory; von Neumann Algebras: Introduction, Modular Theory, and Classification Theory.

Further Reading Chamseddine A and Connes A (1997) The spectral action principle. Communications in Mathematical Physics 186: 731 (hep-th/9606001). Connes A (1994) Noncommutative Geometry. San Diego: Academic Press. Connes A (1995) Noncommutative geometry and reality. Journal of Mathematical Physics 36: 6194. Connes A (1996) Gravity coupled with matter and the foundation of noncommutative geometry. Communications in Mathematical Physics 155: 109 (hep-th/9603053). Gracia-Bondı´a JM, Va´rilly JC, and Figueroa H (2000) Elements of Noncommutative Geometry. Boston: Birkha¨user. Kastler D (2000) Noncommutative geometry and fundamental physical interactions: the Lagrangian level. Journal of Mathematical Physics 41: 3867. Landi G (1997) An Introduction to Noncommutative Spaces and Their Geometry, hep-th/9701078. Berlin: Springer.

Noncommutative Geometry from Strings 515 Madore J (1995) An Introduction to Noncommutative Differential Geometry and Its Physical Applications. Cambridge: Cambridge University Press. Martı´n CP, Gracia-Bondı´a JM, and Va´rilly JC (1998) The standard model as a noncommutative geometry: the low mass regime. Physics Reports 294: 363 (hep-th/9605001). O’Raifeartaigh L (1986) Group Structure of Gauge Theories. Cambridge: Cambridge University Press.

Scheck F, Werner W, and Upmeier H (eds.) (2002) Noncommutative Geometry and the Standard Model of Elementary Particle Physics, Lecture Notes in Physics, vol. 596. Berlin: Springer. Schu¨cker T (2005) Forces from Connes’ geometry. In: Bick E and Steffen F (eds.) Topology and Geometry in Physics, Lecture Notes in Physics, vol. 659, hep-th/0111236. Berlin: Springer. The Particle Data Group, Particle Physics Booklet and http:// pdg.lbl.gov

Noncommutative Geometry from Strings Chong-Sun Chu, University of Durham, Durham, UK ª 2006 Elsevier Ltd. All rights reserved.

Noncommutative Geometry from String Theory The first use of noncommutative geometry in string theory appears in the work of Witten on open-string field theory where the noncommutativity is associated with the product of open-string fields. Noncommutative geometry appears in the recent development of string theory in the seminal work of Connes, Douglas, and Schwarz where they constructed and identified the compactification of Matrix theory on a noncommutative torus. Matrix Theory Compactification and Noncommutative Geometry

The matrix theory (M-theory) is an 11-dimensional quantum theory of gravity which is believed to underlie all superstring theories. Banks, Fischler, Shenker, and Susskind proposed that the large N limit of the supersymmetric matrix quantum mechanics of N D0-branes should describe the M-theory compactified on a lightlike circle. Compactification of the M-theory on a torus can be easily achieved by considering the torus as the quotient space R d =Zd with the quotient conditions j

Ui1 Xj Ui ¼ Xj þ i 2Ri ;

i ¼ 1; . . . ; d

½1

Here Ri are the radii of the torus. The unitary translation generators Ui generate the torus. They satisfy Ui Uj = Uj Ui . T-dualizing the D0 brane system, eqn [1] leads to the dual description as a (d þ 1)-dimensional supersymmetric gauge theory on the dual toroidal D-brane. A noncommutative torus Td is defined by the modified relations Ui Uj ¼ eiij Uj Ui

½2

where ij specify the noncommutativity. Compactification on a noncommutative torus can be easily

accommodated and leads to noncommutative gauge theory on the dual D-brane. The parameters ij can be identified with the components Cij of the 3-form potential in M-theory. Since M-theory compactified on a circle leads to IIA string theory, the components Cij correspond to the Neveu–Schwarz (NS) B-field Bij in IIA string theory. The physics of the D0 brane system in the presence of an NS B-field can also be studied from the viewpoint of IIA string theory. This led Douglas and Hull to obtain the same result that a noncommutative field theory lives on the D-brane. Toroidally compactified IIA string theory has a T-duality group SO(d, d; Z). The T-duality symmetry gets translated into an equivalence relation between gauge theories on the noncommutative torus: a gauge theory on the noncommutative torus Td is equivalent to that on the noncommutative torus Td0 if their noncommutativity parameters and metrics are related by a T-duality transformation. For example, 0 ¼ ðA þ BÞðC þ DÞ1 ;   A B 2 SOðd; d; ZÞ C D

½3

It is remarkable that the T-duality acts within the field theory level, rather than mixing up the field theory modes with the string winding states and other stringy excitations. Mathematically, eqn [3] is precisely the condition for the noncommutative tori Td and Td0 to be Morita equivalent. Open-String in B-Field

It was soon realized that the D-brane does not necessarily need to be toroidal in order to be noncommutative. A direct canonical quantization of the open-string system shows that a constant B-field on a D-brane leads to noncommutative geometry on the D-brane world volume. Consider an open string moving in a flat space with metric gij and a constant NS B-field. In the presence of a Dp brane, the components of the B-field not along the brane can be gauged away; thus, the B-field can

516 Noncommutative Geometry from Strings

have effects only in the longitudinal directions along the brane. The world-sheet (bosonic) action for this part is Z 1 d2  S¼ 40     gij @a xi @ a xj  20 Bij ab @a xi @b xj ½4 where i, j = 0, 1, . . . , p is along the brane. It is easy to see that the boundary condition gij @ xj þ 2i0 Bij @ xj = 0 at  = 0,  is not compatible with the standard canonical quantization [xi (, ), xj (, 0 )] = 0 at the boundary. Taking the boundary condition as constraints and performing canonical quantization, one obtains the commutation relations ½aim ; ajn  ½xi0 ; xj0 

ij

¼ mG mþn ; ¼ i

½xi0 ; pj0 

ij

¼ iG ;

ij

½5

Here, the open-string mode expansion is xi ð; Þ ¼ xi0 þ 20 ðpi0   20 ðg1 BÞij pj0 Þ pffiffiffiffiffiffiffi X ein þ 20 n n6¼0    iain cos n  20 ðg1 BÞij ajn sin n Gij and ij are the symmetric and antisymmetric parts of the matrix (g þ 20 B)1ij :  ij 1 1 ij G ¼ g g þ 20 B g  20 B  ij ½6 1 1 ij 0 2 B  ¼ ð2 Þ g þ 20 B g  20 B It follows from [5] that the boundary coordinates xi  xi (, 0) obey the commutation relation ½xi ; xj  ¼ iij

½7

Relation [7] implies that the D-brane world volume, where the open-string endpoints live, is a noncommutative manifold. One may also start with the closed-string Green function and let its arguments to approach the boundary to obtain the open-string Green function i hxi ðÞxj ð 0 Þi ¼ 0 Gij lnð   0 Þ2 þ ij ð   0 Þ 2

½8

where () is the sign of . From [8], one can again extract the commutator [7]. Gij = gij  (20 )2 (Bg1 B)ij is called the open-string metric since it controls the short-distance behavior of open strings. In contrast, the short-distance behavior for closed strings is controlled by the closed-string metric gij . One may also treat

the boundary B-term in [4] as a perturbation to the open-string conformal field theory and from which one may extract [8] from the modified operator product expansion of the open-string vertex operators. D-branes in the Wess–Zumino–Witten model provide another example of noncommutative geometry. In this case, the background is not flat since there is a nonzero H = dB  k1=2 , where k is the level. Examining the vertex operator algebra, one obtains that D-branes are described by nonassociative deformations of fuzzy spheres with nonassociativity controlled by 1=k. String Amplitudes and Effective Action

The effect of the B-field on the open-string amplitudes is simple to determine since only the xi0 commutation relation is affected nontrivially. For example, the noncommutative gauge theory can be obtained from the tree-level string amplitudes readily. For tree and one loop, the vertex operator formalism can be used. Generally, the vertex operator can be inserted at either the  = 0 or  =  boundary, where the string has zero mode parts xi0 j and yi0  xi0  (20 )2 (g1 B)ij p0 , respectively. The commutation relations are ½xi0 ; xj0  ¼ iij ; ½yi0 ; yj0  ¼ iij

½xi0 ; y0j  ¼ 0; ½9

The difference in the commutation relation for x0 and y0 implies that the two boundaries of the open string have opposite commutativity. This fact is not so important for tree-level calculations since one can always choose to put all the interactions at, for example, the  = 0 boundary. Collecting all these zero mode parts of the vertex operators, one obtains a phase factor PN I J P a 1 2 N ði=2Þ p p I ðÞ sup

½7

where  > 0 and :  ðxÞ¼ log ðe þ jxjÞ;

x 2 R3

½8

We denote by X  the set of the phase points X such that Q (X) < 1. It is possible to prove that for any   1/3, X  has full measure with respect to any Gibbs measure. We define the partial dynamics t 7! X(n) (t) as the solutions to eqn [1] obtained by neglecting all the particles which are initially outside the ball of radius n and centered at the origin. Theorem If X 2 X  there exists a unique flow X ! X(t) 2 X ð3/2Þ satisfying eqn [1] with X(0) = X. Moreover, the partial dynamics locally converges to X(t) as n ! 1. The result has been extended to bounded superstable long-range interactions. The (nontrivial) proof is based on several steps: we introduce a mollified version on the local energy and study its evolution in time under the partial dynamics. The energy conservation allows us to prove that the local energy grows at most as the cube of the maximal velocity. On the other hand, a suitable time average allows us to control the maximal velocity via the local energy in an appropriate way. The result is achieved by letting n ! 1.

Long-Time Behavior Existence and locality of the dynamics is only a first, preliminary, step. The next and much more subtle question concerns the asymptotic (in time) and the statistical properties of the motion. Here, the main problem is the absence of simple but nontrivial models. Let us explain this point by a comparison with the situation in equilibrium statistical mechanics. In this case, even the simpler model, the free-particle system, exhibits all the relevant

542 Nonequilibrium Statistical Mechanics: Dynamical Systems Approach

thermodynamical properties of real systems away from the critical regime. In fact, the effort is often reduced to rigorously proving that the real systems away from the critical region behave as a free-particle system. The presence of the interaction is instead essential to describe phase transitions. In the case of nonequilibrium statistical mechanics there are very few solvable models (free particles, chain of oscillators, hard-core system in one dimension), and typically they do not catch the essential properties of the real systems. For example, let us consider a system which is close to equilibrium and ask whether it converges to the corresponding Gibbs state. Two possible mechanisms usually come together: the dispersive properties of the matter (by which perturbations ‘‘escape’’ to infinity) and the mixing properties (by which perturbations are ‘‘spread’’ and disappear). The former is present also in the free-particle system, being responsible of its ergodic properties. The latter requires a deep analysis of the dynamics of interacting-particle systems and it is too difficult to be analyzed except in rare cases. We just mention the case of systems with instantaneous interaction, which are simple enough to be studied but nevertheless exhibit a nontrivial long-time behavior. We recall in particular the famous Sinai’s billiard: a particle moving freely in a two-dimensional torus except for elastic collisions with the boundary of a convex obstacle. As proved by Sinai (1970), this system has strong ergodic properties. Sinai’s billiard can be proved to be equivalent to the ‘‘Lorentz gas’’ in which the obstacles are dislocated in a periodic way. Bunimovich and Sinai (1981) proved that when the obstacles are close enough to each other, the diffusive (weak) limit of the particle motion is the Wiener process. This remarkable result gives a rigorous derivation of Brownian motion from a Hamiltonian system. More recently, similar questions have been investigated in the case of a charged particle subject to a constant electric field and interacting with a medium described by a particle system. Several rigorous results have been obtained on this subject. We only recall those by Boldrighini and Soloveitchik (1995, 1997). In the context of a simplified model, the asymptotic motion of the charged particle is described as a drift plus a Brownian motion, and the Einstein relation between the drift and the diffusion constant is established.

Mean-Field Limit The validity of any model is related to some approximation limit. In statistical mechanics, we

encounter one of the most important ones, the ‘‘thermodynamical limit,’’ used to stress the effect of large number of particles. Here we briefly discuss the ‘‘mean-field limit.’’ For the kinetic, Boltzmann–Grad limit, see Boltzmann Equation (Classical and Quantum) and Kinetic Equations. We consider N particles of mass m mutually interacting via the force F. The equations of motion are P 8 Fðxi ðtÞ  xj ðtÞÞ xi ðtÞ ¼ < m€ j¼1;...;N:j6¼i

:

½9

ðxi ð0Þ; x_ i ð0ÞÞ ¼ ðxi ; vi Þ i ¼ 1; . . . ; N

We consider a system with N very large, the mass m of each particle very small, and the interaction very weak. An interesting situation arises when the quantities N, m, and F are linked by the relations m¼

M ; N

F¼

G N2

½10

for some function G. Of course, M is the total mass of the system. We are interested in investigating the limit N ! 1. We assume that the initial data are P chosen in a way that the empirical measure N  1 i xi vi weakly converges (as N ! 1) to the absolutely continuous measure f0 (x, v) dx dv with some smooth density f0 (x, v). We ask whether at P some positive time t > 0 the empirical measure N 1 i xi (t) vi (t) weakly converges to f (x, v, t) dx dv with a density f (x, v, t) satisfying some limiting evolution equation. Formally, it is easy to find this equation: by the Liouville theorem, a continuous medium in which each point moves under the action of an acceleration field behaves as an incompressible fluid. The continuity equation becomes @t f ðx; v; tÞ þ v  rx f ðx; v; tÞ þ E  rv f ðx; v; tÞ ¼ 0 f ðx; v; 0Þ ¼ f0 ðx; vÞ

½11

where Eðx; tÞ ¼

Z R3

ðx; tÞ ¼

dy Gðx  yÞðy; tÞ

Z R3

dv f ðx; v; tÞ

½12

½13

This equation can be studied by following the characteristics, for which it suffices to look at the pair of functions ðx; vÞ 7! ðXðx; v; tÞ; Vðx; v; tÞÞ;

f0 ðx; vÞ 7! f ðx; v; tÞ

Nonequilibrium Statistical Mechanics: Dynamical Systems Approach

where (x, v) 2 R 3  R 3 and t 2 R, solutions of

X dr i ¼ "d Gðr i  r j Þ d 2 j:j6¼i

_ _ Xðx; v; tÞ ¼ Vðx; v; tÞ; Vðx; v; tÞ ¼ Eðx; tÞ Xðx; v; 0Þ ¼ x; Vðx; v; 0Þ ¼ v

½14

543

½16

Then eqn [14] is the limiting equation as " ! 0.

f ðXðx; v; tÞ; Vðx; v; tÞ; tÞ ¼ f0 ðx; vÞ This is a weak formulation of eqn [11], in the sense that any smooth solution to eqn [11] satisfies eqn [14], but this last equation in meaningful also for nonsmooth functions. This is a weak version of the Vlasov equation and its measure solutions will play an important role in the sequel. Equations [11]–[14] are called Vlasov equations, after Vlasov, who first introduced them in plasma physics. They have a Hamiltonian structure and conserve several quantities: the total mass, the total energy, the Liouville measure dx dv, and in general each moment of this measure. The existence and uniqueness of the solutions has been studied in many papers. Two cases have to be considered, depending on whether the total mass Z M¼ dx dv f0 ðx; vÞ ½15 R6

is finite or not. We start with the first case. If the interaction G is bounded, the analysis is easy. On the other hand, in plasma physics one deals with the Coulomb interaction, which is singular at the origin. In this case (where eqn [11] is usually called the Vlasov–Poisson equation), existence and uniqueness can still be proved, but it is not straightforward, especially in dimension d = 3. The case with the complete Lorentz force, also taking into account the relativistic effect, is much more difficult. For infinite total mass, the problem has been solved recently in three (or lower) dimensions for bounded, non-negative, finite-range interactions, and in two dimensions for singular Helmholtz interactions. Another way to relate the Vlasov equation with the particle systems is to consider the usual transition from microscopic to macroscopic evolutions based on a separation between microscopic and macroscopic scales. Moreover, the force between the particles is due to a long-range pair interaction of the Kac type, in which the range parameter tends to infinity as the ratio "1 between the macro and the micro spatial scale: F(xi  xj ) = "2dþ1 G("xi  "xj ). Finally, the mass of the particles is proportional to "d : m = "d . After rescaling space and time by a factor ", in the macroscopic variables (, r) = ("t, "x), the equations of motion (eqn [9]) become

Other Models We mention another model of larger interest. We introduce it in the simplest formulation, leaving possible generalizations to the reader. We consider an infinite chain of anharmonic oscillators, with Hamiltonian H given by Hðq; pÞ

2 3 X p2 X 4 i þ aq4 þ b ¼ ðqi  qj Þ2 þ cq2i þ d5 i 2m i2Z j:jijj¼1 ½17

where qi , pi 2 R, a  0, b, c, d > 0. When a = 0, it reduces to the well-known chain of harmonic oscillators, which is integrable and widely studied in the literature. The time evolution defined by the Hamiltonian in eqn [17] exists and it is unique for initial data chosen in a set large enough to be the support of any reasonable thermodynamic (equilibrium or nonequilibrium) state. This can be achieved by proving integral inequalities for the ‘‘Lyapunov function’’  2  pi 1 4 Lðq; pÞ ¼ sup þ aqi þ d jij þ1 2m i2Z It is interesting to note that uniqueness holds only in a class of data such that the position of the ith oscillator does not increase too much as jij ! 1. For example, besides the stationary solution qi (t) = 0, i 2 Z, we can construct a different solution corresponding to the same initial conditions qi (0) = 0, pi (0) = 0, i 2 Z. In fact, by imposing q0 (t) = t2 and qi (t) = qi (t), we can solve recursively the equations of motion and obtain a nonzero solution qi (t), which however increases superexponentially as jij ! 1. The Hamiltonian dynamical systems (classical or quantum) are surely quite faithful descriptions of real systems, but they are too difficult to study. Mainly it is not known how to prove good dynamical mixing for deterministic evolutions with many degrees of freedom. Therefore, stochastic evolutions have been introduced to model the real systems. More precisely, one renounces a full description of the microscopic dynamics, introducing simplified models where the effects of the

544 Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

‘‘hidden degrees of freedom’’ are taken into account by adding suitable stochastic forces. Many useful results have been obtained, which show that these stochastic model systems exhibit a macroscopic behavior much closer to that observed in nature. The main criticism concerns the role of stochasticity, which in these models is introduced ab initio. In other words, if one believes that the statistical properties of the deterministic motion on the small scale determine the collective behavior of systems with many degrees of freedom, then these properties do have to be proved for a true understanding of nonequilibrium phenomena. See also: Adiabatic Piston; Boltzmann Equation (Classical and Quantum); Fourier Law; Kinetic Equations; Nonequilibrium Statistical Mechanics (Stationary): Overview; Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations.

Further Reading Boldrighini C and Soloveitchik M (1995) Drift and diffusion for a mechanical system. Probability Theory and Related Fields 103: 349–379. Boldrighini C and Soloveitchik M (1997) On the Einstein relation for a mechanical system. Probability Theory and Related Fields 107: 493–515. Bunimovich LA and Sinai YG (1981) Statistical properties of Lorentz gas with periodic configuration of scatterers. Communications in Mathematical Physics 78: 479–497.

Caglioti E, Marchioro C, and Pulvirenti M (2000) Non-equilibrium dynamics of three-dimensional infinite particle systems. Communications in Mathematical Physics 215: 25–43. Cornfeld IP, Fomin SV, and Sinai YG (1982) Ergodic theory. In: Artin M et al. (eds.) Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 245. New York: Springer. Dobrushin RL and Fritz J (1977) Non-equilibrium dynamics of onedimensional infinite particle systems with a hard-core interaction. Communications in Mathematical Physics 55: 275–292. Fritz J and Dobrushin RL (1977) Non-equilibrium dynamics of twodimensional infinite particle systems with a singular interaction. Communications in Mathematical Physics 57: 67–81. Lanford OE III (1968) Classical mechanics of one-dimensional systems with infinitely many particles. I An existence theorem. Communications in Mathematical Physics 9: 176–191. Lanford Oscar E, III (1975) Time evolution of large classical systems. In: Ehlers J, Hepp K, and Weidendmu¨ller HA (eds.) Dynamical Systems, Theory and Applications (Recontres, Battelle Res. Inst., Seattle, WA, 1974), Lecture Notes in Physics, vol. 38, pp. 1–111. Berlin: Springer. Neunzert H (1984) An introduction to the nonlinear Boltzmann– Vlasov equation. In: Dold A and Eckmann B (eds.) Kinetic Theories and the Boltzmann Equation (Montecatini, 1981), Lecture Notes in Mathematics, vol. 1048, pp. 60–110. Berlin: Springer. Sinai TG (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. (Russian) Uspehi Mat. Nauk 25: 141–192. Spohn H (1991) Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics. Berlin: Springer. Sza´sz D (ed.) (2000) Hard Ball Systems and the Lorentz Gas. Encyclopædia of Mathematical Sciences, vol. 101. Berlin: Springer.

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations R Livi, Universita` di Firenze, Sesto Fiorentino, Italy ª 2006 Elsevier Ltd. All rights reserved.

Introduction Nonequilibrium statistical mechanics concerns a wide range of fundamental problems and applications. Perturbative methods are quite effective for approaching weakly nonlinear problems, usually relying upon effective coarse-grained equations. The attempt of obtaining a microscopic description of genuine nonlinear problems demands the combined use of theoretical methods and numerical simulations. The proprotypic case is the numerical experiment performed by Fermi, Pasta, and Ulam in 1955. As we discuss in the following section, the main questions, which had inspired this experiment, remained without an answer for a long time, while new puzzling problems emerged. Despite its

apparent failure, the Fermi–Pasta–Ulam (FPU) experiment represents a remarkable example in the history of science of how a good guess may be the source of many fruitful achievements. Part of them are discussed in the section on energy relaxation in nonlinear chains, where we summarize the present understanding of the very slow relaxation mechanism, characterizing the dynamics of nonlinear chains of oscillators, like the FPU model, at low energies. Next, we report one further success of the interplay between theory and numerics, that is, the formulation of a generalized fluctuation–dissipation relation for stationary processes. Finally, we survey the main achievements concerning the study of anomalous transport properties in low-dimensional systems. In particular, we focus our attention on the heat conduction in nonlinear lattices. Lacking a general hydrodynamic theory, also in this case computer simulations and theoretical arguments have greatly

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

contributed to clarify the general scenario, unveiling surprising aspects, which, up to a few years ago, were completely unexpected.

The Numerical Experiment by Fermi, Pasta, and Ulam

H¼

N X p2i !2 þ ðqiþ1  qi Þ2 2m 2 i¼1

þ

  ðqiþ1  qi Þ3 þ ðqiþ1  qi Þ4 3 4

Accordingly, the model contains the minimal basic ingredients, needed for testing the conjecture about the finiteness of thermal conductivity. The equations of motion q_ i ¼

The impressive progress of electronic technology during World War II made possible the design of the first digital computers. The equally impressive budgets for their production and maintenance could only be justified by their employment in classified military research. Nonetheless, some of the outstanding scientists involved in these researches, like E Fermi, immediately realized the great potential of these new machines for tackling also some fundamental problems in basic science. Fermi had in his mind a crucial and still open physical problem. In 1914 the Dutch physicist P Debye had suggested that the finiteness of thermal conductivity in crystals should be due to the nonlinear forces acting among the constituent atoms. Forty years later a microscopic theory of transport processes, including nonlinear effects, was still lacking. Actually, technical difficulties prevented a theoretical approach based on analytic methods. Numerical integration of the equations of motion by a digital machine appeared to Fermi as an effective way for tackling this problem. In collaboration with the mathematician S Ulam and the physicist J Pasta, Fermi used MANIAC 1 (a prototype digital computer installed at Los Alamos National Laboratories, USA) for integrating the dynamical equations of the simplest mathematical model of an anharmonic crystal: a chain of N harmonic oscillators, coupled by nonlinear forces. Its Hamiltonian reads

½1

where ! is the harmonic frequency, while  and  are the positive coupling constants of the nonlinear terms. The integer space index i labels the oscillators along the chain, while qi and pi are the displacement from the equilibrium position and the momentum of the ith oscillator, respectively. The potential energy is the general form taken by any nonlinear interaction potential, when expanded, up to fourth order, around its equilibrium position. This choice guarantees the boundedness of trajectories for any finite energy.

545

@H ; @pi

p_ i ¼ 

@H @qi

½2

were integrated numerically by an algorithm, where space and time derivatives were approximated by proper finite-difference expressions. The choice of the initial conditions was motivated by a further basic question concerning Fermi and his collaborators. In fact, they aimed at verifying also a common belief that had never been proved rigorously: in an isolated mechanical system with many degrees of freedom (i.e., made of a large number of oscillators), a generic nonlinear interaction among them should eventually yield equilibrium through ‘‘thermalization’’ of the energy. On the basis of physical intuition, nobody would object to this expectation if the mechanical system would start its evolution from an initial state very close to thermodynamic equilibrium. Nonetheless, the same should be observed by considering an initial state, where energy is supplied to a small subset of oscillatory modes of the crystal. At variance with a finite system of linear oscillators, where each initially excited mode keeps its energy constant, nonlinear terms should make the energy flow towards all oscillatory modes, until thermal equilibrium is eventually reached. Thermalization corresponds to energy equipartition among all the modes. This statement has to be interpreted in a statistical sense: the time averages of the energies contained in the modes converge to the same constant value. But if this was the case, one further fundamental aspect concerning the evolution towards thermodynamic equilibrium could be checked. In the formulation of his transport equation, L Boltzmann had conjectured that thermodynamic irreversibility can emerge from microscopic reversible dynamics (which is the case of eqns [2]). The paradoxical implication of Boltzmann’s conjecture was pointed out by H Poincare´, who had proved that any isolated Hamiltonian system necessarily evolves towards an almost-recurrent dynamics. This is manifestly incompatible with the second law of thermodynamics, which implies that thermodynamic systems, in the absence of a supplied energy flux, have to evolve irreversibly towards their equilibrium state. In this perspective, the FPU numerical experiment was intended to test also if and how equilibrium is approached by a relatively large number of nonlinearly coupled oscillators, obeying the classical

546 Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

laws of Newtonian mechanics. Furthermore, the measurement of the time interval needed for approaching the equilibrium state, that is, the ‘‘relaxation time’’ of the chain of oscillators, would have provided an indirect determination of thermal conductivity. In fact, according to elementary kinetic theory, the relaxation time, r , represents an estimate of the timescale of energy exchanges inside the crystal: Debye’s argument predicts that thermal conductivity  is proportional to the specific heat at constant volume of the crystal, Cv , and inversely proportional to r , in formulas  / Cv =r . Fermi, Pasta, and Ulam considered relatively short chains, up to 64 oscillators – a size that already challenged the limits of the computational power of MANIAC 1. They imposed fixed boundary conditions (i.e., the particles at the chain boundaries interact with infinite mass walls) and the energy was initially stored just in one of the long-wavelength oscillatory modes. A very surprising and unexpected scenario showed up. Contrary to any intuition, the energy did not flow to the higher modes, but was exchanged only among a small number of longwavelength modes, before flowing back almost exactly to the initial state, thus yielding a recurrent behavior. Although nonlinearities were at work, neither a tendency towards thermalization, nor a mixing rate of the energy could be identified. The dynamics exhibited regular features very close to those of an integrable system. Fermi guessed that they were facing a very important result, but he was also quite disappointed by the difficulties in finding a convincing explanation. This lacking, he had decided not to publish the results in a scientific review, which remained confined into a Los Alamos report for almost one decade. In fact, he died in 1955, the same year of publication of the report. The results were finally published in 1965, in a volume containing his collected papers (Fermi et al. 1965), and they immediately raised a renewed interest in the scientific community. Despite the failure in answering all the questions that had been raised, the FPU numerical experiment represents a crucial scientific achievement, which determined many subsequent scientific progresses. The implications about nonequilibrium will be widely discussed in the following sections. Here, we want to conclude by mentioning the important developments, inspired by the FPU experiment, that led to the discovery of solitons by Zabusky and Kruskal in 1965.

Slow and Fast Energy Relaxation in Nonlinear Chains The results of the FPU numerical experiment indicate that the energy initially supplied to longwavelength oscillatory (Fourier) modes remains localized for a very long time in a small subset of long-wavelength modes. This time can be exceedingly larger than any typical timescale of the model (e.g., !1 , i.e., the inverse of the harmonic frequency in [1]). An explanation of this apparently bizarre scenario has been tackled by combining theoretical approaches with numerical studies. A complete account of the many contributions in this direction being beyond the scope of this text, we shall summarize the two main lines along which this problem has been considered. The Resonance-Overlap Criterion

The almost-recurrent behavior of single-mode excitations studied in the FPU experiment can be explained by the resonance-overlap criterion, introduced in 1959 by the Russian scientist B Chirikov. Moreover, this criterion provides a quantitative estimate of the value of the energy density, above which the regular motion observed in the FPU experiment should be definitely lost. In order to provide the reader with an illustration of this criterion, we have to introduce a few simple mathematical ingredients. The Hamiltonian [1] can be rewritten in terms of linear normal Fourier coordinates, (Qk (t), Pk (t)), as follows:  1 X 2 Pk þ !2k Q2k þ V3 ðfQk gÞ H¼ 2 k þ V4 ðfQk gÞ

½3

Here, we have used the shorthand notation Vn ({Qk }) for the lengthy explicit expressions, in the new set of coordinates, of the nonlinear potentials of [1]. Without prejudice of generality, we can impose periodic boundary conditions to the FPU chain: the frequency of the kth normal mode is given by the expression !k = 2 sin(k=N). The coupling constants  and  control the energy exchange among the normal modes, due to nonlinear interactions. For the sake of space, we give here a brief sketch of Chirikov’s criterion for the FPU -model (this model amounts to take  = 0 in [3], i.e., to exclude the cubic part of the nonlinear potential). By making reference to the initial conditions of the FPU experiment, we can consider a single excited mode, so that the Hamiltonian [3] can be

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

approximated by the expression in action-angle variables H ¼ H0 þ H1  !k Jk þ

 ð!k Jk Þ2 2N

½4

Here, Jk = !k Q2k is the action variable. In practice, this amounts to approximate the original Hamiltonian by the sum of the harmonic and nonlinear selfenergy of the initially excited mode. In this framework, H0 and H1 are the unperturbed (integrable) Hamiltonian and the perturbation, respectively. Indeed, if the energy is initially attributed to mode k, the following relations hold: !k Jk  H0  E. By the approximated Hamiltonian [4], one can compute the nonlinear correction to the linear frequency !k , giving the renormalized frequency !rk : !rk ¼

@H  ¼ !k þ !2k Jk ¼ !k þ k @Jk N

½5

For N  k one has k 

H0 k N2

½6

The distance between two primary resonances, in the harmonic limit, is given by the expression !k ¼ !kþ1  !k  N 1

½7

Consistently with [6], the last approximation is valid only for small wave number (k  N), that is, longwavelength modes. The ‘‘resonance overlap’’ criterion amounts to compare this distance with the frequency shift. In formulas: k  !k

½8

This equation allows to obtain also an estimate of the ‘‘critical’’ energy density, c , above which sizeable chaotic regions develop and a fast diffusion takes place in phase space:   H0 1 ½9 c ¼  N c k with k = O(1)  N. Below c , primary resonances are weakly coupled and determine a slow-relaxation process to energy equipartition. Above c , due to ‘‘primary resonance’’ overlap, fast relaxation to equipartition sets in (Izrailev and Chirikov 1966). This prediction was verified numerically later by Chirikov et al. (1973). The presence of a critical energy density can be tested by measuring the evolution Rof the finite time-averaged quantity  k (t) = t1 t Ek ()d, where Ek = (P2 þ !2 Q2 )=2 is E k k k 0 the harmonic energy of the kth mode. For energy  k (t) exhibits an densities much smaller than c , E

547

extremely slow relaxation towards the equipartition  k = constant. Conversely, for  > c such condition, E a condition is rapidly approached on a relatively short timescale. The slow relaxation below c can be traced back to the overlap of higher-order resonances: its typical timescale has been found to be inversely proportional to a power of the energy density (Shepelyansky 1997). Energy-Equipartition Thresholds

The first paper reporting evidence of the existence of an energy threshold in chains of coupled anharmonic oscillators had already been published in 1970 by Bocchieri et al. (1970). This pioneering numerical experiment concerned a chain of oscillators coupled through a Lennard-Jones interatomic potential. The Italian group observed an energy threshold, separating a high-energy thermalized regime from a regular dynamics regime at low energies (like the one observed by Fermi, Pasta, and Ulam). The main point raised by this experiment concerns the consequences on ergodic theory: the ordered motion observed in the low-energy regime seems to violate ergodicity, although the model is known to be chaotic at any energy. This is quite a delicate and widely debated issue for its statistical implications. Actually, as we have mentioned in the previous section, also Fermi, Pasta, and Ulam expected that a nonlinear dynamical system, made of a large number of degrees of freedom, should naturally evolve towards equilibrium. Further confirmations to the seminal paper by Bocchieri and co-workers came from more refined numerical experiments, showing that, for sufficiently high energies, regular behaviors disappear, while equipartition among the Fourier modes sets in rapidly. Later on, the presence of the energy threshold was characterized P by introducing an appropriate entropy, S =  k pk ln pk with pk = hEk (t)=Ei, which counts the number of effective Fourier modes involved in the dynamics: at equipartition, this entropy is maximal (Livi et al. 1985). Nowadays, we know that the approach to equipartition below and above the energy threshold is a matter of timescales, which turn out to be very different in the two regimes. For instance, the analytic estimate of the maximum Lyapunov exponent  of the FPU -model (Casetti et al. 1995) has definitely pointed out that there is a threshold value of the energy density, T , at which its dependence on  changes drastically:  1=4  if   T ; ðÞ  ½10 2  if   T :

548 Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

This implies that the typical relaxation time, that is 1 , may become exceedingly large for very small values of  below T . It is worth stressing that this result holds in the thermodynamic limit, thus indicating that the presence of T is statistically relevant. A more controversial scenario emerges from the studies of the relaxation dynamics for specific classes of initial conditions. When a few longwavelength modes are initially excited, regular motion may persist over times much longer than 1 (De Luca et al. 1995). The excitation of smallwavelength modes yields an even more complex scenario: solitary wave dynamics is observed, followed by slow relaxation to equipartition. It is also worth mentioning that some regular features of the dynamics persist even at high energies. As we shall discuss in the section ‘‘Heat transport,’’ such regularities still play a crucial role in determining energy transport mechanisms, although they do not affect significantly the equilibrium statistical properties of the FPU model at high energies.

The Generalized Fluctuation–Dissipation Theorem Another fundamental problem of nonequilibrium statistical mechanics concerns the possibility of establishing a fluctuation–dissipation theorem, generalizing the relation valid for equilibrium conditions. In fact, on this basis one might develop a large-deviation formalism, aiming at the identification of an explicit nonequilibrium statistical measure, analogous to the equilibrium Boltzmann–Gibbs measure. Recently, some relevant progresses in this direction have been made. A crucial numerical experiment, which attracted the attention on the problem of formulating a generalized fluctuation–dissipation relation for stationary flows, was performed at the beginning of the 1990s (Evans et al. 1993). Stationary conditions for momentum transport were obtained in the shear flow of a fluid contained between moving walls. The reversibility of the microscopic dynamics yields the heuristic fluctuation relation:  t ¼ AÞ 1 PrðR ln  t ¼ AÞ ¼ A t PrðR

½11

 t = A) is the probability that the average where Pr(R  t , along a trajectory entropy production rate, R segment of duration t, takes the value A. For sufficiently large values of t, this relation was confirmed by numerical analysis. Gallavotti and Cohen (1995a,b) proved a theorem meant to put on a rigorous mathematical

basis eqn [11], that is, the proposed extension to nonequilibrium steady states of the equilibrium fluctuation–dissipation theorem. This theorem concerns the phase-space contraction rate of the dynamics, which equals the entropy production rate in the case of particle systems, whose internal energy is a constant of the motion. The proof of the theorem is based on restrictive hypotheses, which include the existence of an average nonvanishing phase-space contraction rate, the timereversal invariance of the dynamics and a strong form of chaos (the dynamics is assumed to be of the Anosov type, that is, smooth and uniformly hyperbolic). Nonetheless, the prediction of the theorem, that is, 1 t ðpÞ ln ¼ Dh ip t t ðpÞ

½12

is expected to hold much more generally. Here t ðpÞ is the probability that a fluctuation variable takes the value p. The theorem proved by Gallavotti and Cohen states that t ðpÞ has to satisfy the large deviation relation [12], where is the average phase-space contraction rate over a trajectory segment of duration t and D is a suitable constant. It must be pointed out that the rigorous derivation of this relation provided strong motivations for investigating its validity and generality in many other contexts. The first numerical experiment, where almost all the constituent hypotheses of the Gallavotti– Cohen theorem were satisfied, was performed by Bonetto et al. (1997). They studied a Lorentz gas (massive pointlike noninteracting particles bouncing elastically on circular scatterers displaced on a regular lattice without free horizon) of charged particles moving in an uniform external electric field. Numerical simulations were found to be in very good agreement with [11] and [12] (which, in this case, refer to the same quantity). One further test of the fluctuation–dissipation relation was later performed for a different setup (Lepri et al. 1998). The FPU -model is put in contact at its boundaries with thermal heat baths of different temperatures Tþ and T (Tþ > T ). Numerical simulations have been performed for sufficiently large applied thermal gradients, which guarantee sizeable effects of fluctuations, suitable for verifying a relation like [11]. It is worth noticing that many of the constituent hypotheses of the Gallavotti–Cohen theorem are not valid for this setup, but eqn [12] is still expected to hold, although in this case it does not refer to the entropy production rate. Nonetheless, the extension [11] of the fluctuation–dissipation theorem can be tested, thanks to the following useful relation,

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

between the heat flux j and the entropy production rates,  , at the chain boundaries:   1 1 h þ i þ h  i ¼ j  ½13 T Tþ This can be interpreted as a balance relation for the global entropy production. In fact, according to the principles of irreversible thermodynamics, the local rate of entropy production in the bulk is given by   d 1 ½14 ðxÞ ¼ j dx TðxÞ By integrating this equation, one straightforwardly obtains the previous one, which then applies to the entropy production from the heat baths. Careful numerical simulations show that stationary conditions are found to hold over a wide range of temperatures and gradients. Equation [13] indicates that the heat flux is equivalent to the entropy production rate, apart from a multiplicative constant which depends on the amplitude of the applied field. Let us define the finite-time average of the global heat flux Z N 1X 1 t Jt ¼ dji ðÞ ½15 N i¼1 t 0 The normalization of this quantity can be obtained by computing the asymptotic average value J1 ¼ lim Jt t!1

½16

The quantity of statistical interest is the normalized finite-time average global heat flux z¼

J J1

½17

Accordingly, the fluctuation–dissipation relation in this case takes the form:   P ðzÞ 1 1 ln ¼ zj  ½18 P ðzÞ T Tþ The conjecture that such a relation might be valid in this case has been confirmed by numerical analysis. It is worth stressing that, in this out-of-equilibrium setup, the probability distribution, P (z), is not Gaussian and exhibits a peculiar asymmetric shape. Nonetheless, for increasing values of , the asymmetry progressively reduces, while P (z) approaches a Gaussian shape. This observation indicates that, in this case, large fluctuations deviate from the typical statistics of independent events. It should be mentioned that generalized fluctuation– dissipation relations, like those discussed in this

549

section, have been successfully checked in many other situations, where the hypotheses of the Gallavotti– Cohen theorem did not apply. The ‘‘robustness’’ of relations such as [11] and [12] indicates that a more general theory may be possible.

Heat Transport The validity of Debye’s conjecture about the necessity of nonlinear forces for obtaining a finite heat conductivity in crystals still remained an open problem after the unsuccessful FPU numerical experiment. The setup, described in the previous section for testing the generalized fluctuation– dissipation relation in the FPU chain, can be used also for tackling the verification of this conjecture. Actually, the thermal conductivity, , of a chain of oscillators can be measured from the Fourier’s law JQ ¼ rTðxÞ

½19

where JQ is the heat current and rT(x) is the temperature gradient. This problem was solved analytically for a chain of N harmonic oscillators (Rieder et al. 1967). The bulk of the chain is found to reach thermal equilibrium conditions at the average temperature T = (Tþ þ T )=2, corresponding to a constant temperature profile. Only at the chain boundaries the harmonic chain exhibits a steep temperature gradient. This implies that the heat current is proportional to the temperature difference, rather than to the temperature gradient, thus violating Fourier’s law. Accordingly, a harmonic chain, made of N oscillators, in contact with two heat reservoirs at different temperatures, exhibits anomalous transport properties and the effective thermal conductivity is found to diverge in the infinite-chain limit as   N. This peculiar behavior is a consequence of the integrability of the harmonic chain dynamics. Actually, the Fourier modes propagate with finite velocity through the harmonic chain, so that any energy injected from the hot reservoir flows ballistically to the cold one, rather than diffusing, as required for the validity of [19]. It is worth stressing that any integrable system should exhibit a similar scenario. This is the case of the equal-mass hard sphere gas in one dimension and of the Toda chain, where the harmonic potential (!2 =2)(qiþ1  qi )2 is replaced by the nonlinear expression a exp½bðqiþ1  qi Þ In the former case, integrability and ballistic propagation are straightforward consequences of

550 Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

the conservation laws, inherent elastic collisions between hard spheres. In the latter model, the normal nonlinear modes, called ‘‘Toda solitons,’’ are responsible for such anomalous behavior. Debye’s conjecture should be modified accordingly: nonintegrability of the equations of motion has to be invoked as a necessary property for explaining heat transport in real solids. Let us observe that the FPU model is known not to be integrable and it is expected to be a good candidate for confirming Debye’s conjecture, at least in its fully chaotic regime. Careful and extended numerical simulations have shown that the FPU chain maintains anomalous properties (Lepri et al. 1997). In particular, the thermal conductivity, , is found to diverge in the infinite chain limit as   N

½20

with   2=5. This value agrees with independent analytic estimates (e.g., see Lepri et al. (2003)), although renormalization arguments indicate that one should rather find  = 1=3 (Narayan and Ramaswamy 2002). This discrepancy could be due to the peculiar features associated with the presence of a quartic nonlinearity in the FPU problem and also to the fact that in the FPU chain heat can be transported only through longitudinal oscillations. Anyway, this is still an open problem, which requires further theoretical advances to be solved. In a more general perspective, the main outcome of these numerical studies indicates that a powerlaw divergence like [20] is found in all onedimensional nonintegrable models. This general feature must be attributed to the combined effect of low-space dimensionality, with energy and momentum conservation. In such a situation, fluctuations are strongly constrained, so that the evolution of long-wavelength hydrodynamic modes is not sufficiently damped, to be ruled by diffusion (which is a necessary ingredient for the validity of [19]). It must be stressed that these numerical investigations have strongly revived the interest for this problem. In particular, they have also stimulated new theoretical efforts for explaining the power-law divergence of transport coefficients in d = 1. One of the main achievements of these theoretical approaches is that the power-law divergence turns to a logarithmic one in d = 2, while the divergence should disappear in d 3. Despite the difficulty of performing the necessary large-scale simulations for such systems in d > 1, it seems that numerics essentially agree with such predictions. One can find normal transport properties even in d = 1, if suitable models are considered. For

instance, momentum conservation can be broken by adding to the Hamiltonian [1] a local interaction potential, U(qi ), which breaks translation invariance, thus restoring finite heat conductivity (e.g., see Casati et al. 1984). The exception to this case is the harmonic chain with the addition of a local harmonic potential: in this case the dynamics is still integrable and there are as many conserved quantities as degrees of freedom. A further peculiar case is represented by the rotator model in d = 1, which is known to be nonintegrable. Its Hamiltonian contains the interaction potential [1  cos(qiþ1  qi )], replacing the algebraic potentials of the FPU chain. Anyway, such a Hamiltonian still guarantees momentum conservation, since the nearest-neighbor form of the interaction is maintained. Notice that, for small oscillations around the equilibrium position, also the rotator potential admits a Taylor-series expansion, whose first three terms correspond to quadratic, cubic, and quartic contributions, as in the FPU chain. Nonetheless, at variance with the FPU problem, the potential of the rotator model is bounded also from above. Numerical investigations (Giardina et al. 2000) have shown that for any finite energy density and for a sufficiently long finite time, some previously oscillating rotators start to rotate, due to local energy fluctuations, that allow to overtake the potential barrier. These dynamical configurations typically appear in the form of spatially localized, synchronous rotating clusters. Their time evolution is characterized by an intermittent behavior: they are eventually reabsorbed by lattice fluctuations and may reappear afterwards at other lattice positions. In this way they play the role of scattering centers for hydrodynamic modes. It must be pointed out that such a qualitative argument is not sufficient for explaining the onset of a genuine diffusive behavior, compatible with the validity of Fourier’s law. A hydrodynamic theory, still to be developed, could provide a more convincing insight on these results. It is worth concluding this section by mentioning that the overall scenario described above is confirmed by numerical studies, relying upon a different approach, based on equilibrium measurements. Actually, the linear response theory by Green and Kubo (see Kubo (1985)) provides an alternative, but essentially equivalent, definition of the thermal conductivity, according to the expression

¼

1 1 lim lim KB T 2 t!1 N!1 N

Z 0

t

dhJðÞJð0Þi

½21

Nonequilibrium Statistical Mechanics: Interaction between Theory and Numerical Simulations

The crucial quantity to be computed numerically is the heat-flux time-correlation function CJ () = h J()J(0)i, where h i represents the thermodynamic equilibrium average. In practice, numerical simulations can be performed for a chain of N oscillators in contact with boundary heat reservoirs at the same temperature T = Tþ = T . The presence of anomalous transport coefficients can be singled out by analyzing the long-time behavior of CJ (). It has to decay at least as  (1þ") , with " > 0 to yield a finite heat conductivity. In one-dimensional models exhibiting the power-law divergence [20] one rather finds CJ ðÞ   1þ

½22

where the positive exponent  is the same appearing in [20]. This relation between space and time exponents can be easily explained, by considering that space and time variables depend linearly on each other through a proportionality constant, which is the velocity of sound in the lattice. Since 0 <  < 1, the anomalous behavior observed in out-of-equilibrium conditions is recovered. One major problem in performing proper numerical studies concerns the control over finite-size effects, which demands a consistent increase of the integration time with the system size. This may yield very extended and expensive computations, mainly when very slow relaxation processes set in. This is the case of the low-energy regime originally studied by FPU in their pioneering computer simulations. Numerical analysis indicates that in this regime the expected behavior of CJ (), reported in eqn [22], sets in after a crossover time tc , which increases, for decreasing energy density , as tc  2 . This seems to be compatible with the studies described earlier. We conclude this section by pointing out that this result also contributes significantly to clarify one of the basic questions raised by the FPU numerical experiment. See also: Dynamical Systems and Thermodynamics; Ergodic Theory; Fourier Law; Gravitational N-Body Problem (Classical); Lyapunov Exponents and Strange Attractors; Nonequilibrium Statistical Mechanics: Dynamical Systems Approach.

551

Further Reading Bocchieri P, Scotti A, Bearzi B, and Loinger A (1970) Anharmonic chain with Lennard–Jones interaction. Physical Review A 2: 2013. Bonetto F, Gallavotti G, and Garrido P (1997) Chaotic principle: an experimental test. Physica D 105: 226. Casati G, Ford J, Vivaldi F, and Visscher WM (1984) Onedimensional classical many-body system having a normal thermal conductivity. Physical Review Letters 52: 1861. Casetti L, Livi R, and Pettini M (1995) Gaussian model for chaotic instability of Hamiltonian flows. Physical Review Letters 74: 375. Chirikov BV, Izrailev FM, and Tayursky VA (1973) Numerical experiments on the statistical behaviour of dynamical systems with a few degrees of freedom. Computational Physics Communications 5: 11–16. De Luca J, Lichtenberg AJ, and Lieberman MA (1995) Time scale to ergodicity in the Fermi–Pasta–Ulam system. Chaos 5: 283. Evans DJ, Cohen EGD, and Morriss GP (1993) Probability of second law violations in shearing steady state. Physical Review Letters 71: 2401. Fermi E, Pasta JR, and Ulam S (1965) Studies of nonlinear problems. In: Collected Works of E. Fermi, vol. 2, p. 978. Chicago: University of Chicago Press. Gallavotti G and Cohen EGD (1995a) Dynamical ensembles in stationary states. Journal of Statistical Physics 80: 931. Gallavotti G and Cohen EGD (1995b) Dynamical ensembles in nonequilibrium statistical mechanics. Physical Review Letters 74: 2694. Giardina´ C, Livi R, Politi A, and Vassalli M (2000) Finite thermal conductivity in 1D lattices. Physical Review Letters 84: 2144. Izrailev FM and Chirikov BV (1966) Statistical properties of a nonlinear string. Soviet Physics Doklady 11: 30. Kubo R, Toda M, and Hashitsume N (1985) Statistical Physics II. Berlin: Springer. Lepri S, Livi R, and Politi A (1997) Heat conduction in chains of nonlinear oscillators. Physical Review Letters 78: 1896. Lepri S, Livi R, and Politi A (1998) Energy transport in anharmonic lattices close to and far from equilibrium. Physica D 119: 140. Lepri S, Livi R, and Politi A (2003) Thermal conduction in classical low-dimensional lattices. Physics Reports 377: 1. Livi R, Pettini M, Ruffo S, Sparpaglione M, and Vulpiani A (1985) Physical Review A 31: 1039, 2740. Narayan O and Ramaswamy S (2002) Anomalous heat conduction in one-dimensional momentum-conserving systems. Physical Review Letters 89: 200601. Rieder Z, Lebowitz JL, and Lieb E (1967) Properties of a harmonic crystal in a stationary nonequilibrium state. Journal of Mathematical Physics 8: 1073. Shepelyansky D (1997) Low energy chaos in the Fermi–Pasta– Ulam problem. Nonlinearity 10: 1331.

552 Nonlinear Schro¨dinger Equations

Nonlinear Schro¨dinger Equations M J Ablowitz, University of Colorado, Boulder, CO, USA B Prinari, Universita` degli Studi di Lecce, Lecce, Italy ª 2006 Elsevier Ltd. All rights reserved.

Historical Background Ginzburg–Landau Equations

Nonlinear Schro¨dinger (NLS) equations have become one of the most important nonlinear systems studied in mathematics and physics. Actually, one can find the essence of NLS equations in the early work of Ginzburg and Landau (1950) and Ginzburg (1956) in their study of the macroscopic theory of superconductivity, and also of Ginzburg and Pitaevskii (1958), who subsequently investigated the theory of superfluidity. By minimizing the free energy of a superconductor near the superconducting transition, Ginzburg and Landau arrived at what are now called the Ginzburg–Landau equations: 1  e 2 ihr  A þ  þ j j2 ¼ 0 2m c

J¼

ie h ½ mc



r  r





e2 j j2 A mc

½1

½2

where ,  are phenomenological parameters, A the electromagnetic vector potential, and  denotes complex conjugate of . The first equation determines the field based on the applied magnetic field. The second equation provides the superconducting current J. The equation describing the behavior of superfluid helium near the transition point in the stationary case derived in Ginzburg and Pitaevskii (1958) is completely analogous to eqn [1] in the phenomenological theory of superconductivity. Equation [1] contains all the ingredients of the NLS equations which are discussed below. However, it was not until the 1960s that the wide physical importance of NLS equation became evident. The next section discusses how the NLS equation historically first appeared in the context of nonlinear optics. Nonlinear Optics: Self-Focusing of Optical Beams in Nonlinear Media

In the mid-1960s, Chiao et al. (1964) and Talanov (1964) investigated the conditions under which an

electromagnetic beam can produce its own dielectric waveguide and propagate without spreading. This is a reflection of the phenomenon of self-focusing. In fact, self-focusing of optical beams may occur in materials whose dielectric constant increases with field intensity. In the general situation, a beam of uniform intensity in a dielectric broadens due to diffraction. However, the refractive index of many physically important materials (the so-called Kerr materials, such as silica) depends on the field intensity as follows: n ¼ n0 þ n2 jEj2 þ    If the term n2 jEj2 is large enough, the critical angle for total internal reflection at the beam’s boundary can be greater than the angular divergence due to diffraction; thus, spreading does not occur as a result of diffraction. As a consequence, a beam above a certain critical power level is trapped and does not spread. In a remarkable contribution, Kelley (1965) observed, using computational methods (years before computational methods became easy to implement and, consequently, so popular) that when the self-focusing effect due to the increase in the nonlinear index is not compensated by diffraction, there is a buildup in intensity of part of the beam as a function of the distance in the direction of propagation. Consequently, the intensity of the self-focused regions tended to become ‘‘anomalously large,’’ that is, a singularity appeared to develop. Consider as starting equation the electromagnetic wave equation in the presence of nonlinearities derived earlier by Chiao et al. (1964): 0 2 r2 E  2 @ t2 E  2 @ t2 ðE2 EÞ ¼ 0 ½3 c c where 2 jEj2  1. One assumes a linearly polarized wave of frequency !, propagating along the z-axis, so that E ¼ 12ðEeiðkz!tÞ þ c:c:Þbe 1=2

where c.c. denotes complex conjugation, k = 0 !=c, the factor exp(ikz  !t) represents the propagating part, that is, the ‘‘carrier,’’ of the wave, and E is the slowly varying part. Substituting the above expression for E into eqn [3], neglecting the third-harmonic term and the term @z2 E from r2 E (assuming it to be small), yields   3 2 2ik@z E þ @x2 þ @y2 E þ k2 jEj2 E ¼ 0 ½4 4 0

Nonlinear Schro¨dinger Equations

or, with a suitable rescaling of the dependent and independent variables (E ! =((3=4)k2 2 =0 )1=2 , z ! 2kz), i@z þ

r2?

þ 2j j

2

¼0

½5

553

Indeed, consider a scalar nonlinear wave equation written symbolically as Lð@t ; rÞu þ GðuÞ ¼ 0

which is the NLS equation in standard nondimensional form. It should be remarked here that the name NLS equation for equations of the form of [5] is natural due to the formal analogy with the Schro¨dinger equation in quantum mechanics:

where L is a linear differential operator with constant coefficients and G a nonlinear function of u and its derivatives. For a real, smallamplitude solution of magnitude   1, the nonlinear effects can first be neglected, and the equation admits approximate monochromatic wave solutions

i@t þ r2 þ V ¼ 0

u ¼  eiðkx!tÞ þ c.c.

½6

2

If one sets V = 2j j in eqn [6], the result is the NLS equation. In the context of quantum mechanics, a nonlinear potential arises in the ‘‘mean-field’’ description of interacting particles. Modifications of [6] also arise as mean-field descriptions of Bose–Einstein condensates which is of keen interest in physics (see Pethick and Smith (2002) and references therein). The normalized equation is   i@t  r2 þ Vðx; yÞ þ 2j j2 ¼0 ½7 where V is an external potential. This is generally referred to as the Gross–Pitaevskii equation. Talanov (1965) (see also Zakharov et al. (1971)) investigated the behavior of stationary light beams in a self-focusing nonlinear medium and found that for a purely cubic nonlinearity, ‘‘collapse’’ of the beam can take place. The proof that there is a singularity in eqn [5] is remarkably straightforward. This is discussed in the section ‘‘Wave collapse.’’ In order to avoid wave collapse, other physical effects (e.g., saturable nonlinearity or dissipation) are required. Universal Character of the NLS Equation

It turns out that almost any dispersive, energypreserving system gives rise, in an appropriate limit, to the NLS equation. For instance, one can derive the NLS from other physically significant equations such as the Klein–Gordon equation utt  uxx þ u þ ku3 ¼ 0 and the Korteweg–de Vries (KdV) equation ut þ 6uux þ uxxx ¼ 0 Actually, the NLS equation provides a ‘‘canonical’’ description for the envelope dynamics of a quasimonochromatic plane wave (the carrier wave) propagating in a weakly nonlinear dispersive medium when dissipative processes are negligible.

½8

with small amplitude j j. Substituting [8] into the linear equation, one can find that the frequency ! and the wave vector k are related by the dispersion relation Lði!; ikÞ ¼ 0 Let ! ¼ !ðkÞ be one of the solutions of the previous equation. Suppose one is interested in a solution which is not constant, but slowly varying in space and time. This has the interpretation of k having a ‘‘sideband’’ wave vector and ! a ‘‘sideband’’ frequency. More precisely, restricting discussion, for simplicity, to the (1 þ 1)-dimensional case, the slowly varying amplitude assumption corresponds to letting ðx; tÞ ¼ ðX; TÞ ¼

0e

iðKxtÞ

where X = x and T = t. Note that K = k and  = ! are sometimes referred to as the sideband wave number and frequency, respectively, because they correspond to a deviation from the central wave number k and central frequency !. Looking at these deviations from the point of view of operators, whereby ! ! i@t , k ! i@x and  ! i@T , K ! i@X , one has !tot  ! þ  ¼ ! þ i@T ktot  k þ K ¼ k  i@X Then !(k) can be expanded in a Taylor series around the central wave number as !ðk  i@X Þ  !ðkÞ  i!0 @X  2

!00 2 @ þ  2 X

Therefore, !tot ðkÞ  ½!ðkÞ þ i@T    00 0 2! 2  !ðkÞ  i! @X   @ 2 X

554 Nonlinear Schro¨dinger Equations

which shows that, to the leading order,   @ !00 @ 2 0 @ þ! i ¼0 þ 2 @T @X 2 @X2

½9

In the moving frame  = X  !0 (k)T,  = T  2 t, eqn [9] transforms to   !00 2 i  þ ¼0  2 which is the linear Schro¨dinger equation with the canonical !00 (k)=2 coefficient. On the other hand, if one considers rather general conservative nonlinear wave problems with leading quadratic or cubic nonlinearity, asymptotic analysis (e.g., multiple scale analysis which yields the so-called Stokes– Poincare´ frequency shift) shows that a wave solution of the form uðx; tÞ ¼  ðÞeiðkx!tÞ þ c:c: with  = 2 t has () satisfying i

@ þ nj j2 ¼ 0 @

½10

where the constant coefficient n depends on the particular equation under study. It should be remarked here that cubic nonlinearity yields an O(3 ) contribution, which is balanced by a slow timescale of order 2 . Putting the linear and nonlinear effects together (i.e., eqns [9] and [10]) implies that an NLS equation of the form i

@ !00 @ 2 þ þ nj j2 ¼ 0 @t 2 @2

naturally arises. The NLS equation is viewed as a ‘‘universal’’ equation as it generically governs the slowly varying envelope of a monochromatic wave train (see also Benney and Newell (1969)).

Physical Applications The nonlinear propagation of wave packets is governed by NLS-type systems in several different branches of scientific and technological applications, beyond what has been mentioned earlier. Some of these applications are discussed below. NLS equation in Water Waves

The NLS equation in the context of small-amplitude water waves was derived by Zakharov (1968) (infinite depth) and Benney and Roskes (1969) (finite depth). The procedure for deriving the NLS equation from the Euler–Bernoulli equations of fluid dynamics in one horizontal direction will now be discussed, under the assumption of small-amplitude

waves and deep water. The interested reader can also find the details of the derivation in Ablowitz and Clarkson (2006). The relevant equations are xx þ zz ¼ 0; z ¼ 0; t þ

1 < z < ðx; tÞ z ! 1

  2 x þ 2z þ g ¼ 0; 2 t þ x x ¼ z ;

½11 ½12

z ¼ 

z ¼ 

½13 ½14

where  is the velocity potential of an ideal (i.e., incompressible, irrotational, and inviscid) fluid, (x, t) is the free surface of the fluid, which is to be found, in addition to (x, z; t). Equation [11] expresses the ideal nature of the fluid; the condition [12] expresses the requirement that there is no vertical flow at infinity; and eqn [13] is the Bernoulli equation of energy conservation. Finally, eqn [14] is a kinematic condition stating that no flow occurs transverse to the free surface. At the free boundary, for small amplitudes, one can expand  = (t, x, ) for   1 as  ¼ ðt; x; 0Þ þ z ðt; x; 0Þ þ

ðÞ2 zz ðt; x; 0Þ þ    2

and similarly for the derivatives. Second, one introduces slow temporal and spatial scales (one expects the slowly varying envelope of the wave to depend on slow variables X = x, Z = z, T = t). Finally, because of the quadratic nonlinearity one expects second harmonics to be generated; hence,      ¼ Aeiþjkjz þ c:c: þ  A2 e2iþ2jkjz þ c:c: þ       ¼ Bei þ c:c: þ  B2 e2i þ c:c: þ  where A, A2 ,  depend on X, Z, T and B, B2 ,  depend on X, T ( and  are mean contributions, which are real) and  = kx  !t with the dispersion relation !2 = gjkj. Substituting this ansatz into the equations, one obtains from the order-2 terms ! v2g 2k4 2 A þ 2i!A  ½15 jAj A ¼ 0 2! ! where vg = !0 (k) = g=2! is the group velocity and the new variables  = T,  = X  vg T. Equation [15] is the typical formulation of the (1 þ 1)-dimensional NLS equation found in water wave theory for large depth. In the section ‘‘NLS in nonlinear optics,’’ a special solution to (a rescaled version of) eqn [15], namely a soliton solution, is discussed in the

Nonlinear Schro¨dinger Equations

context of nonlinear optics. It should be remarked here that the coefficients of both terms A and jAj2 A have the same sign. This is necessary for a decaying soliton solution to exist (see, e.g., Lighthill (1965)). NLS in Nonlinear Optics

The NLS equation also describes self-compression and self-modulation of electromagnetic wave packets in weakly nonlinear media. Hasegawa and Tappert (1973a, b) first derived the NLS equation in the context of fiber optics. Light-wave propagation in a fiber is mainly affected by: (1) group velocity dispersion (GVD), that is, the frequency dependence of the group velocity originating from the refractive index of the fiber and (2) fiber nonlinearity (the so-called Kerr effect), originating from the dependence of the refractive index on the intensity of the optical pulse. In the presence of GVD and Kerr nonlinearity, the refractive index is expressed as nð!; EÞ ¼ n0 ð!Þ þ n2 jEj2

½16

where ! and E represent the frequency and electric field of the light wave, respectively, n0 (!) is the frequency-dependent linear refractive index, and the constant n2 , referred to as the Kerr coefficient, is ‘‘small’’ but can have significant impact since the nonlinear effects accumulate over long distances. Normally, the electric field is modulated into a slowly varying amplitude of a carrier wave: Eðz; tÞ ¼ Eðz; tÞeiðk0 z!0 tÞ þ c:c:

½17

where z denotes the distance along the fiber, t the time, k0 = k0 (!0 ) the wave number, !0 the frequency, and E(z, t) the envelope of the electromagnetic field. A Taylor series expansion of the dispersion relation (see also the section ‘‘Universal character of the NLS equation’’) kð!; EÞ ¼

! ðn0 ð!Þ þ n2 jEj2 Þ c

around the carrier frequency ! = !0 yields k  k0 ¼ k0 ð!0 Þð!  !0 Þ þ !0 n2 2 jEj þ c

k00 ð!0 Þ ð!  !0 Þ2 2

using k  k0 = (!=c)n0 (!) and letting eqn [18] operate on E yields   @E @E k00 ð!0 Þ @ 2 E þ k00 ð!0 Þ i þ jEj2 E ¼ 0 ½19  0 @z @t 2 @t2 where = !0 n2 =cAeff , with Aeff being the effective cross-section area of the fiber (the factor 1=Aeff comes from a more detailed derivation which takes into account the finite size of the fiber; the factor 1=Aeff is needed in order to account for the variation of field intensity in the cross section of the fiber). Note that k00 (!0 ) = 1=vg , where vg represents the group velocity of the wave train. Introducing dimenpffiffiffiffiffi sionless variables t0 = tret =t , z0 = z=z , q = E= P yields the NLS equation i

@q sgnðk000 ð!0 ÞÞ @ 2 q þ þ jqj2 q ¼ 0 @z0 2 @t02

where the prime represents derivative with respect to ! and k0 = k(!0 ). Replacing k  k0 and !  !0 by their Fourier operator equivalents, i@z and i@t resp.,

½20

where t , P are the characteristic time and power, respectively, and tret = t  k00 (!0 )z = t  z=vg , z = 1= P , with the constraint that the ‘‘nonlinear length’’ is balanced by the linear dispersion time, that is, t = ðz j  k00 (!0 )jÞ1=2 . There are two cases of physical interest depending on the sign of k000 . The so-called focusing case occurs when k000 < 0; this is called ‘‘anomalous’’ dispersion. The defocusing case obtains when the dispersion is ‘‘normal’’: k000 > 0. Now write eqn [20] in the form iqt þ qxx 2jqj2 q ¼ 0

½21

with corresponding to the focusing (þ) and defocusing () case, respectively. The focusing NLS equation admits special solutions called ‘‘bright’’ solitons (solutions that are traveling localized ‘‘humps’’). A pure one-soliton solution in the focusing (þ) case has the form qðx; tÞ ¼  sech½ðx þ 2t  x0 Þ ei 2

½22

2

where  = x þ (   )t þ 0 . The parameters  and  are such that = =2 þ i=2 is an eigenvalue from the inverse scattering transform analysis. The defocusing () NLS equation does not admit solitons that decay at infinity. However, it does admit soliton solutions which have a nontrivial background intensity (called ‘‘dark’’ and ‘‘gray’’ solitons). A darksoliton solution has the form qðx; tÞ ¼  tanhðxÞ e2i

½18

555

2

t

½23

Note that q !  as x ! 1. A gray-soliton solution is h i1=2 qðx; tÞ ¼  1  B2 sech2 ðBðx  x0 ÞÞ eiðx;tÞ ½24

556 Nonlinear Schro¨dinger Equations

with pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðx; tÞ ¼   2  B2 t þ  1  B2 x   B tanhðBxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ tan1 þ 0 1  B2  2

and jBj < 1. Note that as B ! 1 , the gray soliton becomes a dark soliton, taking 0 = =2. Recall that the solutions [23] and [24] can be allowed to travel uniformly by making a Galilean transformation, that is, taking into account that if q1 (x, t) is a solution of [21], then so is q2 ðx; tÞ ¼ q1 ðx  vt; tÞ eiðkx!tÞ with k = v and ! = k2 =2. It should also be remarked that Ablowitz et al. (1997) have shown that, in quadratically nonlinear optical materials, more complicated NLS-type equations arise. These equations are analogous to the finite-depth multidimensional nonlocal NLS-type systems derived in the context of water waves by Benney and Roskes (1967) and later by Davey and Stewartson (1974).

be reliably transmitted to. Soliton control mechanisms were introduced in the early 1990s in order to deal with these difficulties (cf. Mecozzi et al. (1991) and Kodama and Hasegawa (1992)). By the mid-1990s, the development of all optical transmission systems began to take great advantage of wavelength-division-multiplexing (WDM), that is, the simultaneous transmission of multiple signals in different frequency (or equivalently wavelength) ‘‘channels’’ (Hasegawa 2000). However, it was found that a serious problem affected WDM systems. Namely, the interactions of solitons traveling at different velocities cause resonant amplifier-induced instabilities in adjacent frequency channels (four-wave mixing (Mamyshev and Mollenauer 1996, Ablowitz et al. 1996)). In order to avoid these instabilities, researchers developed and analyzed dispersion-managed (DM) transmission systems (cf. Hasegawa (2000)). In a DM transmission system, the fiber is composed of alternating sections of positive (normal) and negative (anomalous) dispersion fibers. The (dimensionless) NLS equation that governs this phenomenon is

Optical Communications

Hasegawa and Tappert (1973) first suggested using solitons as the ‘‘bit’’ format for transmission of information in optical fiber systems. Motivated by this, in 1980, scientists at Bell Laboratories observed solitons (described by the NLS equation) in optical fibers (Mollenauer et al. 1980). The development of optical amplifiers (erbium-doped amplifiers) in the mid-1980s provided a mechanism to compensate fiber loss, and this permitted the transmission of information entirely optically over long distances. With damping and amplification included (see, e.g., Hasegawa and Kodama (1995)), the NLS equation [20] takes the form i

@q sgnðk000 ð!0 ÞÞ @ 2 q þ þ gðzÞjqj2 q ¼ 0 @z 2 @t2

i

@q dðzÞ @ 2 q þ þ gðzÞjqj2 q ¼ 0 @z 2 @t2

½26

where d(z) is usually taken to be a periodic, large, rapidly varying function of the form d(z) = a þ (z), with j(z)j 1 and having zero average in the period za (generally the same as that of the amplifier). In fact, asymptotic analysis of [26] yields a nonlocal NLS-type equation (Gabitov and Turitsyn 1996, Ablowitz and Biondini 1998). It has also been shown that eqn [26] admits various types of optical pulses, such as DM solitons (Ablowitz and Biondini 1998), and quasilinear modes (Ablowitz et al. 2001). NLS Equation in Other Settings

½25

g(z) = a20

exp( 2z=za ), 0 < z < za , and periwhere odically extended thereafter, and a20 is determined by Z 1 za < g >¼ gðz=za Þdz ¼ 1 za 0 with za = la =z , la being the amplifier length. Remarkably, asymptotic analysis (za  1) shows that, to leading order, q(z, t) still satisfies the NLS equation [20]. Amplifiers, however, introduce small amounts of noise to the system, which causes the temporal position of the soliton to fluctuate (cf. Gordon and Haus (1986)) and thus limits the distance signals can

Many other interesting applications of the NLS equations exist in such different areas of physics as magnetic spin waves (see, e.g., the work by Zvezdin and Popkov (1983) and also by Kalinikos et al. (1997)), plasma physics (cf. the work by Zakharov (1972) on collapse of Langmuir waves), other areas of fluid dynamics, etc. (the interested reader can find an overview in the monograph by Ablowitz (1981)).

Mathematical Framework Mathematically, the NLS equation had attained broad significance since it is integrable via

Nonlinear Schro¨dinger Equations

inverse-scattering transform (IST), admits multisoliton solutions, has an infinite number of conserved quantities, and possesses many other interesting properties. Some of these are discussed below. The Inverse-Scattering Transform

The IST method allows one to linearize a large class of nonlinear evolution equations and can be considered as a nonlinear version of the Fourier transform. An essential prerequisite of IST method is the association of the nonlinear evolution equation with a pair of linear problems (Lax pair), a linear eigenvalue problem, and a second associated linear problem, such that the given equation results as a compatibility condition between them. A key research breakthrough on NLS systems appeared in 1972, in the papers of Zakharov and Shabat (1972, 1973), who first analyzed the scalar NLS equation in the form

557

data S(k, 0) are evolved via eqn [29] to get S(k, t) at an arbitrary time t > 0. Finally, by employing the methods of inverse scattering, eqn [28] allows one to reconstruct the evolved solution q(x, t) from S(k, t). One can easily note the ‘‘formal’’ resemblance to the well-known method of Fourier transform for linear differential equations. There is considerable literature on the subject and the interested reader is encouraged to consult, for instance, some of the following references: Ablowitz and Segur (1981), Calogero and Degasperis (1982), Novikov et al. (1984), Ablowitz and Clarkson (1991), Ablowitz et al. (2004). Linear Stability Analysis

Consider a special solution of eqn [27] in the focusing (þsign) case: q = a exp(2ia2 t). If this solution is perturbed as 2

iqt ¼ qxx 2jqj2 q

( correspond to the focusing/defocusing case, respectively) and found the associated Lax pair   ik q v ½28 vx ¼ q ik  vt ¼

2ik2  ijqj2 2kq  iqx

 2kq  iqx v 2ik2 ijqj2

qðx; tÞ ¼ ae2ia t ð1 þ ðx; tÞÞ

½27

where jj  1, it is found that  satisfies the condition it ¼ xx þ 2a2 ð þ  Þ On the periodic spatial domain 0 < x < L,  has the Fourier expansion ðx; tÞ ¼

½29

where v(x, t) is a two-component vector. The compatibility of [28] and [29] yields eqn [27], assuming that the eigenvalue parameter k is constant in time (so that [27] is often said to be isospectral). The solution of the initial-value problem of a nonlinear evolution equation by IST proceeds in three steps, as follows: 1. the forward problem – the transformation of the initial data from the original ‘‘physical’’ variables to the transformed ‘‘scattering’’ variables; 2. time dependence – the evolution of the transformed data according to simple, explicitly solvable evolution equations; and 3. the inverse problem – the recovery of the evolved solution in the original variables from the evolved solution in the transformed variables. The implementation of steps 1–3 described above is more concretely carried out as follows. The initial (Cauchy) datum q(x, 0) for eqn [27] is mapped into scattering data S(k, 0) (comprising, in general, discrete eigenvalues and associated normalization constants, and reflection coefficients) by means of eqn [28]. The

1 X

^n ðtÞei n x

1

where

n ¼

2n L

½30

Assuming a solution of the form     ^n  in t ¼ e  ^n  one finds that n satisfies   2n ¼ 2n 2n  4a2

½31

It then turns out that when aL= < n the system is unstable. Note that there are only a finite number of unstable modes (i.e., for fixed a, L, sufficiently high mode numbers n will not satisfy the above inequality). In the context of water waves, this corresponds to the famous experimental and theoretical result by Benjamin and Feir that the Stoke’s water wave is unstable. Later, Benney and Roskes (1969) showed that all periodic wave solutions of the generalized nonlocal NLS equation resulting from water waves in (2 þ 1)-dimensions are unstable. Also, in (2 þ 1)dimensions soliton solutions are unstable to weak transverse modulations.

558 Nonlinear Schro¨dinger Equations Wave Collapse

The equation i

t

þ  þ j j2 ¼ 0;

x  ðx; yÞ 2 R2

½32

has the following conserved quantities: Z P ¼ j j2 dx Z M¼ r dx

Z 1 H¼ jr j2  j j4 dx 2 that is, mass (power), momentum, and energy (Hamiltonian) are conserved. Remarkably, Talanov (1965) showed that eqn [32] satisfies the following equation: @2V ¼ 8H @t2

½33

where V¼

Z

ðx2 þ y2 Þj j2 dx dy

Equation [33] is also known as the ‘‘virial’’ theorem. Hence, it follows that V ¼ 4Ht2 þ c1 t þ c2 and if H < 0 initially, then a singularity in eqn [32] results since V must be positive. Actually, one can further show (see, e.g., C Sulem and P L Sulem (1999), and references therein) that there exists a time t such that Z jr j2 dx becomes infinite as t ! t , which in turn implies that also becomes infinite as t ! t (blowup in finite time). Note also that for the more general equation i

t

þ d þ j j2 ¼ 0;

x 2 Rd

where d is the d-dimensional Laplacian, one has the following types of solutions:

Supercritical (d > 2): the solution blows up. Critical (d = 2): blowup can occur or global solution can exist.

Subcritical (d < 2): global solutions exist. Vector NLS Systems

In many applications vector NLS (VNLS) systems are the key governing equations. Physically, the VNLS

arise under conditions similar to those described by NLS with the additional proviso that there are multiple wave trains moving nearly with the same group velocities (Roskes 1976). Importantly, VNLS also models systems where the field has more than one component. For example, in optical fibers and waveguides, the propagating electric field has two components transverse to the direction of propagation. The nondimensional system   ð1Þ ð1Þ 2 ð2Þ 2 qð1Þ ¼ q þ 2 jq j þ jq j ½34a iqð1Þ z xx   ð2Þ ð1Þ 2 ð2Þ 2 ¼ q þ 2 jq j þ jq j iqð2Þ qð2Þ z xx

½34b

is an asymptotic model which governs the propagation of the electric field in a waveguide, where z is the normalized distance along the waveguide and x a transversal spatial coordinate. It was first examined by Manakov (1974) (see also Anastassiou et al. (1999) and Soljacic´ et al. (2003)). Subsequently, this system was derived as a key model for light-wave propagation in optical fibers. More precisely, in optical fibers with constant birefringence (i.e., constant phase and group velocities as a function of distance) Menyuk (1987) has shown that the two polarization components of the electromagnetic field E = (u, v)T which are orthogonal to the direction of propagation, z, along the fiber asymptotically satisfy the following nondimensional equations (assuming anomalous dispersion): iðuz þ ut Þ þ 12 utt þ ðjuj2 þ jvj2 Þu ¼ 0

½35a

iðvz  vt Þ þ 12 vtt þ ðjuj2 þ jvj2 Þu ¼ 0

½35b

where represents the group velocity ‘‘mismatch’’ between the u, v components of the electromagnetic field,  is a constant that depends on the polarization properties of the fiber, z the distance along the fiber, and t a retarded temporal frame. In deriving eqn [35], it is assumed that the electromagnetic field is slowly varying (as in the scalar problem); certain nonlinear (four-wave mixing) terms are neglected in the derivation of eqn [35], because the light wave is rapidly varying due to large, but constant, linear birefringence. In this context, birefringence means that the phase and group velocities of the electromagnetic wave in each polarization component are different. In a communications environment, due to the distances involved (hundreds to thousands of kilometers), the polarization properties evolve rapidly and randomly as the light wave evolves along the propagation distance, z. Not only does the birefringence evolve, but it does so randomly, and on a scale much faster than the distances required for

Nonlinear Schro¨dinger Equations

communication transmission (birefringence polarization changes on a scale of 10–100 m). In this case, the relevant nonlinear equation is eqn [35] above, but with = 0 and  = 1. Indeed, this is the integrable VNLS equation first derived by Manakov (1974). It should be remarked that the VNLS equation [34] and its generalization to an arbitrary number of components, 2

iqt ¼ qxx 2kqk q

½36

where q is an N-component vector and kk is the Euclidean norm, are integrable by the IST. One has to suitably extend the analysis discussed earlier in this article (cf. e.g., Ablowitz et al. (2004)). Discrete NLS Systems

Both the NLS and the VNLS equations discussed above admit integrable discretizations which, besides being used as the basis for constructing numerical schemes for the continuous counterparts, also have physical applications as discrete systems. A natural discretization of NLS [27] is the following: i

d 1 qn ¼ 2 ðqnþ1  2qn þ qn1 Þ dt h jqn j2 ðqnþ1 þ qn1 Þ

½37

which is referred to as the integrable discrete NLS (IDNLS). It is an O(h2 ) finite-difference approximation of [27] which is integrable via the IST and has soliton solutions on the infinite lattice (Ablowitz and Ladik 1975, 1976). Note that if the nonlinear term in [37] is changed to 2jqn j2 qn , the equation, which is often called the discrete NLS (DNLS) equation, is apparently no longer integrable. It should be remarked that the (apparently nonintegrable) DNLS equation arises in many important physical contexts. Correspondingly, one can consider the discretization of VNLS given by the following system: i

 d 1 qn ¼ 2 qnþ1  2qn þ qn1 dt h   kqn k2 qnþ1 þ qn1

½38

where qn is an N-component vector. Equation [38] for qn = q(nh) in the limit h ! 0, nh = x gives VNLS [36]. The discrete vector NLS system [38] is also integrable (Ablowitz et al. 1999, Tsuchida et al. 1999). The interested reader can find further details in Ablowitz et al. (2004). See also: Boundary-Value Problems for Integrable Equations; Dynamical Systems in Mathematical Physics: An Illustration from Water Waves; Evolution Equations: Linear and Nonlinear; Ginzburg–Landau Equation;

559

Integrable Systems and Discrete Geometry; Integrable Systems: Overview; Partial Differential Equations: Some Examples; Riemann–Hilbert Methods in Integrable Systems; Schro¨dinger Operators.

Further Reading Ablowitz MJ and Biondini G (1998) Multiscale pulse dynamics in communication systems with strong dispersion management. Optics Letters 23: 1668–1670. Ablowitz MJ, Biondini G, and Blair S (1997) Multi-dimensional propagation in non-resonant (2) materials. Physics Letters A 236: 520–524. Ablowitz MJ, Biondini G, Chakravarty S, Jenkins RB, and Sauer JR (1996) Four-wave mixing in wavelength-division multiplexed soliton systems: damping and amplification. Optics Letters 21: 1646. Ablowitz MJ and Clarkson PA, Nonlinear Waves, Solitons and Symmetries, London Mathematical Society Lecture Notes Series. Cambridge: Cambridge University Press (to be published). Ablowitz MJ and Clarkson PA (1991) Solitons Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Notes Series, vol. 149. Cambridge: Cambridge University Press. Ablowitz MJ, Hirooka T, and Biondini G (2001) Quasi-linear optical pulses in strongly dispersion managed transmission systems. Optics Letters 26: 459–461. Ablowitz MJ and Ladik JF (1975) Nonlinear differential-difference equations. Journal of Mathematical Physics 16: 598–603. Ablowitz MJ and Ladik JF (1976) Nonlinear differentialdifference equations and Fourier analysis. Journal of Mathematical Physics 17: 1011–1018. Ablowitz MJ, Ohta Y, and Trubatch AD (1999) On discretizations of the vector nonlinear Schro¨dinger equation. Physics Letters A 253: 253–287. Ablowitz MJ, Prinari B, and Trubatch AD (2004) Continuous and Discrete Nonlinear Schro¨dinger Systems, London Mathematical Society Lecture Notes Series, vol. 302. Cambridge: Cambridge University Press. Ablowitz MJ and Segur H (1981) Solitons and the inverse scattering transform. Soceity for Industrial and Applied Mathematics 4. Anastassiou C, Segev M, Steiglitz K, Giordmaine JA, Mitchell M et al. (1999) Energy-exchange interactions between colliding vector solitons. Physical Review Letters 83: 2332. Benney DJ and Newell AC (1967) The propagation of nonlinear wave envelopes. Journal of Mathematical Physics 46: 133–139. Benney DJ and Roskes GJ (1969) Wave instabilities. Studies in Applied Mathematics 48: 377–385. Calogero F and Degasperis A (1982) Spectral Transform and Solitons I. Amsterdam: North-Holland. Chiao RY, Garmire E, and Townes CH (1964) Self-trapping of optical beams. Physical Review Letters 15: 479–482. Davey A and Stewartson K (1974) On three-dimensional packets of surface waves. Proceedings of the Royal Society of London Series A 338: 101–110. Gabitov I and Turitsyn S (1996) Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation. Optics Letters 21: 327. Ginzburg VL (1956) On the macroscopic theory of superconductivity. Soviet Physics – JETP 2: 589 (russian: Journal of Experimental and Theoretical Physics USSR 29: (1955) 748–761.). Ginzburg VL and Pitaevskii LP (1958) On the theory of superfluidity. Soviet Physics – JETP 7: 858–861 (russian: Journal of Experimental and Theoretical Physics. USSR 34: 1240–1245).

560 Non-Newtonian Fluids Gordon JP and Haus HA (1986) Random walk of coherently amplified solitons in optical fiber transmission. Optics Letters 11: 665. Gross EP (1961) Structure of quantized vortex. Nuovo Cimento 20: 454. Hasegawa A (ed.) (2000) Massive WDM and TDM Soliton Transmission Systems. Dordrecht: Kluwer Academic. Hasegawa A and Kodama Y (1995) Solitons in Optical Communications. Oxford: Oxford University Press. Hasegawa A and Tappert F (1973a) Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I. Anomalous dispersion. Applied Physics Letters 23: 142. Hasegawa A and Tappert F (1973b) Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers II. Normal dispersion. Applied Physics Letters 23: 171. Kalinikos BA, Kovshikov NG, and Patton CE (1997) Decay-free microwave envelope soliton pulse trains in yittrium iron garnet thin films. Physical Review Letters 78: 2827–2830. Kelley PL (1965) Self-focusing of optical beams. Physical Review Letters 15: 1005–1008. Kodama Y and Hasegawa A (1992) Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect. Optics Letters 17: 31. Landau LD and Ginzburg VL (1950) Journal of Experimental and Theoretical Physics USSR 20: 1064. Lighthill MJ (1965) Contribution to the theory of waves in nonlinear dispersive media. Journal of the Institute for Mathematics and Its Applications 1: 269–306. Mamyshev PV and Mollenauer LF (1996) Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexed transmission. Optics Letters 21: 396. Manakov SV (1974) On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Soviet Physics – JETP 38: 248–253. Marcuse D, Menyuk CR, and Wai PKA (1997) Applications of the Manakov-pmd equations to studies of signal propagation in fibers with randomly-varying birefringence. Journal of Lightwave Technology 15: 1735–1745. Mecozzi A, Moores JD, Haus HA, and Lai Y (1991) Soliton transmission control. Optics Letters 16: 1841. Menyuk CR (1987) Nonlinear pulse propagation in birefringent optical fibers. IEEE Journal of Quantum Electronics 23: 174–176. Mollenauer LF, Stolen LF, and Gordon JP (1980) Experimental observation of picoseconds pulse narrowing and solitons in optical fibers. Physical Review Letters 45: 1095.

Novikov SP, Manakov SV, Pitaevskii LP, and Zakharov VE (1984) Theory of Solitons. The Inverse Scattering Method. New York: Plenum. Pethick CJ and Smith H (2002) Bose–Einstein Condensation in Dilute Gases. Cambridge: Cambridge University Press. Pitaevskii LP (1961) Soviet Physics JETP 13: 451 (russian: Zhurnal Ekspermentalnoi i Teoreticheskoi Fiziki 40: 646.). Roskes GJ (1976) Some nonlinear multiphase interactions. Studies in Applied Mathematics 55: 231. Soljacic´ M, Steiglitz K, Sears SM, Segev M, Jakubowski MH et al. (2003) Collisions of two solitons in an arbitrary number of coupled nonlinear Schro¨dinger equations. Physical Review Letters 90(25): 254102. Sulem C and Sulem PL (1999) The Nonlinear Schro¨dinger Equation – Self-Focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139. Springer. Talanov VI (1964) Radiophysics 7: 254. Talanov VI (1965) Self-focusing of wave beams in nonlinear media. Soviet Physics – JETP Letters 109: 138. Tsuchida T, Ujino H, and Wadati M (1999) Integrable semidiscretization of the coupled nonlinear Schro¨dinger equations. Journal of Physics A: Mathematical and General 32: 2239–2262. Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. Soviet Physics Journal of Applied Mechanics and Technical Physics 4: 190–194. Zakharov VE (1972) Collapse of Langmuir waves. Soviet Physics – JETP 35: 908–914. Zakharov VE and Shabat AB (1972) Exact theory of twodimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics – JETP 34: 62–69. Zakharov VE and Shabat AB (1973) Interaction between solitons in a stable medium. Soviet Physics – JETP 37: 823–828. Zakharov VE, Sobolev VV, and Synakh VC (1971) Behavior of light beams in nonlinear media. Soviet Physics – JETP 33: 77–81. Zvezdin AK and Popkov AF (1983) Contribution to the nonlinear theory of magnetostatic spin waves. Soviet Physics – JETP 57: 350–355.

Non-Newtonian Fluids C Guillope´, Universite´ Paris XII – Val de Marne, Cre´teil, France ª 2006 Elsevier Ltd. All rights reserved.

A fluid is said to be Newtonian if it satisfies the simplest constitutive equation, which gives the stress tensor  as a linear function of the rate of deformation tensor D = (1=2)(ru þ ruT ), namely  ¼ ð tr D  pÞI þ 2D

Introduction The flow of a fluid, liquid or gas, is described by three conservation laws, the conserved physical quantities being the mass, the linear momentum, and the energy, and by constitutive equations. The constitutive equations are specific to each fluid, and link deformations to stresses.

½1

where u is the fluid velocity, p is the hydrostatic pressure (p 0), and and  are the Lame´ viscosity coefficients of the fluid, satisfying  0 and þ 2=3 0. The superscript T designates the transpose operation, the abbreviation ‘‘tr’’ the trace operator of a tensor, and I the unit tensor. Water and glycerin are examples of Newtonian liquids.

Non-Newtonian Fluids 561

Non-Newtonian fluids are fluids for which the behavior is not described by eqn [1]. Silicone oils, polymers (melted or in solution), egg yolks, and blood are examples of non-Newtonian liquids. Other examples include liquid crystals, rubbers, suspensions, paints, etc. In the following we shall first describe flows which show Newtonian or non-Newtonian behaviors. Then we shall describe the requirements a constitutive equation needs to satisfy to be considered, introducing the notions of continuum mechanics we need. After giving the most commonly used constitutive equations, we will give a few ideas about the mathematical study of the set of equations, and their numerical study, in the particular case of viscoelastic fluids. Numerous kinds of materials are already known to exist, and more might exist in the future. This report, however, will be limited to the most commonly materials used nowadays, which are polymers, liquid crystals and polymeric liquids crystals, and paints. Moreover, we shall only consider isothermal flows, even though temperature might be an important parameter in experiments or in industry, because in particular most theoretical or numerical studies concern isothermal problems. Non-Newtonian fluids will always be liquids, and we shall use the terms liquid or fluid indifferently.

force and that a dip on the surface of the liquid near the rod results. On the contrary, if we make the same experiment with a polymer, the fluid climbs along the rod. Moreover, for comparable rotation speed, the difference in behaviors might be quantitatively considerable. This is explained by totally different pressure repartitions in both fluids, Newtonian or non-Newtonian: in particular, the pressure in the polymer along the rod is much larger than that along the beaker, so that this pressure difference fights the centrifugal force; this is in contrast with the situation in a Newtonian fluid. Extrudate Swell

If a fluid is forced to flow from a large reservoir out of a circular tube of small diameter, the swell at the exit is much larger for a polymer solution than for a Newtonian fluid. A polymer flowing out of a die might also show a delayed die well, which means that the swell is not at the exit but on the jet at a certain distance of the exit. The explanation of this phenomenon is not unique: it is due partly to memory effects (the fluid remembers its former shape, the one in the reservoir), partly to the release of normal stresses, to interfacial forces, compressibility, viscous heating, and the complicated flow near the die exit. Difference in Normal Stresses

We describe a few experiments to show how differently both types of fluids, Newtonian or nonNewtonian, might react in some experimental situations. We also give some mechanical explanation when possible.

In a shearing flow of a Newtonian fluid, the two normal stress differences are both zero, whereas for a polymer the first normal stress difference might be very large, the second one being nearly zero. These differences in stresses in shearing flow might be a partial answer to the extrudate swell and to rod climbing experienced by polymers.

Shear Thinning or Shear Thickening

Presence of a Yield Stress

In a Poiseuille experiment, where a fluid flows in a tube under the action of a pressure drop, the volumetric flow rate of a Newtonian fluid is inversely proportional to the constant fluid viscosity. Under the same pressure-drop condition, a polymer melt flows much faster out of the tube, which means that there is a decreasing apparent viscosity with increasing shear rate: this is referred to as shear thinning effect. Other fluids might exhibit the opposite behavior and flow out of the tube more slowly: this is called the shear thickening effect.

Some materials, when subjected to shear stress, flow only after a critical value is attained. Such fluids are referred to as Bingham fluids: some cements, slurries, paints, and biological fluids might exhibit such a behavior. It is actually a well-known property of paints: if put in large quantities on a vertical wall, the paint will flow, whereas if put as a very thin film on the same wall, the paint will not flow, but stay in place, and dry to form a nice colored covering.

Non-Newtonian Behaviors

Preferred Orientation of the Particles of Fluid Rod Climbing

When a rotating rod is inserted in a beaker filled with a Newtonian fluid, it is observed that the liquid near the rotating rod is pushed outwards by centrifugal

Fluids with properties as above, Newtonian or non-Newtonian, are isotropic in nature, even though they are constituted of atoms, or of long chains of material. They are the same everywhere, optically,

562 Non-Newtonian Fluids

magnetically, or electrically. Some fluids, liquid crystals, or polymeric liquid crystals in particular, have remarkable properties of nonanisotropy, being able to orient themselves, on average, along a particular direction: this is the nematic phase, which is used in many devices (screens for clocks, hand calculators, and cell phones), because the average orientation may be changed by applying an electric field. Other phases of liquid crystals include smectic A, C, and C phases, where one sees a preferred orientation (tilted for C phases) of the fluid, and also a layer-like structure. As an example, let us mention discotic nematic liquid crystals, which are precursors for carbon-based materials, such as fibers, composites, and films, which possess excellent mechanical and thermal properties. Sails for race sailing boats are made of Kevlar, which is one of these new materials with remarkable properties.

Modeling The flowing fluid will be described by its (Eulerian) velocity at time t and position x, say u(x, t), for x belonging to the domain of the flow  and the time t to R þ , by its mass density (x, t), its pressure p(x, t) (p > 0 defined up to an additive constant), and its stress (x, t) – which is a symmetric tensor. The partial differential equations describing the flow are satisfied in the domain of the flow and read as follows: @ þ divðuÞ ¼ 0  @t  @u þ ðu  rÞu ¼ div  þ f  @t

is closed, its interior is connected and dense everywhere, its boundary is piecewise regular, C0 at least. A mapping  : 0 ! t is a deformation if  is a bijection from 0 onto t and is a C1 –diffeomorphism from the interior of 0 onto the interior of t , with positive Jacobian. The motion of a body S is given by a set of deformations (t, t0 ) : t ! t0 , satisfying ðt; tÞ ¼ Id;

ðt00 ; tÞ ¼ ðt00 ; t0 Þ  ðt0 ; tÞ

The trajectory of the material point which is in X at t0 is the set fðt; t0 ÞðXÞ; t  t0 g A body is said to be rigid if the deformation (t, t0 ) is an isometry for all times t and t0 . A material point p is said to be attached to the rigid body S if the body p [ S is rigid. The motion of a fluid might be described in terms of the Lagrangian coordinates X 2 0 of each particle of fluid: 0 is called the reference configuration and is the fixed configuration occupied by the body of fluid at the time of reference, say t0 . The motion of the fluid might also by described in terms of the Eulerian coordinates x = (X, t), which represent the position of a particle at time t which has position X at t0 . The Lagrangian and Eulerian coordinates of the same particle of fluid are linked by the differential equation _ ðX; tÞ ¼ uððX; tÞ; tÞ;

½2

where f denotes some external forces applied to the fluid. These equations describe the conservation of mass and the conservation of linear momentum. To close the system, we need a constitutive equation for the stress  as well as initial conditions and boundary conditions. Moreover, most non-Newtonian fluids are practically incompressible in most regions of the flow, so that we shall only consider this case: the first equation in [2] is replaced by condition div u = 0 in the domain of the flow.

for t  t0

ðX; t0 Þ ¼ X For defining the constitutive equations, we shall use a few tensors that we define now. The deformation gradient is defined by F(X, t) = @ (X, t)=@X, and the right Cauchy–Green tensor by C = FT F (also called Cauchy strain). To define relative tensors, we denote by  = t (x, s) the position at time s  t of the material point, which is at x at time t. The relative tensors are defined in the following way:

the relative deformation gradient F t (s) = rt (x, s), the relative right Cauchy–Green tensor Ct (s) = FTt (s) F t (s), and

the relative Finger tensor Ct (s)1 . Notions of Continuum Mechanics

At time t, a body S occupies a region t of the Euclidean space E 3 , called the configuration at time t, of the body. Points p of S are called material points or particles of fluids. The configuration t is assumed to be regular in the following sense: t

Note that the rate of deformation tensor is obtained as the time derivative of the relative Cauchy strain tensor: D¼

1 @Ct ðsÞ j 2 @s s¼t

Non-Newtonian Fluids 563 Principle of Objectivity and Frame Invariance

A frame of reference is defined in the spacetime E 3 R attached to the observer by giving a chronology and a system of reference. The chronology is a timescale, which will be assumed to be the same for all observers. The system of reference is a set of at least four points attached to a rigid body (this is the observer), which are not coplanar. The constitutive equation needs to satisfy the principle of frame invariance and of frame indifference (or objectivity), which means that the equation does not depend on rigid motions of the observer. In the mathematical framework, it means that the equation has to be invariant under a change of orthonormal frame of reference x = Q(t)x, where Q(t) is an orthogonal tensor: the transformed equation has to have the same expression, and also to be frame indifferent. We define a scalar quantity ’, a vector field u, or a tensor field , as being frame indifferent if, under the change of variables x = Q(t)x, they satisfy the relations ’(x, t) = ’ (x, t), u(x, t) = Q(t)T u (x , t), and (x, t) = Q(t)T   (x , t)Q(t), respectively. The velocity gradient ru is not frame indifferent, but its symmetric part is. The vorticity, which is the antisymmetric part W = (ru  ruT )=2 of the velo_ = QT W  Q  city gradient, satisfies the equation W T _ Q Q, where the dot denotes the convective derivative d=dt = @=@t þ (u  r). Note that the convective derivative of a scalar function ’ is frame indifferent, which means that @’ @’ þ ðu  rÞ’ ¼ þ ðu  r Þ’ @t @t but the convective derivative of a vector or a tensor is not frame indifferent. It can be easily checked that the derivative D0  d ¼ þ W  W  Dt dt

½3

of a (frame-indifferent) tensor  is frame indifferent, which means that D0  D   ¼ QT 0 Q Dt Dt To obtain another frame-indifferent derivative of a tensor , we need to start with the expression [3], to which we may add other terms containing frameindifferent quantities, for example, combinations of  and D. A derivative which is often considered is the Oldroyd derivative, as introduced by Oldroyd in 1958:

Da  d ¼ þ W  W   aðD þ DÞ Dt dt

½4

where a is a real parameter, chosen in the interval [1, 1]. (This restriction on a is necessary for viscometric reasons, and obtained when simple flows, such as Couette or Poiseuille flows, are studied.) The case a = 1 corresponds to the upper convected derivative, and the case a = 1 to the lower convected derivative. The case a = 0 refers to the corotational or Jaumann derivative. Derivatives corresponding to cases a = 1, 0, or 1 might actually be obtained by derivating  in a frame fixed locally to the body of fluid, and which rotates and/or deforms with the body. Moreover, we shall see that the derivatives corresponding to a = 1 or 1 have very simple integral expressions. Constitutive Equations

The constitutive equation of a non-Newtonian fluid is a nonlinear relationship between the stress tensor and objective variables depending on the flow, such as the pressure, the rate of deformation, frameindifferent derivatives of such quantities, etc. Analogously to the constitutive equation for an incompressible Newtonian fluid, we may also write the stress tensor in the form  = pI þ . The extra stress tensor  could be either a function of objective variables, which characterize the flow, or defined by a differential equation or by an integral equation. The point here is to model the fact that the fluid might have some elasticity or some memory, or might experience, for example, yield stress or orientational properties. Shear dependent viscosity fluids A very simple generalization of the incompressible Newtonian fluid consists in making the viscosity dependent on the rate of deformation tensor,  = (D). This generalization has been introduced by O A Ladyzhenskaya in 1970 and, if the function is chosen properly, this model reproduces the behavior of existing fluids, at least in certain parts of their flow. For power-law fluids, the viscosity depends on the second invariant ID = (1=2)tr D2 of the symmetric tensor D (the first invariant tr D is zero because of incompressibility), and reads as ðDÞ ¼ 0 þ mIn1 D

½5

where 0  0, m > 0, and n  0. If n = 1, we recover the Newtonian case, whereas for n < 1 this equation describes a shear thinning fluid, and for n > 1 a shear thickening fluid. The power law is not valid for ID

564 Non-Newtonian Fluids

close to 0, so that the Carreau–Yasuda law is preferred:  ðn1Þ=ð2Þ   1 ¼ 1 þ ðID Þ2 ½6 0  1 where 0 is the zero-shear rate viscosity, 1 is the infinite-shear rate viscosity,  a time constant, n a dimensionless power-law index, n  0, and  > 0 a parameter (generally equal to 1 for a monomolecular polymer). Oldroyd models and related models Oldroyd models are differential models built with one of the Oldroyd derivatives, and are very commonly used for polymer solutions or melts. The stress tensor is given as a solution of a differential equation in the following way:   Da  Da D þ gð; DÞ ¼ 2 D þ 2  þ 1 ½7 Dt Dt where 1 > 0 is a relaxation time, 2 is a retardation time, 0  2 < 1 , and g(, D) is a tensor-valued function, constrained to certain restrictions due to objectivity, and which is at least quadratic. The Johnson–Segalman model has g = 0, and 1  a  1. Other models of differential type often suppose the parameter a to be 1, because it has been noticed that with a close to 1 the model is able to reproduce some experimental behavior, whereas for a =  1 or close to 1, the model does not work at all. Among the models with a = 1, the following ones are fairly popular: the model of Phan-Thien and Tanner has g(, D) =  tr , where  is a constant; this model can be generalized by defining g(, D) =  2 þ ,  and being functions of the trace of  and of its determinant; the model of Giesekus is the particular case where  is a constant and = 0. The Oldroyd eight-constant model is given by gð; DÞ ¼ 0 ðtr ÞD þ 1 trðDÞ I 2

2

þ 2 D þ 2 trðD Þ I where 0 , 1 , 2 , and 2 are constants. In [7], the limit case 2 = 0 corresponds to Maxwell’s type models, where there is no Newtonian viscosity, while the case 2 > 0 corresponds to the Jeffreys’ type models. The cases where a = 1 and g = 0, are often considered in mathematical or numerical studies: this is the upper convected Maxwell (UCM) model for 2 = 0, and the Oldroyd B model for 2 > 0. The parameters 1 , 2 , and  might also depend on ID : such a model where the upper convected derivative (a = 1) is chosen is referred to as the White–Metzner model, and reads as follows:

   D1  D1 D ¼ 2 I D þ 1 D þ  I  þ I Dt Dt where 1 is also the Newtonian viscosity. Integral equations Other constitutive equations for viscoelastic fluids include integral equations. Actually, some differential equations have integral counterparts: this is the case for the differential equations associated with the upper or lower convected frame-indifferent derivatives. For the upper convected derivative (a = 1), the extra stress is given by the integral expression 2 1  2 Dðx; tÞ þ 2 1 21 Z t

eðtsÞ=1 ðrX xÞ DðX; sÞðrX xÞT ds

ðx; tÞ ¼ 2

1

where X is the position, at time s, of the point which is at x at time t. A similar expression might be obtained for the lower convected derivative. A very common integral equation is the K–BKZ equation (introduced independently by Kaye and Bernstein, Kearsley, and Zapas in 1962–63). In a simplified form, the extra-stress tensor is given as the integral of a combination of the relative Cauchy strain tensor Ct and its inverse:  Z t @WðI1 ; I2 Þ 1 ðx; tÞ ¼ 2 Gðt  sÞ Ct ðsÞ @I1 1  @WðI1 ; I2 Þ  Ct ðsÞ ds @I2 where I1 = tr C1 t (s) and I2 = tr Ct (s). The function G is a given kernel, and W a given scalar potential. The upper convected Maxwell model is obtained from the K–BKZ model by setting W(I1 , I2 ) = I1 and G(s) = (1 2 =2) e1 s . Models issued from kinetic theories or micro–macro models Polymeric fluids could also be modeled by coupling a macroscopic viewpoint – the one of continuum mechanics, as described above – and a microscopic viewpoint. A polymer is, in general, made of long chains of molecules. Rather than trying to represent the polymer behavior by a sophisticated constitutive equation, one describes the mean behavior of the molecules by using their microscopic description. To take an example, we consider a dilute solution of polymer, where each chain of polymer is modeled as a collection of dumbbells, each of them consisting of two beads connected by a spring. The configuration of the spring, namely its length and orientation, is described by a random vector field Q 2 R3 . The dumbbells are convected and stretched by the flow.

Non-Newtonian Fluids 565

The probability (x, Q, t) dQ of finding a dumbbell with a configuration Q at (x, t) is governed by a Fokker–Planck equation: d þ divQ ððruÞQ Þ dt 2 2kT Q ¼ divQ ððrQ WÞ Þ þ where is the friction coefficient of the dumbbell beads, T the temperature, and k the Planck constant, and W the spring potential. The extra stress is given by the constitutive equation Z  ¼  ðrQ W  QÞ ðx; Q; tÞ dQ The simplest potential is the linear one (also called Hookean potential) W(Q) = HjQj2 , where jQj is the length of Q, and H the elasticity constant. In fact, in the case of the Hookean potential, this set of equations is equivalent to the Oldroyd B model. Another potential corresponds to finitely extendable nonlinear elastic (FENE) chain of dumbbells, ! HQ20 jQj2 WðQÞ ¼  log 1  2 2 Q0 for jQj  Q0 , and gives the FENE model, for which there is no macroscopic constitutive equation known. We have only made here a short incursion in these micro–macro models: research is in progress, both ¨ ttinger 1996, Suen et al. analytical and numerical (O 2002, Keunings 2004).

Liquid crystals and polymeric liquid crystals As an example, we present the constitutive equations for a uniaxial nematic liquid crystal. In the theory of Leslie and Ericksen, established in the 1960s and the 1970s, the stress tensor is given as a function of the orientation unit vector n, through the Oseen–Frank elastic energy, 2Wðn; rnÞ ¼ 1 ðdivnÞ2 þ 2 ðn  curl nÞ2 þ 3 jn curl nj

2

where 1 > 0, 2 > 0, and 3 > 0 are the three basic modes (splay, twist, and bend, respectively). The extra stress tensor is precisely given by the relation @W þ 1 ðn  DnÞn  n @rn þ 2 N  n þ 3 n  N

 ¼  ðrnÞT

þ 4 D þ 5 Dn  n þ 6 n 

where N = n_  W n is the corotational derivative of the director, and i , i = 1, . . . , 6, the six Leslie viscosity coefficients. The director satisfies a differential equation derived from continuum mechanics, € ¼ G þ g þ div 1 n where 1 is the moment of inertia per unit volume, G the external director body force (torque per unit volume),  the director stress tensor, and g the intrinsic director body force. Precisely, g ¼ n  ðrnÞ  ¼n þ

@W  1 N  2 Dn @n

@W @rn

where is a Lagrange multiplier vector, and  = 2 =1 is the reactive parameter, with 1 = 3  2 the rotational viscosity, and 2 = 6  5 = 3 þ 2 the irrotational torque coefficient. Polymeric liquid crystals might have other variables entering in the modeling, such as order parameters, order tensors, etc. Because of the complexity of modeling, most studies concern either very simple flows, such as Couette or Poiseuille flows, or steady flows, or flows for which the coefficients satisfy specific relationships. Reports about earlier studies, theoretical as well as numerical, can be found in Coron et al. (1991), and references therein. The study of polymeric liquid crystals, or of the smectic phase of liquid crystals is at its very early stage and one could look into it in specialized journals, such as the Journal of NonNewtonian Fluid Mechanics, or see Liquid Crystals.

Yield stress fluids Bingham materials have the property of flowing only when the stress magnitude is greater than a critical value, and being a solid otherwise. Precisely, in the simplest and the most widely used model, the Bingham model, the extra stress tensor  is given by the relations  ¼ 2D þ  jj  

D ID

if ID 6¼ 0

½8

if ID ¼ 0

where  > 0 is the yield limit. The Bingham model is generalized in taking the viscosity  to be a function of the shear stress:  is given by the relation  1=2   ¼1þ2 ID

566 Non-Newtonian Fluids

for the Casson law, and by the power law [5] for the Herschel–Bulkley model. The mathematical study was started by Duvaut and Lions (1976), and regained interest recently (Malek and Rajagopal 2005), especially in relation with other recent studies in polymeric liquids. Theoretical and Numerical Problems for Viscoelastic Flows

The mathematical study of viscoelastic fluid flows amounts to studying systems of partial differential equations, which all include either the incompressible Euler equation or the incompressible Navier– Stokes equation as particular cases. In particular, it means that the results obtained from such a study are similar to the ones obtained for Euler or Navier– Stokes equations, and, because of the complexity of the system, the results are expected to be qualitatively as good, actually more often less good, than for these equations. For example, the existence of weak three-dimensional solutions to the Navier– Stokes system is known, while for non-Newtonian flows, this result will be true only in very specific cases. Moreoever, when a result is not known for the Navier–Stokes problem, such as the uniqueness of solution for all data in a three-dimensional problem, there is no hope something similar could be proved for non-Newtonian fluid flows. As an example, we consider the case of Johnson– Segalman fluids, which are described by constitutive equation [7] with g = 0. Recall that the limit case 2 = 0 corresponds to the purely elastic case, and 2 = 1 to the purely Newtonian case. Equation [7] is coupled with the equations of motion: 

du þ rp ¼ div  þ f dt div u ¼ 0

½9

Equations [7] and [9] have to be solved in the domain of the flow, which might be the whole space R3 (or R or R2 in case of symmetries), or a domain , bounded or not, in Rn , n = 1, 2, or 3. These equations are supplemented by appropriate boundary conditions and initial conditions for the velocity u and the extra stress  (no boundary condition on  is needed if the homogeneous nonslip boundary condition u = 0 is chosen). We first make explicit the Newtonian contribution to the stress by setting  =  s þ  p and  s = 2s D. The differential equation for  p is then  p þ 1

Da  p ¼ 2p D Dt

where p = (1  2 =1 ) is the so-called polymeric viscosity, s = (2 =1 ) the so-called Newtonian viscosity (or solvent viscosity). We then use nondimensional variables, so as to make explicit the characteristic parameters, which the flow depends on. The non-Newtonian fluid considered in this model will always be homogeneous: its density  is a constant independent of x and t. The dimensional variables are now asterisked. We define quantities which are characteristic of the flow: a length L, a velocity magnitude U, a stress magnitude T, a force magnitude F, and a pressure P. We operate the change of variables and functions x = x =L, u = u =U, t = Ut =L, and also introduce the nondimensional functions ¼

 ; T

p¼

p ; P

f ¼

f F

After choosing the parameters T, P, and F in an appropriate way, namely T = P = U=L, and F = U=L2 , we obtain the following system du þ rp ¼ ð1  !Þu þ div  þ f dt div u ¼ 0 Da  þ  ¼ 2!D We Dt Re

½10

Here the three nondimensional parameters which the flow depends on are the usual Reynolds number Re = 0 UL= and two other numbers: the Weissenberg number We = U=L measures the elasticity per unit time (sometimes also called the Deborah number), and the parameter ! = p = is the ratio of elastic viscosity to total viscosity (! = 0 corresponds to the Newtonian case, while ! = 1 corresponds to the purely elastic case). System [10] couples a transport equation (the equation for the stress ), and either a Navier– Stokes type equation when ! < 1, or a Euler type equation when ! = 1 (for the velocity u). This system is not hyperbolic, parabolic, or elliptic. Maxwell’s type models (! = 1) display two striking phenomena. First, the Cauchy problem (with initial data) can present Hadamard instabilities, that is, instabilities to short waves. It means, in particular, that the Cauchy problem is not well posed in any good class but analytic. Moreover, the partial differential system for Maxwell’s type steady flows may experience a change of type, analogous to the situation in gas dynamics, if the ‘‘Mach number’’ Re We is larger than 1. Jeffreys’ type models (! < 1), because of the presence of a Newtonian viscosity, do not exhibit such phenomenon, but their study does not enter in

Non-Newtonian Fluids 567

the theory of parabolic equations either, the type of the system being composite. Problems of interest for rheologists, as well as for mathematicians, include in particular the high Weissenberg asymptotics, the high Weissenberg boundary layers, the singularity of flows near a reentrant corner, and the stability of flows. We give a few details about stability questions. Instabilities are seen in experimental extrusion of melted polymers from a pipe: melt fracture designates different phenomena appearing at different stages of the experiment, when the speed of the extrusion is increased, such as sharkskin instability, slight distortions of the extrudate, large distortions and wavyness of the extrudate. One may distinguish two kinds of instabilities. First, constitutive instabilities are associated with nonmonotonicity of constitutive functions and loss of evolutionary property of the equations of motion. Other kinds of instabilities are close to classical hydrodynamic instabilities at increasing Re. Note that in viscoelastic flows the Re is usually very small, and might even be set to zero in some studies. Other mathematical questions for system [10] include existence of weak solutions (for the very special case of Oldroyd model with the Jaumann derivative where (a = 0) in [5]), existence of regular solutions defined on some time interval, depending on the magnitude of the data, and existence of regular solutions for all times. Other studies concern the existence, uniqueness, and stability of steady solutions. Another field of study is the numerical simulation of such flows. In summary, there have been numerous computations made in the field of steady or unsteady viscoelastic fluids, and especially models using continuum mechanics. Standard test problems include the cavitydriven flow, flows inside a 4 : 1 contraction, extrusion flows, flows between eccentric cylinders, and flows in ‘‘wiggly’’ pipes. As mentioned already, the type of the sytem of partial differential equations is composite, neither elliptic nor hyperbolic. The numerical codes have to take into account the precise nature of the set of partial differential equations, so as to be able to obtain noncatastrophic results. One of the main challenges has been to deal with the high-We problem: with increasing We, the results would become totally incoherent, and the numerical algorithms would diverge. Nowadays, with the power of computers increasing, molecular simulations of flows are proposed, using the macro–micro modeling mentioned above. Also, simulations of flows of colloidal suspensions and reacting flows have been undertaken with success.

See also: Compressible Flows: Mathematical Theory; Fluid Mechanics: Numerical Methods; Incompressible Euler Equations: Mathematical Theory; Interfaces and Multicomponent Fluids; Inviscid Flows; Liquid Crystals; Newtonian Fluids and Thermohydraulics; Partial Differential Equations: Some Examples; Stability of Flows; Stochastic Hydrodynamics; Viscous Incompressible Fluids: Mathematical Theory.

Further Reading Baranger J, Guillope´ C, and Saut J-C (1996) Mathematical analysis of differential models for viscoelastic fluids. In: Piau J-M and Agassant J-F (eds.) Rheology for Polymer Melt Processing, pp. 199–236. Amsterdam: Elsevier. Bird RB, Armstrong RC, and Hassager O (1987a) Dynamics of Polymeric Liquids. Volume 1: Fluid Mechanics, 2nd edn. New York: Wiley-Interscience. Bird RB, Curtiss CF, Armstrong RC, and Hassager O (1987b) Dynamics of Polymeric Liquids. Volume 2. Kinetic Theory, 2nd edn. New York: Wiley-Interscience. Coron J-M, Ghidaglia J-M, and He´lein F (eds.) (1991) Nematics: Mathematical and Physical Aspects, NATO Series, Series C Mathematical and Physical Sciences, Dordrecht: Kluwer. de Gennes P-G and Prost P (1995) The Physics of Liquid Crystals, The International Series of Monographs on Physics, vol. 83, 2nd edn. Oxford: Oxford University Press. Doi M and Edwards SF (1988) The Theory of Polymer Dynamics, The International Series of Monographs on Physics, vol. 73. Oxford: Oxford University Press. Duvaut G and Lions J-L (1976) Inequalities in Mechanics and Physics, Springer Grundlehren, vol. 219. Berlin: Springer. Joseph DD (1990) Fluid Dynamics of Viscoelastic Liquids, Applied Math Sciences, vol. 84. Berlin: Springer. Keunings R (2004) Micro–macro methods for the multiscale simulation of viscoelastic flow using molecular models of kinetic theory. In: Binding DM and Walters K (eds.) Rheology Reviews 2004. British Society of Rheology, pp. 67–98. Malek J and Rajagopal KR (2005) Mathematical issues concerning the Navier–Stokes equations and some of their generalizations. In: Dafermos C and Feireisl E (eds.) Handbook of Differential Equations. Evolutionary Equations: Volume 2. Amsterdam: North-Holland. ¨ ttinger HC (1996) Stochastic Processes in Polymeric Fluids. O Berlin: Springer. Renardy M (2000) Current issues in non-Newtonian flows: a mathematical perspective. Journal of Non-Newtonian Fluid Mechanics 90: 243–259. Renardy M, Hrusa WJ, and Nohel JA (1987) Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35. Harlow: Longman Scientific and Technical. Suen JKC, Joo YL, and Armstrong RC (2002) Molecular orientation effects in viscoelasticity. Annual Review of Fluid Mechanics 34: 417–444. Tanner RI and Walters K (1998) Rheology: An Historical Perspective, Rheology Series, vol. 9. Amsterdam: Elsevier.

568 Nonperturbative and Topological Aspects of Gauge Theory

Nonperturbative and Topological Aspects of Gauge Theory R W Jackiw, Massachusetts Institute of Technology, Cambridge, MA, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction Classical fields that enter a classical field theory provide a mapping from the ‘‘base’’ manifold on which they are defined (space or spacetime) to a ‘‘target’’ space over which they range. The base and target spaces, as well as the map, may possess nontrivial topological features, which affect the fixed-time description and the temporal evolution of the fields, thereby influencing the physical reality that these fields describe. Quantum fields of a quantum field theory are operator-valued distributions whose relevant topological properties are obscure. Nevertheless, topological features of the corresponding classical fields are important in the quantum theory for a variety of reasons: (1) Quantized fields can undergo local (spacetime-dependent) transformations (gauge transformations, coordinate diffeomorphisms) that involve classical functions whose topological properties determine the allowed quantum field theoretic structures. (2) One formulation of the quantum field theory uses a functional integral over classical fields, and classical topological features become relevant. (3) Semiclassical (WKB) approximations to the quantum theory rely on classical dynamics, and again classical topology plays a role in the analysis. Topological effects of gauge fields in quantum theory were first appreciated by Dirac in his study of the quantum mechanics for (hypothetical) magnetic point monopoles. Although here one is not dealing with a field theory, the consequences of his analysis contain many features that were later encountered in field theory models. The Lorentz equations of motion for a charged (e) massive (M) particle in a monopole magnetic field (B = mr=r3 ) are unexceptional, r_ ¼

p_ ¼

p M

e pB M

½1a

ðc ¼ 1Þ

½1b

and completely determine classical dynamics. But knowledge of the Lagrangian L Rand of the action I – the time integral of L: I = dt L – is further needed for quantum mechanics, either in its functional integral formulation or in its Hamiltonian

formulation, which requires the canonical momentum p  @L=@ r_ . The Lorentz-force action is expressed in terms R of the vector potential R The A, B = Ñ  A: ILorentz = e dt r_  A = e dr  A. magnetic monopole vector potential is necessarily singular because Ñ  B = 4m3 (r) 6¼ 0. The singularity (Dirac string) can be moved, but not removed, by gauge transformations, which also are singular, and do not leave the Lorentz action invariant. Noninvariance of the action can be tolerated provided its change is an integral multiple of 2, since the functional integrand involves exp (iI) (with  h = 1). The quantal requirement, which is not seen in the equations of motion, is met when eg ¼ N=2

½2

The topological background to this (Dirac) quantization condition is the fact that 1 (U(1)) is the group of integers, that is, the map of the unit circle into the gauge group, here U(1), is classified by integers. Further analysis shows that only point magnetic sources can be incorporated in particle quantum mechanics, which is governed by the particle Hamiltonian H = p2 =2M (magnetic fields do no work and are not seen in H). Quantum Lorentz equations are regained by commutation with H: r_ = i[H, r], p_ = i[H, p], provided i½ri ; rj  ¼ 0

½3a

i½pi ; rj  ¼ ij

½3b

i½pi ; pj  ¼ e"ijk Bk

½3c

But [3c] implies that the Jacobi identity is obstructed by magnetic sources Ñ  B 6¼ 0. j k 1 ijk i 2" ½p ; ½p ; p 

¼ eÑ  B

½4

This obstruction is better understood by examining the unitary operator U(a)  exp (ia  p), which according to [3b] implements finite translations of r by a. The commutator algebra [3] and the failure of the Jacobi identity [4] imply that these operators do not associate. Rather one finds Uða1 ÞðUða2 ÞUða3 ÞÞ ¼ ei ðUða1 ÞUða2 ÞÞUða3 Þ ½5 R where  = e d3 x Ñ  B is the total flux emerging from the tetrahedron formed from the three vectors ai with vertex at r (see Figure 1). But quantum mechanics realized by linear operators acting on a Hilbert space requires that operator multiplication

Nonperturbative and Topological Aspects of Gauge Theory

A ! A þ @  þ ½A ;   A þ D  Aa ! Aa þ @ a þ fbc a Ab c  Aa þ ðD Þa a3

½8b

(In a quantum field theory, A becomes an operator but the gauge transformations U,  remain c-number functions.) The field strength F given by

a2

F ¼ @ A  @ A þ ½A ; A  r

569

½9a

is also given by

a1

½D ; D  . . . ¼ ½F ; . . .

½9b

(coupling strength g has been scaled to unity). The definition [9] implies the Bianchi identity Figure 1 Tetrahedron pierced by magnetic flux that obstructs associativity.

be associative. This can be achieved, in spite of [5], provided  is an integral multiple of 2, hence invisible in the exponent. This then needs that (1) Ñ  B be localized at points, so that the volume integral of Ñ  B retain integrality for arbitrary ai and (2) the strengths of the localized poles obey Dirac quantization. The points at which Ñ  B is localized can now be removed from the manifold and the Jacobi identity is regained. The above argument, which rederives Dirac’s quantization, makes no reference to gauge variance of magnetic potentials. In the remainder we shall discuss related phenomena for selected gauge field theories in four, three, and two dimensions that describe actual physical events occurring in nature. We shall encounter in generalized form, analogs to the above quantum mechanical system. Some definitions and notational conventions: Nonabelian gauge potentials Aa carry a spacetime index () (metric tensor g = diag(1, 1, . . . )) and an adjoint group index (a). When contracted with anti-Hermitian matrices Ta that represent the group’s Lie algebra (structure constants fab c ) ½Ta ; Tb  ¼ fab c Tc

½6

½7

Gauge transformations transform A by group elements U: 1 1 A ! AU   U A  U þ U @ U

½8a

For infinitesimal gauge transformations, U  I þ ,   a Ta ; this leads to the covariant derivative D :

½10

Here F is gauge covariant F ! F U ¼ U1 F U

½11a

or, infinitesimally, F ! F þ ½F ; 

½11b

In the gauge invariant Yang–Mills action IYM , the Yang–Mills Lagrange density LYM is integrated over the base space,

IYM ¼

Z

LYM ¼ 12 tr F F Z 1 tr F F LYM ¼ 2

½12

The trace is evaluated with the convention tr Ta Tb ¼ 12 ab

½13

and henceforth there is no distinction between upper and lower group indices. The Euler–Lagrange condition for stationarizing IYM gives the Yang–Mills equation D F ¼ 0

½14a

Should sources J be present, [14a] becomes D F ¼ J

½14b

and J must be covariantly conserved: D J ¼ D D F ¼ 12½D ; D F ¼  12½F ; F  ¼ 0

they become Lie algebra-valued. A  Aa Ta

D F! þ D! F þ D F! ¼ 0

½15

All this is a nonabelian generalization of familiar Maxwell electrodynamics.

Gauge Theories in Four Dimensions Gauge theories in four-dimensional spacetime are at the heart of the standard particle physics model. Their topological features have physical consequences and merit careful study.

570 Nonperturbative and Topological Aspects of Gauge Theory Yang–Mills Theory

In four dimensions, we define nonabelian electric E and magnetic Ba fields, Eia ¼ F a0i ;

a Bia ¼ 12 "ijk Fjk

a

½16

Canonical analysis and quantization is carried out in the Weyl gauge (Aa0 = 0), where the Lagrangian and Hamiltonian (energy) densities read LYM ¼ 12ðEa  Ea  Ba  Ba Þ

½17

HYM ¼ 12ðEa  Ea þ Ba  Ba Þ

½18

The first term is kinetic, with Ea = @t Aa also functioning as the (negative) canonical momentum p a , conjugate to the canonical variable Aa ; the second magnetic term gives the potential. In the Weyl gauge, the theory remains invariant against time-independent gauge transformations. The time component of equation [14] (Gauss law) is absent (because there is no Aa0 to vary); rather it is imposed as a fixed-time constraint on the canonical variables Ea and Aa . This regains the Gauss law: ðD  EÞa ¼ 0

ðin the absence of sourcesÞ

½19a

In the quantum theory D  E annihilates ‘‘physical’’ states. Explicitly, in a functional Schro¨dinger representation, where states are functionals of the canonical fixed-time variable Aji ! (A), [19a] requires    a D ðAÞ ¼ 0 ½19b A that is, physical states must be invariant against infinitesimal gauge transformation, or equivalently, against gauge transformations that are homotopic (continuously deformable) to the identity (the so-called ‘‘small’’ gauge transformations) ðA þ DÞ ¼ ðAÞ

½20

But homotopically nontrivial gauge transformation functions that cannot be deformed to the identity (the so-called ‘‘large’’ gauge transformations) may be present. Their effect is not controlled by Gauss’ law, and must be discussed separately. Fixed-time gauge transformation functions depend on the spatial variable r : U(r). For a topological classification, we require that U tend to a constant at large r. Equivalently, we compactify the base space R3 to S3 . Thus, the gauge functions provide a mapping from S3 into the relevant gauge group G, and for nonabelian compact gauge groups such mappings fall into disjoint homotopy classes

labeled by an integer winding number n: 3 (G) = Z. Gauge functions Un belonging to different classes cannot be deformed into each other; only those in the ‘‘zero’’ class are deformable to the identity. An analytic expression for the winding number !(U) is !ðUÞ ¼

1 242

Z

d3 x "ijk trðU1 @i UU1 @j UU1 @k UÞ ½21

This is a most important topological entity for gauge theories in four-dimensional spacetime, that is, in 3-space, and we shall meet it again in a description of gauge theories in three-dimensional spacetime, that is, on a plane. Various features of ! expose its topological character: (1) ! (U) does not involve a metric tensor, yet it is diffeomorphism invariant. (2) !(U) does not change under local variations of U: Z 1 !ðUÞ ¼ 2 d3 x @i "ijk trðU1 UU1 @j UU1 @k UÞ 8 Z 1 ¼ 2 dSi "ijk trðU1 UU1 @j UU1 @k UÞ 8 ¼0 ½22 The last integral is over the surface (at infinity) bounding the base space and vanishes for localized variations U. In fact, the entire ! (U), not only its variation, can be presented as a surface integral, but this requires parametrizing the group element U on R3 . For example, for SU(2), U ¼ exp ;  ¼ a a =2i ðs  Pauli matricesÞ Z 1 ! ðUÞ ¼ dSi "ijk "abc ^a @j ^b @k ^c ðsin jj  jjÞ 162 pffiffiffiffiffiffiffiffiffiffi ^a  a =jj ½23 jj  a a ;

! 2n (so that U r!! Specifically, with jj r I), 1 !1 !(U) =  n. As befits a topological entity, !(U) is determined by global (here large distance) properties of U. Since all gauge transformations, small and large, are symmetry operations for the theory, [20] should be generalized to ðAUn Þ ¼ ein ðAÞ

½24

where  is an universal constant. Thus, Yang–Mills quantum states behave as Bloch waves in a periodic lattice, with large gauge transformations playing the role of lattice translations and the Yang–Mills vacuum angle  playing the role of the Bloch momentum. This is further understood by noting that the profile of the potential energy density, 12 Ba  Ba possesses a periodic structure symbolically depicted in Figure 2.

Nonperturbative and Topological Aspects of Gauge Theory

Energy density

Z 1 d4 x trð F F Þ 162 Z 1 d4 x " trðF F Þ ¼ 322

571

P

½27

This again is an important topological entity: –2

0

–1

+1 Instanton

A

+2

Figure 2 Schematic for energy periodicity of Yang–Mills fields.

Thanks to Gauss’ law, potentials A that differ by small gauge transformations are identified, while those differing by large gauge transformations give rise to the periodicity. Zero energy troughs correspond to pure gauge vector potentials in different homotopy classes n: A = Un1 ÑUn . The  angle (Bloch momentum) arises from quantum tunneling in A space. Usually, in field theory tunneling is suppressed by infinite energy barriers. (This gives rise to spontaneous symmetry breaking.) However, in Yang–Mills theory there are paths in field space that avoid such barriers. Quantum tunneling paths are exhibited in a semiclassical approximation by identifying classical motion in imaginary time (Euclidean space) that interpolates between classically degenerate vacua and possesses finite action. In Yang–Mills theory, continuation to imaginary time, x0 ! ix4 , places a factor of i on Ea . Zero (Euclidean) energy is maintained when Ea = Ba , or with covariant notation in Euclidean space,

1. The diffeomorphism invariant P does not involve the metric tensor. 2. P is insensitive to local variations of A , Z 1 P ¼  2 d4 x trð F F Þ 8 Z 1 ¼  2 d4 x trð F D A Þ 4 Z 1 ¼ 2 d4 x trðD F A Þ ¼ 0 4

½28

3. P may be presented as a surface integral owing to the formula 1  4 tr F F

¼ @ K

  K  " tr 12A @ A þ 13A A A where K is the Chern–Simons current, Z 1 P ¼  2 dS K 4

½29 ½30

½31

½25

The integral [31] is over the base space boundary, S3 . The Chern–Pontryagin index of any gauge field configuration with finite (Euclidean) action (not only instantons) is quantized. This is because finite action requires F to vanish at large distances; equivalently, A ! U1 @ U. Using this in [30] renders [31] as Z 1 dS "  P¼ 242  trðU1 @ UU1 @ UU1 @ UÞ ½32

Euclidean finite action field configurations that satisfy [25] are called self-dual or anti-self-dual instantons. By virtue of the Bianchi identity [10], instantons also solve the field equation [14a] in Euclidean space. Since the Euclidean action may also be written as

which is the same as [20] and, for the same reason, is given by an integer [3 (G) = Z]. Alternatively, for instantons in the (Euclidean) Weyl gauge (A4 = 0), which interpolate as x4 passes from 1 to þ1 between degenerate, classical vacua Ai = 0 and Ai = U1 ri U, P becomes

1 

F

2"

IYM

1 ¼ 4

Z

4

 F ¼ F







d x trðF  F ÞðF  F Þ Z 1 d4 x tr F F

2

½26

and the first term vanishes for instantons, we see that instantons are characterized by the last term, the Chern–Pontryagin index,

Z 1 P ¼ 2 dx4 d3 x ð@4 K4 þ Ñ  KÞ 4 Z 1 ¼ 2 d3 x K4 jx4 ¼1 4 Z 1 d3 x "ijk trðU1 @i UU1 @j UU1 @UÞ ¼ 242 ¼ !ðUÞ

½33

572 Nonperturbative and Topological Aspects of Gauge Theory

We have assumed that the potentials decrease at large arguments sufficiently rapidly so that the gradient term in the first integrand does not contribute. This rederivation of [32] relies on the ‘‘motion’’ of an instanton between vacuum configurations of different winding numbers. An explicit 1-instanton SU(2) solution (P = 1) is A ¼

2i ðx  Þ2 þ 2

 x

½34

(Upon reinserting the coupling constant g, which has been scaled to unity, the field profiles acquire the factor g1 .) In [34],   (1/4i)(y   y  ),   (i, I). is the ‘‘location’’ of the instanton, is its ‘‘size,’’ and there are three more implicit parameters fixing the gauge, for a total of eight parameters that are needed to specify a single SU(2) instanton. One can show that there exist N instanton/anti-instanton solutions (P = N=N) and in SU(2) they depend on 8N parameters. From [26] we see that at fixed N, instantons minimize the (Euclidean) action. Explicit formulas exist for the most general N = 2 solution, while for N  3 explicit formulas exhibit only 5N þ 7 parameters. But algorithms have been found that construct the most general 8N-parameter instantons. The 1-instanton solution is unchanged by SO(5) rotations, the maximal compact subgroup of the SO(5, 1) conformal invariance group for the Euclidean 4-space Yang–Mills equation [14a]. The Chern–Pontryagin index also appears in the Yang–Mills quantum action, for the following reason. Since all physical states respond to gauge transformations Un with the universal phase n [24], physical states may be presented in factorized form, ðAÞ ¼ eiWðAÞ ðAÞ

½35

where (A) is invariant against all gauge transformations, small and large, while the phase response is carried by W(A), WðAUn Þ ¼ WðAÞ þ n

½36

An explicit expression for W(A) is given by R (1/42 ) d3 x K0 , where K0 is the time (fourth) component of K , with dependence on the fourth variable suppressed, that is, K0 is defined on 3-space, WðAÞ ¼ 

1 42

Z

  d3 x "ijk tr 12Ai @j Ak þ 13Ai Aj Ak

½37

The gauge transformation properties of W(A) are WðAU Þ

Z 1 ¼ WðAÞ þ 2 d3 x "ijk @i trð@j UU1 Ak Þ 8 Z 1 d3 x "ijk trðU1 @i U U1 @j U U1 @k UÞ þ 242 ½38

The middle surface term does not contribute for well-behaved A; the last term is again !(U), the winding number of the gauge transformation U. Thus, [36] is verified. The universal gauge-varying phase eiW(A) , which multiplies all gauge-invariant functional states, may be removed at the expense of subtracting from the action Z Z   d4 x @t WðAÞ ¼  2 d4 x @t K0 ¼ P 4 (as in [33]). Thus, the Yang–Mills quantum action extends [12] to   Z 1    4 quantum IYM ¼ d x tr F F þ F F ½39 2 162 The additional Chern–Pontryagin term in [35] does not contribute to equations of motion, but it is needed to render all physical states invariant against all gauge transformations, large and small. With this transformation, one sees that the -angle is a Lorentz invariant, but CP noninvariant effect. Evidently, specifying a classical gauge theory requires fixing a group; a quantized gauge theory is specified by a group and a -angle, which arises from topological properties of the gauge theory. The energy eigenvalues depend on , and distinct ’s correspond to distinct theories. Note that the reasoning leading to [24] and [39] relies on exact quantum-mechanical arguments, while the instanton-based tunneling discussion is semiclassical. Adding Fermions

When fermions couple to the gauge fields, the previously described topological effects are modified by action of the chiral anomaly. Dirac fields, either noninteracting but quantized, or unquantized but interacting with a gauge potential through a covariantly conserved current Ja , LI = Ja Aa , also possess a chiral current j5 =    5 , which satisfies @ j5 ¼ 2m i  5

½40

Nonperturbative and Topological Aspects of Gauge Theory

Here m is the mass, if any, of the fermions. j5 is conserved for massless fermions, which therefore enjoy a chiral symmetry: ! ei 5 . However, when the interacting fermions are quantized, there arises correction to [40]; this is the chiral anomaly:   a @ j5 A ¼ 2imh  5 iA þ C F a F ½41 C is determined by the fermion quantum numbers and coupling strengths. (For a single charged (e) fermion and a U(1) gauge potential, C = e2 =82 .) h j iA signifies the fermionic vacuum matrix element in the presence of A . The modified equation [41] indicates that even in the massless limit chiral symmetry remains broken due to the anomaly, which    arises with quantized fermions. j5 A may also be presented as   j5 A ¼ tr 5   h iA ½42 In Euclidean space h iA is the coincident-point limit of the resolvent R(x, y; ) for the Dirac equation, X

Rðx; y; Þ ¼

 ðxÞ

y  ðyÞ

 þ i



½43

Here  is an eigenfunction of the massless, Euclidean Dirac operator in the presence of the gauge field A , i   ð@ þ A Þ



¼



½44

The coincident-point limit is singular, so R must be regulated: R ! R  RReg (we do not specify the regularization procedure). It then follows that @ hj5 i ¼ 2i

X 

¼ 2i

X 

y  ðxÞ5

 ðxÞ

 þ i y  ðxÞ5



 þ i

ðxÞ

 tr 5   @ RReg a þ C F a F

½45

The first term on the right-hand side is the (Euclidean space) analog of the mass term in [40] or [41], while the second survives even after the regulators are removed, giving the anomaly tr F F . The anomaly formula [41], or more explicitly [45], is also the local form of the Atiyah–Singer index theorem, which follows after [45] is integrated over all space: The left-hand side integrates to zero. The integral of the first term on the right-hand side, R dx  5  , vanishes for  6¼ 0 by orthogonality, because 5  is an eigenfunction of [44] with eigenvalue . Only zero modes contribute to the  sum since these can be chosen to be eigenfunctions of 5 , n of them satisfying 0 = 5 0 . For a single multiplet, the normalizations work out so that

nþ  n  ¼

1 162

Z

d4 x tr F F

573

½46

The result that the (signed) number of zero modes is the Chern–Pontryagin index is an instance of the Atiyah– Singer theorem. (In specific applications, one can frequently show that nþ or n vanishes.) It, therefore, follows that in the background field of instantons, the Euclidean Dirac equation possesses zero modes. Another viewpoint on the chiral anomaly arises within the functional integral formulation, where the exponentiated action is constructed from unquantized fields, over which the functional integration is performed. Here the classical action retains chiral symmetry ! ei 5 , but the Grassmann fermion measure d d , once it is properly regularized, looses chiral invariance and acquires the anomaly, d d  ! d d  exp iC

Z

d4 x tr F F

½47

Evidently, the chiral anomaly involves the gaugetheoretic topological entity, the Chern–Pontryagin density. Not unexpectantly, the anomaly phenomenon affects significantly the topological properties of the gauge theory that are connected to P and were described previously. When there is (at least) one massless fermion coupling to the Yang–Mills fields, the Yang–Mills -angle looses physical relevance. This is because a chiral transformation that redefines the massless Dirac field does not modify the classical action, but owing to the chiral noninvariance of the functional measure, [47], an anomaly term is induced in the (effective) quantum action. The strength of this induced term can be fixed so that it cancels the -term in [39]. Since field redefinition cannot affect physics, the elimination of the -term indicates that it had no physical relevance in the first place. In particular, energy eigenvalues no longer depend on . An alternate argument for the same conclusion is based on the functional determinant that arises when the functional integral is performed over the massless Dirac field: det [  (@ þ A )]. The semiclassical tunneling analysis of the -angle is based on instantons, but in the presence of instantons the Dirac equation has a zero mode [46]. Consequently the determinant vanishes, tunneling is suppressed and so is the -angle. However, in the standard model for particle physics, there are no massless fermions, so the presence of the -angle entails the following physical consequences. The tunneling amplitude  in leading semiclassical approximation is determined by the Euclidean action, namely the continuation of iIYM in

574 Nonperturbative and Topological Aspects of Gauge Theory

[39] to imaginary time. This results in the same expression except that the topological -term acquires a factor of i. Only the 1-instanton and anti-instanton give the dominant contribution,  / cos  e

82 =g2

½48

where the coupling constraint g has been reinserted; the proportionality constant has not been computed, owing to infrared divergences. (Higher-instantonnumber configurations contribute at an exponentially subdominant order and have thus far played no role in physics.) The tunneling leads to baryon decay, but fortunately at an exponentially small rate. More useful is the fact that instanton tunneling gives semiclassical evidence for the removal of an unwanted chiral U(1) Goldstone symmetry, which would be present in the standard model if the chiral anomaly did not interfere. Furthermore, the chiral anomaly facilitates the decay of the neutral pion to two photons; a process forbidden by other apparent chiral symmetries of the standard model, which in fact are modified by the chiral anomaly. Gauge fields in four dimensions must interact with anomaly free currents. This necessitates a precise adjustment of fermion content and charges so that the anomaly coefficients (analogs of ‘‘C’’ in [41]) vanish for currents coupled to gauge fields. Finally,  provides a tantalizing source of CP violation in the strong-interaction sector of the standard model. But no experimental signal (e.g., neutron electric dipole moment) for this effect has been seen. At present, we do not know what mechanism is responsible for keeping  vanishingly small. These are the physical consequences of topological effects in four-dimensional gauge theories. Although they have provided experimentalists with only a few numbers to measure (e.g., 0 ! 2 decay amplitude, prediction of anomaly-free arrangements of quarks and leptons in families), they have added enormously to our appreciation of the complexities of quantized gauge theories. That chiral anomalies are an obstruction to consistent gauge interactions can be established within perturbation theory. A similar, but nonperturbative effect is seen in an SU(2) gauge theory with N Weyl fermion (5 =  ) SU(2) doublets, which lead upon functional integration to det [  (@ þ A )]N=2 . But because 4 (SU(2)) = Z2 , there exists a single homotopy class of gauge transformations which are not deformable to the identity. One shows that the determinant changes sign when such a gauge transformation is performed. Thus, the theory is ill-defined for odd N. Consistent SU(2) gauge theories must possess an even number of Weyl

fermion doublets, but such models have not found a place in physical theory. Adding Bosons

Instantons are finite-action solutions to classical equations continued to imaginary time; they provide a semiclassical description of quantum-mechanical tunneling. A field theory may also possess finiteenergy, time-independent (static) solutions to the real-time equations of motion. When these solutions are stable for topological reasons, they are called ‘‘solitons.’’ Solitons give semiclassical evidence for the existence in the quantum field theory of a particle sector disjoint from the particles obtained by quantizing field fluctuations around the vacuum state. The soliton particles are heavy for weak coupling g. (Their energy is O(1=g2 ); the field profiles are O(1=g).) They do not decay owing to the conservation of ‘‘charges’’ that do not arise from Noether’s theorem but are topological. Yang–Mills theory does not possess soliton solutions (except in five-dimensional spacetime, where the static solitons are just the four-dimensional instantons discussed previously). However, when a gauge theory, based on a simple group is coupled to a scalar field that undergoes symmetry breaking to U(1), soliton solutions exist. These are the ‘t Hooft– Polyakov magnetic monopoles, found in a SU(2) gauge theory with scalar fields in the adjoint representation, as well as various generalizations. The topological consideration that arises here concerns finite energy of the static, scalar field multiplet ’, which in the Weyl gauge is Z

Eð’Þ ¼ d3 x jðD’Þa  ðD’Þa j2 þ Vð’Þ ½49 V is non-negative and possesses non trivial symmetry breaking zeroes. On the sphere S2 at spatial infinity, ’ must tend to such a zero. Thus, the fields belong to G=H, where G is the gauge group and H the unbroken subgroup. For the ‘t Hooft–Polyakov monopole these are SU(2) and U(1), respectively, and the scalar field provides a mapping of the sphere at infinity S2 to S2  SU(2)=U(1). One now considers 2 (S2 ) = 2 (SU(2)=U(1)) = 1  (U(1)) = Z, and one shows that the magnetic flux is determined by the winding number. Hence, the magnetic charge is quantized. Explicitly, the electromagnetic U(1) gauge field is given by a ^a F ^a ðD ’Þ ^ b ðD ’Þ ^ c f  ’  "abc ’

¼ @  a  @  a ^a Aa  cos @

a  ’

½50

Nonperturbative and Topological Aspects of Gauge Theory

^a is the unit isovector, parametrized as where ’ a ^ = ( sin cos , sin sin , cos ). The manifestly ’ conserved magnetic current jm ¼ @ f 

½51a

is rearranged to read ^ a @ ’ ^ b @ ’ ^c jm ¼ 12"  "abc @ ’

½51b

and is nonvanishing because ’a possesses zeroes, where @ ’ˆ a acquires localized singularities. The magnetic charge Z Z 1 1 3 0 d x jm ¼ d3 x Ñ  b ½52 m¼ 4 4 (bi = U(1) magnetic field:  12 "ijk fjk = f i0 ) is given by the topological entity (Kronecker index of the mapping) Z 1 ^a @j ’ ^ b @k ’ ^c d3 x "ijk "abc @i ’ m¼ 8 Z 1 ^ a @j ’ ^b @k ’ ^c dSi "ijk "abc ’ ¼ 8 Z 1 dSi "ijk @j cos @k

¼ ½53 4 which readily evaluates the integer winding number. The theory also supports charged magnetic monopole solutions called ‘‘dyons.’’ Here the profiles involve time-periodic gauge potentials, where the time variation is just a gauge transformation @t A = D . (Gauge-equivalent, static expressions have slow large-distance fall-off, which is removed by the time-dependent gauge function.) For dyons, the integer valued Chern–Pontryagin index, with the integration taken over all space and in time over the dyon period, reproduces the magnetic monopole strength. Regrettably, these fascinating structures are not found in nature. Nor do they arise in the standard model, whose structure group is not simple, although speculative grand unified models, with simple G and H = SU(3)  U(1), would support magnetic monopoles and dyons. While challenged physically, the magnetic monopole phenomena have produced extensive and interesting mathematical analysis.

575

reflection of topological behavior in the physically important four-dimensional theory. Abelian Gauge Theory

Take the spatial interval to be [L, L]. Homotopically nontrivial gauge transformations satisfy (L)  (L) = 2n (1 Uð1Þ= Z). States (A) of the free gauge theory that satisfy Gauss’ law and respond with a -angle are Z i dx A ðAÞ ¼ exp 2 ½54 ðA þ @Þ ¼ ein ðAÞ In this model,  has the interpretation of a constant background electric field E = =2, EðAÞ ¼ EðAÞ; E  F01   ðAÞ ¼  ðAÞ i A 2 This also gives the energy eigenvalue: Z Z 1 1 2 dx E ðAÞ ¼ dx E 2 ðAÞ 2 2

½55

½56

The phase may be Rremoved by adding to the Lagrangian (=2) dx @t A; equivalently, the action becomes   Z 1    quantum 2 " F ¼ d x  F F þ IEM ½57a 4 4 which apart from a constant is also given by a formula with the background field: Z 1 quantum dxðE þ EÞ2 ¼ ½57b IEM 2 Because of gauge invariance, there is only one R state, annihilated by E and carrying energy 12 dx E 2 . Distinct  (different E) correspond to distinct theories. We recognize in [57a] the two-dimensional Chern–Pontryagin density, contributing a total derivative to the action, Z 1 d2 x " F P¼ ½58 4 the Chern–Simons current, whose divergence is P,

Gauge Theories in Two Dimensions Two-dimensional gauge theories have only a few physical applications; edge states of the planar quantum Hall effect can be described by excitations moving on a line. However, the abelian model with fermions is useful in that it provides a very accurate

K ¼

1  " A 2

½59

and the Chern–Simons term, which carries the phase of  Z Z 1 0 dx K ¼ dx A ½60 2

576 Nonperturbative and Topological Aspects of Gauge Theory

For Euclidean-space gauge potentials, which are given at large distance by the pure gauge 2n tan1 y=x, P = n. All this is just as in the fourdimensional theory, except there are no instantons and no tunneling. Adding Fermions

The addition of massless fermions to the U(1) gauge theory results in the Schwinger model of massless quantum electrodynamics in two-dimensional spacetime. The equation of motion becomes @ F ¼ J

½61

with the vector current constructed from the Dirac fields as J =    . This current remains conserved in the quantized version because it couples to the gauge field. But the axial vector current j5 =    5 acquires an anomaly that involves the Chern– Pontryagin density in [58], @ j5 ¼

1  " F 2

½62

The model is readily solved, and shows no -angle (background field) dependence in physical quantities. The solution is directly obtained by combining [61] with [62] into a second-order differential equation and using the matrix identity of twodimensional Dirac (= Pauli) matrices: "  5 =   . It follows that   1 &þ E¼0 ½63  So the theory describes a free massive photon (mass squared = 1= in units of h and the coupling constant, which have been scaled to unity), with no sign of a -angle (background field). However, in parallel with four-dimensional behavior, the model with massive fermions regains a  dependence in the particles’ energy spectrum; a result that is established perturbatively, because a complete solution is not available. Note that in the Schwinger model, the gauge particle (‘‘photon’’) acquires a mass, even though local gauge invariance is preserved. This happens essentially for topological/anomaly reasons. Such topological mass generation is met again in three dimensions. Adding Bosons

Scalar electrodynamics with a negative mass squared term in (3 þ 1)-dimensional spacetime leads to the Higgs mechanism and short-range interactions due to the massive photons. In (1 þ 1) spacetime dimensions, the model possesses instantons – scalar and

gauge field profiles that solve the imaginary-time equations of motion – labeled by 1 (U(1)) = Z. These disorder the Higgs condensate so that the force between charged particles remains long-range, like in the positive mass-squared case. This is a vivid example of how excitations arising from nontrivial topological issues significantly effect physical content.

Gauge Theories in Three Dimensions Gauge theories on three-dimensional spacetime, that is, evolving on a plane, have physical application to planar phenomena, like the quantum Hall effect. Also, the high-temperature limit of four-dimensional field theories is governed by the corresponding field theory in three Euclidean dimensions. In three (more generally, odd) dimensions, there are no Chern–Pontryagin quantities, no Chern– Simon currents, no axial vector currents or anomalies (there is no 5 matrix). These are replaced by odd-dimensional entities that can modify Yang– Mills dynamics. Yang–Mills and Other Gauge Theories

Using the three-index Levi-Civita tensor, one can construct a gauge-covariant, covariantly conserved vector, which can be added to the Yang–Mills equation. Thus, [14] can be modified to m D F þ " F ¼ J ½64a 2 or, equivalently, in terms of the dual-field strength F  12 " F ,



" D F þ m F ¼ J

½64b

For dimensional balance, m carries dimension of mass. Indeed, in the source-free case [64] implies ðD D þ m2 Þ F ¼ " ½ F ; F 

½65

This shows that excitations are massive, even though local gauge invariance is preserved. Otherwise, as in the Dirac monopole case, the equations of motion are unexceptional. However, for the quantum theory we need the action, whose variation produces the mass term in [64]. This is just the Chern–Simons term W(A) in [37], multiplied by 82 m and now defined on (2 þ 1)-dimensional spacetime: Z   ICS ¼ 2m d3 x "  tr 12A @ A þ 13A A A ½66 Everything holds also in the abelian theory; the last term in [66] is then absent.

Nonperturbative and Topological Aspects of Gauge Theory

In this model, the mass is generated by a topological mechanism since ICS possesses the usual attributes for a topological entity: it is diffeomorphisms invariant without a metric tensor; when the potentials are appropriately parametrized, it is given by a surface term. (In the abelian case, the appropriate parametrization is in terms of Clebsch decomposition, A = @  þ @ .) Most importantly, in the nonabelian theory [66] changes by 82 mn with three-dimensional gauge transformations carrying winding number n. Hence, for consistency of the nonabelian quantum theory, m must be quantized as n=4 (in units of  h and the coupling constant, which have been scaled to unity). All this is a clear field-theoretic analog to the quantum mechanics of the Dirac monopole, and just as for the magnetic monopole, a Hamiltonian argument for quantizing m can be constructed, as an alternative to the above action-based derivation. The time component of [64] relates the electric and magnetic fields to the charge density: D  E  mB ¼

½67

In the abelian case, the first term involves a total derivative and its spatial integral vanishes, leaving a formula that identifies magnetic flux with a total charge. At low energy, the mass term dominates the conventional kinetic term in [64], and the flux– charge relation becomes a local field-current identity, m F   J 

½68

These formulas have made Chern–Simons-modified gauge theories relevant to issues in condensed matter physics, for example, the quantum Hall effect. In the abelian case, m need not be quantized. Adding Fermions

Three-dimensional Dirac matrices are minimally realized by 2  2 Pauli matrices. As a consequence, a mass term is not parity invariant; also, there is no 5 matrix, since the product of the three Dirac (= Pauli) matrices is proportional to I. While there are no chiral anomalies, there is the so-called parity anomaly: integrating a single doublet of massless SU(2) fermions one obtains (A)  det[  (i@ þ A )], which should preserve parity and gauge invariance. Since there are no anomalies in current divergences, (A) is certainly invariant against infinitesimal gauge transformations. But for finite gauge transformations (categorized by 3 (SU(2) = Z) one finds that (A) is not invariant: when the gauge transformation belongs to an odd-numbered homotopy class, (A) changes sign. To regain gauge invariance, one must either work

577

with an even number of fermion doublets or, if only one doublet (more generally, odd number) is to be used, one must add to the gauge Lagrangian a parityviolating Chern–Simons term with half the correctly quantized coefficient, to neutralize the gauge noninvariance of (A). Alternatively, (A) can be regularized in a gauge-invariant manner. But this requires massive, Pauli–Villars regulator fields, which produce a parityviolating expression for (A). One cannot avoid the parity anomaly. Adding Bosons

There are a variety of bosonic field models that one may consider: Abelian or nonabelian; with conventional kinetic term or supplemented by the Chern– Simons topological mass; or, for low energy, no kinetic term but only the Chern–Simons term, as in [68]. Abelian charged Bose fields in a Maxwell theory lead to vortex solitons, based on 1 (U(1)) = Z. These are just the instantons of the (1 þ 1)-dimensional bosonic gauge theory discussed previously. With Maxwell kinematics there are no charged vortices, but these appear when the Chen–Simons mass is added; see [67]. Pure Chern–Simons kinematics, with no Maxwell term, can produce completely integrable soliton equations (Liouville, Toda) when the Bose field dynamics is appropriately chosen.

Conclusion Topological effects in field theory are associated with the infinities and regularization that beset quantum field theories. These give rise to the chiral anomaly, parity anomaly (and scale symmetry anomalies, not discussed here). Yet the anomalies themselves are finite quantities that have topological significance (Atiyah– Singer, Chern–Pontryagin, Chern–Simons). This paradoxical pairing has not been understood. Nor can we explain why the anomalies interfere in a topological manner with symmetries associated with masslessness. Although the range of topological effects in gauge theory is large, and even larger in non-gauge theories (sigma models, Skyrme models) the relevance to actual fundamental physics is confined to the -angle phenomenon, which is analyzed accurately and abstractly by reference to 3 (G) and to the interplay with fermions through the chiral anomaly. Instantons are relevant only to an approximate, semiclassical discussion. Although after much mathematical work, general instanton configurations are well understood, only the 1-instanton solution enjoys physical significance. Other topological entities that fascinate are either nonexistent in fundamental physics or are relevant to

578 Normal Forms and Semiclassical Approximation

condensed matter physics (vortices, Chern–Simons effects). But here too, we note that the fundamental equation of condensed matter physics – the many-body Schro¨dinger equation – carries no evident topological structure. Only the phenomenological equations, which replace the fundamental one, give rise to topological intricacies.

Acknowledgment This work is supported in part by funds provided by the US Department of Energy (DOE) under cooperative research agreement DE-FC02-94ER40818. See also: Abelian and Nonabelian Gauge Theories Using Differential Forms; Abelian Higgs Vortices; Anomalies; BF Theories; Gauge Theories from Strings; Gauge Theory: Mathematical Applications; Quantum Field Theory: A Brief Introduction; Seiberg–Witten Theory.

Further Reading Adler SL (1970) Perturbation theory anomalies. In: Deser S, Grisaru M, and Pendleton H (eds.) Lectures on Elementary Particles and Quantum Field Theory, vol. 1, pp. 3–164. Cambridge: MIT Press. Adler SL (2005) Anomalies to all orders (ArXiv: hep-th/0405040). In: ‘t Hooft G (ed.) Fifty Years of Yang–Mills Theory. Singapore: Word Scientific.

Bertlmann RA (1996) Anomalies in Quantum Field Theory. Oxford: Clarendon. Coleman S (1985) Classical lumps and their quantum descendants and the uses of instantons. In: Coleman S Aspects of Symmetry, pp. 185–350. Cambridge: Cambridge University Press. Fujikawa K and Suzuki H (2004) Path Integrals and Quantum Anomalies. Oxford: Oxford University Press. Jackiw R (1977) Quantum meaning of classical field theory. Reviews of Modern Physics 49: 681–706. Jackiw R (1979) Introduction to the Yang–Mills quantum theory. Reviews of Modern Physics 52: 661–673. Jackiw R (1985) Field theoretic investigations in current algebra and topological investigations in quantum gauge theories. In: Treiman S, Jackiw R, Zumino B, and Witten E (eds.) Current Algebra and Anomalies, pp. 81–359. Princeton: Princeton University Press; Singapore: World Scientific. Jackiw R (1995) Diverse Topics in Theoretical and Mathematical Physics. Singapore: World Scientific. Jackiw R (2005) Fifty years of Yang–Mills theory and our moments of triumph (ArXiv: physics/0403109). In: ‘t Hooft G (ed.) Fifty Years of Yang–Mills Theory. Singapore: World Scientific. Jackiw R and Pi S-Y (1992) Chern–Simons solitons. Progress of Theoretical Physics 107: 1–40. Rajaraman R (1982) Solitons and Instantons. Amsterdam: North-Holland. Shifman M (1994) Instantons in Gauge Theories. Singapore: World Scientific. ‘t Hooft G (1976) Symmetry breaking through Bell–Jackiw anomalies. Physical Review Letters 37: 8–11. Weinberg S (1996) The Quantum Theory of Fields. vol. II, chs. 22 and 23. Cambridge: Cambridge University Press.

Normal Forms and Semiclassical Approximation D Bambusi, Universita´ di Milano, Milan, Italy ª 2006 Elsevier Ltd. All rights reserved.

Introduction Quantum mechanics was born at the beginning of the twentieth century with the quantization rules for the harmonic oscillator and for the hydrogen atom. Such rules were almost immediately extended to more general systems by the so-called Bohr–Sommerfeld quantization rule: ‘‘the actions of the classical system can assume only those values which are integer multiples of h.’’ However, the actions are defined only in some special situations and, moreover, at the present time the Schro¨dinger equation is the paradigm of quantum mechanics. A question naturally arises: is there any relation between the eigenvalues of the Schro¨dinger operator and the numbers obtained by Bohr–Sommerfeld quantization rule (when available)? According to common wisdom, the ‘‘Bohr– Sommerfeld numbers’’ are a first approximation to the eigenvalues of the Schro¨dinger operator in the so-called

semiclassical limit. However, precise mathematical results on the subject were obtained only in the 1980s and a good understanding of the problem has been achieved only recently. In particular it is now clear how to compute higher-order corrections to the eigenvalues: this is done through suitable normal form procedures. In the present article we will discuss the above questions for the case of perturbed harmonic oscillators, a case which, on the one hand, is physically relevant and, on the other, is well understood. We will only briefly discuss the quantization of perturbations of integrable systems.

A Statement On L2 (R n ), consider the Schro¨dinger operator 2

^ ¼  h  þ V H 2

½1

where  is the n-dimensional Laplacian and V is a smooth real potential having an absolute nondegenerate minimum at the origin. We are interested in

Normal Forms and Semiclassical Approximation

the eigenvalues of [1] close to zero. Introduce coordinates adapted to the normal modes, namely such that VðxÞ ¼

n X !2 x2 i

i¼1

i

2

þ Oðkxk3 Þ

Assume (H1) Nonresonance: There exist  > 0 and  2 R such that, for any k 2 Zn  {0} one has j!  kj 

 jkj

½2

(H2) V(x) > 0 for x 6¼ 0, and lim inf VðxÞ > 0 jxj!1

(H3) V 2 C1 (Rn ) and for any r  0 there exists Cr such that   jj  @ V   Cjj hxim ; 8 2 Nn  ðxÞ   @x where we used the notation hxi := ð1 þ kxk2 Þ1=2 . Theorem 1 Assume that (H1)–(H3) hold. Then, for any positive N, M there exist positive constants hN, M , N, M , C1N, M , C2N, M , and a smooth function

function Z N, M . One could choose  = (h) = h with some positive  < 1, obtaining a simpler statement valid for the eigenvalues in [0, h ). It is also possible to weaken the nonresonance condition (H1) to the condition !  k 6¼ 0 for k 2 Zn  {0}. A theorem very close to [1] was proved by Sjo¨strand (1992) by a method different from the one that will be presented here (see also Graffi and Paul (1987)). In the analytic or Gevrey case (recall that a C1 function f(x) is Gevrey in some domain if there exist constants C,  such that, for all multiindexes  2 N n one has  jj  @ f   jj    @x   C ð!Þ in the whole domain), the error can be reduced to be exponentially small with the parameters (Bambusi et al. 1999). Previous results dealing with compact perturbations of the harmonic oscillator were obtained by Bellissard and Vittot (1990). It is possible to deal also with the resonant case in which (H1) is violated. In this case the spectrum of the complete system is qualitatively different from the spectrum of the harmonic one. As discussed later, the normal form allows one to compute the main qualitative differences.

Birkhoff Normal Form

Z N;M ðI1 ; . . . ; In ;  hÞ hN, M , the such that, 80 <   N, M and 0 < h   eigenvalues of [1] in [0, ) have the representation      k ¼ k þ 12  ! h þ Z N; M k þ 12  h;  h hÞ; þ RNM ðk; 

579

k 2 Nn ; kj  1

½3

In this section we recall the procedure leading to classical Birkhoff normal form, whose quantization leads to the proof of Theorem 5. Birkhoff’s Theorem

The operator [1] is the quantization of the classical Hamiltonian

where hÞj  jRNM ðk; 

C1N; M N

þ

C2N; M

 M h  

More precisely, for any k 2 N n such that      k þ 12  ! h þ Z N; M k þ 12 h;  h 2 ½0; Þ

n X 2 i

i¼1

2

þ VðxÞ

½5

Denote ½4

there exists an eigenvalue k 2 [0, ) for which [3] holds, and vice versa, for any eigenvalue in [0, ) there exists a k satisfying [3] and [4]. The function Z N, M (I1 , . . . , In ; 0) coincides with the classical Birkhoff normal form of the system computed up to order N. The proof of the theorem is constructive, in the sense that it provides an algorithm allowing to construct explicitly, by elementary operations, the

H0 ð ; xÞ :¼

n X j¼1

!j I j ;

Ij :¼

j2 þ !2j x2j 2!j

½6

then we have Theorem 2 For any positive integer N  2 there exist a neighborhood U N of the origin and a canonical transformation T N : R2n  U N ! R2n which puts the system [5] in Birkhoff normal form up to order N, namely such that H  T N ¼ H0 þ ZN þ RN

½7

580 Normal Forms and Semiclassical Approximation

where ZN Poisson-commutes with H0 , namely {H0 ; ZN } 0 and RN is small, that is,

So H  3 is in normal form up to O(2 ) provided

3 fulfills the so-called homological equation:

jRN ð ; xÞj  CN kð ; xÞkNþ1

W3 þ f 3 ; H0 g ¼ Z3

½8

Moreover, if the frequencies are nonresonant, namely n

!  k 6¼ 0;

8k 2 Z nf0g

½9

the function ZN depends on the actions Ij only. We recall that the Poisson bracket of two functions f and g is defined by  n  X @f @g @f @g ff ; gg :¼  ¼ fg; f g @ j @xj @xj @ j j¼1 and coincides with the Lie derivative of g with respect to the Hamiltonian vector field of f. Remark 1 In the case where the frequencies fulfill (H1) and the potential V is analytic (or of Gevrey class) the remainder can be reduced to be exponentially small with k( , x)k.

where the unknown function Z3 has to be in normal form. Note that, since the operator

7! f ; H0 g maps linearly polynomials of degree l into polynomials of degree l, eqn [13] can be interpreted as a linear equation in the finite-dimensional space of polynomials of degree 3 in the phase-space variables. Lemma 1 The homological equation [13] admits a solution ( 3 , Z3 ). Proof

Scheme of the Proof

Make the rescaling =  0 , x = x0 . In terms of the primed variables, the Hamiltonian of the system [5] takes the form H ð 0 ; x0 Þ ¼ H0 ð 0 ; x0 Þ þ Wðx0 Þ

½10

with 0

Wðx Þ :¼

Vðx0 Þ  2 0

Pn

j¼1 3

!2j ðx0j Þ2 =2 ½11

and Wl is the Taylor polynomial of order l of V. In what follows we will omit primes from the scaled variables. Given an auxiliary Hamiltonian 3 , denote by 3t the flow of the corresponding Hamiltonian vector field. We construct 3 so that H  3 is in normal form up to order 2 . Remark P1 l 2 Given a C l = 0  gl , with g0 :¼ g;

function g one has g 

1 gl ¼ f 3 ; gl1 g; l

Introduce the canonical coordinates (, ) by ! j 1 pffiffiffiffiffi j :¼ pffiffiffi pffiffiffiffiffi þ ixj !j !j 2 ! ½14 j 1 pffiffiffiffiffi j :¼ pffiffiffi pffiffiffiffiffi  ixj !j !j i 2

In these variables the unperturbed Hamiltonian H0 P reads H0 = j1 i!j j j and W3 is transformed in a different polynomial, again of third order. The important fact is that in these coordinates the eigenvectors of the linear operator {H0 ; .} are the monomials  k l 1k1    nkn 1l1    nln

0

¼ W3 ðx Þ þ W4 ðx Þ þ   

1

½13

3



Indeed, one has {H0 ;  k l } = i!  (k  l) k l . As a consequence, writing X W3 ð; Þ ¼ Ck; l  k l k; l

one can define the resonant set R :¼ fðk; lÞ : !  ðk  lÞ ¼ 0g and Z3 ð; Þ :¼

l1

½12

where denotes the fact that the left-hand side is asymptotic to the right-hand side (a precise definition appears later in the article). If both g and 3 are analytic then the series of g  3 can be shown to converge in a neighborhood of the origin. Using [12] to compute H  3 , we get H  3 ¼ H0 þ ½W3 þ f 3 ; H0 g þ Oð2 Þ

X

Ck;l  k l

k; l2R

3 ð; Þ :¼

X

Ck;l  k l i!  ðk  lÞ k; l62R

½15

Going back to the original variables, one has the & solution of the homological equation. Definition 1 The function Z3 solving [13] will be called the resonant part of W3 and will be denoted by hW3 i.

Normal Forms and Semiclassical Approximation

Using the function 3 , one can transform the Hamiltonian to the form H0 þ Z3 þ 2 R3 Remark 3 Equation [12] allows to construct directly the Taylor expansion of R3 in terms of the Taylor expansion of W and of its Poisson brackets with 3 . Iterating the construction (which however slightly changes due to the presence of Z3 ), one gets the proof of Theorem 2. Remark 4 In the nonresonant case !  (k  l) = 0 implies that k = l; therefore, the resonant part of a polynomial is the sum of monomials of the form k k

 ¼

I1k1

   Inkn

that is, it is a function of the actions only. Moreover, in this case one has Z3 = 0, while in general Z4 6¼ 0.

It is useful to extend such a definition to functions explicitly depending also on h. This can be done in a straightforward way by asking the constants Cr to be independent of h in a neighborhood of the origin. Different classes of symbols can also be defined, but for our purpose this class is enough. Theorem 3 Let f 2 S(hzim ), m 2 R, and 2 S(Rn ); then the formal expression [16] is a well-defined oscillatory integral. Example 1

Under Weyl quantization rule, one has

^j ¼ ih@xj ;

^j ¼ xj x

Definition 4 A sequence (fj )j0 with fj 2 S(hzim ) will be called the asymptotic expansion of f 2 S(hzim ) if, for any integer N, there exist two positive constants CN , hN such that  j fj þ RN h

Nþ1

To understand how to quantize the procedure of Birkhoff normal form, we consider the classical– quantum correspondence. It is well known that there are different procedures in order to associate an operator with a classical observable. Here we concentrate on the Weyl quantization rule. To a function f 2 S(R 2n ) (Schwartz class), we associate an operator fˆ acting on functions 2 S(Rn ), which is defined by   Z 1 xþy ^ ; f ½f ðxÞ :¼ 2 ð2 hÞn Rn Rn ðyÞ dy d

Using the method of oscillatory integrals, the Weyl quantization rule can be extended to much more general observables f. We recall that, roughly speaking, the method of oscillatory integrals consists in giving meaning to a formal expression of the form [16] by using successive integration by parts (see, e.g., Martinez (2001)). 1

with jRN (z, h)j  CN h

2n

Definition 3 A function f 2 C (R ) will be called a smooth symbol of class S(hzim ) if, for any r  0, there exists Cr such that  jj  @ f  m 2n    @z ðzÞ  Cjj hzi ; 8 2 N

hzim , and h 2 (0, hN ).

The key point for the quantization of the normal form procedure is the following. Theorem 4 Let f 2 S(hzim1 ) and g 2 S(hzim2 ); then there exists a unique F 2 S(hzim1 þm2 ) such that ^ ¼ ^f ^g F

ðoperator product!Þ

moreover, one has  F ¼ exp

 ih ð@x  @  @y  @ Þ 2

 ðf ðx; Þgðy; ÞÞjy¼x; ¼

½16

Definition 2 The operator [16] is called the Weyl ˆ quantization of f and in turn f is called the symbol of f.

Where hzi is as defined earlier.

N X j¼0

Some Symbolic Calculus

iðxyÞ h

ðmultiplication operatorÞ

1 ^^ ^j ^j Þ d j xj ¼ 2 ð j x jþx

f ¼

e

581

½17

Finally, F admits an asymptotic expansion in  h which coincides with the formal expansion of [17]. The proof is obtained by using eqn [16] to write down an expression for fˆgˆ and obtain a formula for F. Then, one shows that the formula is well defined and therefore the result is not formal. Definition 5

In the above context, the symbol G of i h^ i ^ f ; ^g ¼: G h

will be called the ‘‘Moyal bracket’’ of f and g and will be denoted by {f ; g}M . By formula [17], one has in particular ff ; ggM ¼ ff ; gg þ h2 1 ðf ; gÞ þ Oðh4 Þ

½18

582 Normal Forms and Semiclassical Approximation

where

step of the approximation.) Equivalently, the symbol P of X gX ˆ 1  is formally given by l l gq, l with

 1 @3f @3g @3f @3g 3 2 1 ðf ; gÞ ¼  3 3 24 @ @x @ @x @x2 @  @3f @3g @3f @3g  þ3 @ @x2 @x@ 2 @x3 @ 3

gq; 0 :¼ g;

where we used a vector notation for the derivatives. If either f or g are polynomials of degree 2, then ff ; ggM ¼ ff ; gg

½19

Given a self-adjoint operator A and a smooth function G : R ! R, it is well known how to define by spectral theorem the operator G(A). Suppose now that A = fˆ for some symbol f. In ˆ 6¼ Gd general, one has G(f)  f . However, by symbolic calculus (i.e., using eqn [17]), one has: Lemma 2 Denote Ij (x, ) = (!2j x2j þ j2 )=2!j . Then, for any positive integer k there exists a function Fk (Ij ,  h) such that d ðIj Þk ¼ Fk ð^Ij ;  hÞ where the right-hand side is defined by spectral calculus. Moreover, Fk can be computed explicitly by the recursion formula Fkþ1 = Ij Fk þ Fk1  h2 (k2  k þ 1)=4. As a consequence of this fact and of the fact that [Iˆ j , Iˆ l ] = 0, one has that the Weyl quantization of a polynomial function of the actions is a function of the action operators.

1 gq; l :¼ f ; gq; l1 gM ; l  1 l

from which one sees a remarkable similitude with the classical equation. Moreover, [21] converges to [12] when h ! 0. Applying the unitary transformation generated by ˆ  (cf. eqn [10]), one

ˆ to the Hamiltonian operator H c1 with ˆ  X1 = H has X H q  Hq1 ¼ H0 þ ½W3 þ f ; H0 gM  þ Oð2 Þ H0 þ ½W3 þ f ; H0 g þ Oð2 Þ

Let be a smooth symbol such that ˆ is self-adjoint, and consider the group of unitary operators X : = exp ((i=h) ). ˆ Let g be a smooth symbol; apply the unitary transformation X to g, ˆ namely compute X gX . Noting that (on a suitable domain) ˆ 1  d i ^ ^ ðX ^ gX1 gX1  Þ ¼ X ½ ;  d h one has (formally!) the expansion of X gX ˆ 1  in : X l gd gX1 X ^ q; l  ¼ l0

where gd g; q; 0 :¼ ^

gd q; l ¼

1i ^ gd ½ ; q; l1 ; l  1 l h

½20

(Such a series can be interpreted as an asymptotic expansion provided one restricts the domain at each

½22 ½23

where we used the fact that H0 is a quadratic polynomial, so that [19] holds. It is thus clear that Lemma 1 allows to solve also the quantum homological equation appearing in this context and to determine the symbol of the operator generating the unitary transformation putting the Hamiltonian operator in normal form up to corrections of order 2 . Moreover, one can compute in terms of Moyal brackets (of polynomials!) the expansion of the symbol of the new remainder and of the normal form. Iterating the construction, one generates a well-defined semiclassical normal form of the quantum system. Example 2 Denote by Zq, l , l = 1, 2 . . . , the term added to the semiclassical normal form at the lth step of the iterative construction. Explicitly, the first terms are given by Zq;1 ¼ hW3 i ¼ Z3

Semiclassical Normal Form

½21

½24

  Zq; 2 ¼ hW4 i þ 12 f 3 ; W3 gM þ 12 f 3 ; Z3 gM

½25

  Zq; 3 ¼ hW5 i þ f 4 ; Z3 gM þ 13 f 3 ; H2 gM   þ 12 f 3 ; W3;1 gM þ f 3 ; W4 gM

½26

where, according to Definition 1, h . i is the resonant part of its argument, j is (formally) the symbol of the operator generating the jth unitary transformation, and H2 :¼ 12f 3 ; Z3  W3 gM ;

W3;1 :¼ f 3 ; W3 gM

Note that all the Moyal brackets involved contain polynomials of degree at most 4, so that they can be computed exactly using formula [18] which in this case does not contain corrections of order h4 . The problem in making previous construction rigorous is that all the series involved are in general

Normal Forms and Semiclassical Approximation

divergent. Moreover, it is not possible to show that the remainders appearing when truncating such series are small in a reasonable sense. Nevertheless, it is possible, using the tools of microlocal analysis, to show that the semiclassical normal form contains essentially all the information on the part of the spectrum close to zero. The precise relation between the spectrum of the original Hamiltonian and the spectrum of the semiclassical normal form is captured by the following definition. Let H1 (,  h), H2 (,  h) be two families of self-adjoint operators; set Spec (H1, 2 ) := Spec(H1, 2 ) \ [0, ). Definition 6

We say that 1

1

Spec ðH1 Þ ¼ Spec ðH2 Þ modð þ ðh=Þ Þ if for any N, M > 0 there exist C1N, M and C2N, M such that for any 1 2 Spec (H1 ) there exists 2 2 Spec (H2 ) such that 1 = 2 þ RN, M with jRN j  C1N;M N þ C2N;M ðh=ÞM

½27

and conversely. Equation [27] has to hold for any couple ( h, ) with  and (h=) small enough. Theorem 5 Assume (H2) and (H3); assume also: (H10 ) There exist  > 0 and  2 R such that, for any k 2 Zn , one has either !  k ¼ 0

or j!  kj 

 jkj

½28

Then there exists a polynomial function Z q such that one has ^ Spec ðHÞ

 1   h  1 ^ c ¼ Spec ðH0 þ Z q Þ mod  þ ½29 

The polynomial Z q coincides with the semiclassical normal form defined at the beginning of the section. Scheme of the proof It consists of six steps. (1) Make the unitary transformation (U )(x) := n=4 (1=2 x) which transforms the Hamiltonian operator [1] into the Weyl quantized of H := (H0 þ 1=2 W), but a Weyl quantization where h is substituted by  h0 := h=. (2) Make a cutoff of H , namely, fix R and consider a smooth function t such that t(s) 1 for jsj  R, t(s) 0 for jsj  2R, define a(x, ) := W(x)t(k( , x)k). (3) Compare the ˆ  with the spectrum spectrum of the Hamiltonian H t of H := H0 þ a. By microlocal analysis, one has that, in any fixed bounded interval such spectra coincide modulo  h1 (see, e.g., Martinez (2001)).

583

(4) Rescale back the variables, namely apply the transformation U1 to H t . (5) Apply the normal form algorithm to the so-obtained Hamiltonian showing that all the series involved are convergent in suitable norms. (6) Use again microlocal analysis to show that the spectrum of the semiclassical normal form coincides with the spectrum of the normalized & operator with compactly supported symbol. Remark 5 Fix an arbitrary 1 >  > 0 and link  to h by  := h . Then one obtains a simplified statement according to which the spectrum of [1] in [0,  h ] 1 ˆ coincides modulo h with the spectrum of H0 þ Zbq in the same interval. Remark 6 In the case where the frequencies are nonresonant one has that the symbol of the normal form depends on the actions only. By Lemma 2 one has that also the quantization of the normal form is a function of the action operators only (explicitly computable), and therefore the spectrum of the normal form is given by a quantization formula as claimed in Theorem 1.

The Resonant Case In the case where the frequencies are nonresonant, due to the particular structure of the normal form, one obtains a very precise information on the spectrum. In the case where there are some resonances, the situation is more difficult. In order to illustrate what happens we concentrate on the completely resonant case, that is, the case where all the frequencies are integer multiples of a single fundamental frequency . ˆ 0 form a subset of In this case, the eigenvalues of H Nh þ ð1=2Þj!jh and are degenerate. One expects the nonlinear part to break such a degeneracy and to transform each eigenvalue in a small band. One can use the normal form to study the structure of the soobtained band. To this end, the most relevant contribution is due to the first nonvanishing term of the normal form. For the sake of definiteness, we assume that this is the term of order 4, namely Z 4 . Denote 

 N :¼ Z 4 H1 ð1Þ ; BðEÞ E  13h; E þ 13h 0

h is Theorem 6 Fix 1 > 1 > 1=2, then, provided  small enough, one has [ ^ \ ðh; h1 Þ BðEÞ ½30 SpecðHÞ ^ 0Þ E2SpecðH

Moreover, denote by E þ 1 ðE; hÞ      E þ m ðE; hÞ

½31

584 Normal Forms and Semiclassical Approximation

ˆ in B(E) counted with multithe eigenvalues of H plicity, then 1 ðE; hÞ ¼ E2 Min N þ E2 ðOð h=EÞ þ OðE1=2 ÞÞ ½32

one has that both K0  T 0 and K1  T 0 are real analytic on D  Tn . Then, the KAM theory applies. To state the corresponding result, denote by D0 D a domain whose closure is contained in D. Theorem 7

and similarly hÞ ¼ Max N þ E2 ðOð h=EÞ þ OðE1=2 ÞÞ ½33 E2 m ðE;  This statement is due to Bambusi, Charles, and Tagliaferro (see Bambusi 2004); for previous results, see Vu˜ Ngoc (1998). _ Equation [30] shows that the spectrum has a band structure, while eqns [32] and [33] allow one to compute the minimum and the maximum of each band. The idea of the proof is as follows. First forget highorder terms of the normal form, whose effect is included in the error terms. Then, due to the commutation ˆ 0 , one has that Z 4 property of the normal form with H ˆ 0. restricts to an operator acting on the eigenspaces of H On the classical side, one has that by Marsden– Weinstein procedure Z 4 defines a classical Hamiltonian system on the manifold obtained by symplectic reduction of the original phase space. By the methods of geometric quantization, it turns out that the quantum ˆ 0 is a Toeplitz operator acting on an eigenspace of H operator whose principal symbol is exactly the above reduced classical Hamiltonian. Then, the proof follows by classical properties of Toeplitz operators. We point out that results of this kind are useful in the computation of the molecular spectra (Michel and Zhilinskii 2001, Zhilinskii 2001).

Quantization of KAM Tori In this section we present a result on the quantization of KAM tori. It allows one to construct part of the spectrum of a close-to-integrable system. We recall that a classical Hamiltonian system with n degrees of freedom is said to be integrable if it has n integrals of motion independent and in involution. If the energy surface is compact, then, by Arnol’d–Liouville theorem there exists a canonical transformation T 0 : Rn  Tn  D  Tn ! R2n introducing action-angle variables, namely such that, denoting by K0 the original integrable Hamiltonian, K0  T 0 is independent of the angles  2 Tn . Here, D is an open bounded domain. Consider now a close-to-integrable analytic Hamiltonian system, namely a Hamiltonian system with Hamiltonian K ¼ K0 þ K1 where  is a small parameter. We assume that, denoting again by T 0 the canonical transformation introducing action-angle variables for the system K0 ,

Assume that 8I 2 D one has  2  @ ðK0  T 0 Þ det 6¼ 0 @I2

½34

then there exists a positive constant  and, for any  with jj <  , there exists a Gevrey canonical transformation T  : D0  Tn ! R2n and a Cantor set D D0 with the following properties: K  T  ¼ ZðIÞ þ RðI; ; Þ

½35

where R(I, , ) vanishes at infinite order on D , that is, for any multi-index  there exists Cjj such that one has  jj     @ R  c   ½36 @ðI; Þ ðI; ; Þ  Cjj exp  jI  D j  with a suitable  > 0 and jI  D j denoting the distance from D . Moreover, as  tends to zero, the measure of D tends to the measure of D0 . A particular consequence is that the set D is foliated in invariant tori. From the proof, it also turns out that the motion on each torus is quasiperiodic with frequencies fulfilling the assumption (H1) stated earlier. Moreover, the tori are linearly stable and even more: they are stable in an exponential sense (namely, a solution starting O() close to a torus takes at least a time O( exp (c= )) to double its distance from the torus). Quantizing the normalizing transformation T  by using the theory of Fourier integral operators, one can also put the quantum Hamiltonian in a suitable normal form which allows to deduce some spectral information on the system. To fix ideas we restrict to the case where K is a natural system, namely it has the form (3.1), and is close to integrable in the above sense. Fix two parameters E1 < E2 ; assume (1) that K1 ([1, E2 þ ]) is compact for some positive  and (2) that the domain D0 can be constructed in such a way that T 0 : D0  Tn ! K1 0 ([E0 , E1 ]) is a bijection and, moreover, the KAM condition [34] holds. Denote by  2 Zn the Maslov class of the tori of K0 (see, e.g., Lazutkin (1993)) and, having fixed some 0 <  < 1, define the set of indexes I :¼ fk 2 Zn : jD  hðk þ =4Þj  h g

½37

Theorem 8 There exist positive constants h , c, C, and  < 1, and a function Kq : D0  (0, h ) ! R with the following property: for any k 2 I there exists at least one eigenvalue of Kˆ in the interval

N-Particle Quantum Scattering

h  Zq ð hðkþ=4Þ;  hÞ  Cec=h ; Zq ð hðkþ=4Þ;  hÞ þ Cec=h

i



½38

One can also show that a large part of the spectrum is constructed in this way. This is obtained by comparing the semiclassical estimate of the number of eigenvalues in [E1 , E2 ] to the number of eigenvalues thus constructed. Theorem 8 is due to Popov (2000); the quantization of KAM tori was initiated by Lazutkin and widely developed by Colin de Verdie`re, who obtained a result similar to Theorem 8 for the case where K is C1 and describes the geodesic flow on a compact Riemannian manifold (Colin de Verdie`re 1977). See also: Central Manifolds, Normal Forms; h-Pseudodifferential Operators and Applications; Optical Caustics; Quantum Mechanics: Foundations; Schro¨dinger Operators; Stationary Phase Approximation.

Further Reading Bambusi D (2004) Semiclassical normal forms. In: Multiscale Methods in Quantum Mechanics, pp. 23–39. Trends in Mathematics. Boston, MA: Birkha¨user.

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Bambusi D, Graffi S, and Paul T (1999) Normal forms and quantization formulae. Communications in Mathematical Physics 207: 173–195. Bellissard J and Vittot M (1990) Heisenberg’s picture and noncommutative geometry of the semiclassical limit in quantum mechanics. Annales de l’Institut Henri Poincare´. Physique The´orique 52: 175–235. Colin de Verdie`re Y (1977) Quasi Modes sur les varie´te´s Riemanniennes. Inventiones Mathematichaes 3: 15–52. Graffi S and Paul T (1987) The Schro¨dinger equation and canonical perturbation theory. Communications in Mathematical Physics 108: 25–40. Lazutkin V (1993) KAM Theory and Semiclassical Approximations to Eigenfunctions. Berlin: Springer. Martinez A (2001) An Introduction to Semiclassical and Microlocal Analysis. New York: Springer. Michel L and Zhilinskii BI (2001) Symmetry, invariants, and topology. I–III. Physics Reports 341: 11–84, 175–264. Popov G (2000) Invariant tori effective stability and quasimodes with exponentially small error terms I–II. Annales Henri Poincare´ 1: 223–248, 249–279. Robert D (1987) Autour de l’approximation semiclassique. Basel: Birkha¨user. Sjo¨strand J (1992) Semi-excited levels in non-degenerate potential wells. Asymptotic Analysis 6: 29–43. Vu˜ Ngoc S (1998) Sur le spectre des syste`mes comple`tement _ inte´grables semi-classiques avec singularite´s. Ph.D. thesis, Institut Fourier. Zhilinskii BI (2001) Symmetry, invariants, and topology. II: Physics Reports 341: 85–171.

N-Particle Quantum Scattering D R Yafaev, Universite´ de Rennes, Rennes, France ª 2006 Elsevier Ltd. All rights reserved.

Introduction The present article relies heavily on Quantum Mechanical Scattering Theory in this Encyclopedia and can be considered as its continuation. We use here freely the notation and results discussed in this article. An important problem of scattering theory concerns the Schro¨dinger H operator of N, N  3, interacting particles. Since the potential energy of pair interactions between particles depends on their relative positions only, it does not tend to zero at infinity in the configuration space of a system, even if the center-of-mass motion is removed. This is qualitatively different from the two-particle case. It turns out that asymptotically (for large times t ! þ1 or t ! 1) an N-particle system splits up into clusters, C1 [    [ Cn ¼ f1; . . . ; Ng;

Ck \ Cl ¼ ; if k 6¼ l

½1

Particles from the same cluster Ck , k = 1, . . . , n, form a bound state, and different clusters do not interact with each other. In particular, if n = 1 and C1 = {1, 2, . . . , N}, then we have a bound state of the system. In another extreme case n = N, all particles are free. The asymptotic evolution determined by clusters C1 , . . . , Cn where n  2, and bound states of all these clusters is called a scattering channel. Physically it is natural to expect that the list of all such channels is exhaustive, that is, no other scattering process is possible. This statement is called asymptotic completeness. We emphasize that an N-particle system may be in different scattering states as t ! þ1 and t ! 1 and different rearrangement processes are possible. For example, a three-particle system may asymptotically consist of free particles or a pair of particles may be in a bound state, whereas the third particle may be asymptotically free. If particles are free at both 1 and þ1, then one speaks about elastic scattering; we have a capture if particles free at 1 form a bound state of a couple after the interaction; an opposite process, when a bound state at 1 gives three free particles, is known as a breakup. It is also possible that a bound state of one couple yields a bound state of

586 N-Particle Quantum Scattering

another pair (a rearrangement) or a bound state of a couple transforms into another bound state of the same couple (an excitation). All these processes are described by the scattering operator. On the contrary, if the whole system forms a bound state at 1 (i.e., n = 1), then it remains in the same state for all t. As far as monographic literature on N-particle scattering is concerned, we mention Derezin´ski and Ge´rard (1997), Faddeev (1965), Reed and Simon (1979), and Yafaev (2000).

Let us recall the definition of the N-particle Schro¨dinger operator (Hamiltonian) H ¼ H0 þ V

½2 d

If the configuration space of each particle is R , then the operator H acts in the space L2 (RdN ). The operator of kinetic energy (the ‘‘unperturbed’’ Hamiltonian) is N X ð2mj Þ1 xj

½3

j¼1

where xj and mj are the position and mass of the particle labeled by j. The operator of potential energy of pair interactions of particles (the perturbation) V is the operator of multiplication by the function X VðxÞ ¼ V ij ðxj  xi Þ; i; j ¼ 1; . . . ; N ½4 i1

½11

t!1

exist and are isometric on the ranges Ran Pa of projections Pa . The subspaces Ran Wa are mutually orthogonal, and scattering is asymptotically complete: M Ran Wa ¼ HðacÞ a

j2Cl

describes positions of centers of masses of the clusters. The ‘‘internal’’ variable xa is the set of numbers xj  xl for all j 2 Cl and all l = 1, . . . , n. Of course, for each l only jCl j  1 (jCl j is the number of particles in a cluster Cl ) of variables xj  xl are independent. Set Ka ¼ xa ¼ 

jV  ðx Þj  Cð1 þ jx jÞ ;

The singular continuous spectrum of H is empty, so the absolutely continuous subspace H(ac) of the operator H can be replaced by H H(p) , where H(p) is spanned by all eigenvectors of H. These results can be reformulated in terms of scattering theory in a couple of spaces. Suppose that, for every a, eigenvectors a, n are normalized and orthogonal if the corresponding eigenvalues a, n coincide. Let us introduce an auxiliary space M ^ ¼ H Ha ; Ha ¼ Ha ¼ L2 ðXa Þ ½12 a

Then Ha ¼ Ka  I þ I  H a Note that eigenvalues a, n of the operator H a are sums over l = 1, . . . , n of eigenvalues of the operators X HðCl Þ ¼ H0 ðCl Þ þ V 2Cl

describing each cluster. Similarly, eigenfunctions a, n of H a are products of eigenfunctions of these operators. We usually write a instead of a couple {a, n}. In the following, the index a labels all cluster decompositions with #(a)  2. The eigenvalues a of the operators H a (a = 0 if #(a) = N) are called thresholds of the Schro¨dinger operator [8]. If all functions V  (x ) ! 0 as jx j ! 1, then the essential spectrum of the operator H consists of the interval [0 , 1), where 0 ¼ min a a

and an auxiliary operator M ^ ¼ Ka ; Ka ¼ Ka þ a H

½13

a

in this space. Here and below, the sums are taken ˆ ! H by over all a. We define an identification ^J : H the relations X ^J ¼ Ja ; Ja fa ¼ fa  a ½14 a

where the tensor product is the same as in [10]. In particular, J0 = I. Since Ha Ja = Ja Ka , the wave ^ ^J) exist and are isometric and operators W  (H, H; complete, that is, ^ ^JÞ ¼ HðacÞ Ran W  ðH; H; Thus, for states orthogonal to eigenvectors of H, evolution of an N-particle system decomposes

588 N-Particle Quantum Scattering

asymptotically into a sum of evolutions which are ‘‘free’’ in external variables xa and are determined by eigenvalues and eigenfunctions of the Hamiltonians H a in internal variables xa . To be more precise, we have that, for all f 2 H(ac) and t ! 1, X expðiKa tÞfa  a þ oð1Þ ½15 expðiHtÞf ¼ a

where fa ¼ W  ðH; Ka ; Ja Þ f and the term o(1) tends to zero in H. The wave operator W  (H, Ka ; Ja ) describes the scattering channel where a system of N interacting particles splits up asymptotically (for t ! 1) into noninteracting clusters C1 , . . . , Cn , n  2, and particles from the same cluster Cl are in the bound state (if there are more than one particle in Cl ) given by the function a (xa ). Somewhat loosely speaking, this implies that the continuous spectrum of the operator H consists of branches starting from all its thresholds. Note that the scattering problem can equivalently be formulated without the separation of centerof-mass motion. In this case, a trivial decomposition with #(a) = 1 should be added, and the set of thresholds of the operator H includes eigenvalues of the operator H. The existence of the wave operators and their isometricity can be obtained by the Cook method. Only the asymptotic completeness is a difficult mathematical problem. It can be solved within the framework of the smooth method, which requires a study of boundary values of resolvents as the spectral parameter z approaches the continuous spectrum or, equivalently, a study of a large-time behavior of evolution operators. The scattering operator ^ ^JÞ W  ðH; H; ^ ^JÞ S ¼ W þ ðH; H; is unitary on the space H^ and commutes with the ^ Its component Sab : Hb ! Ha describes operator H. a process where a system in a state b as t ! 1 goes over in a state a as t ! þ1. Diagonalizing ^ the operator H by a unitary operator ^ ^ ^ ^ F, (F H f )() = (F f )(),  > 0 , we obtain the scattering matrix S() defined by the equation ^ )() = S()(F ^ f )(). In its turn, the scattering (FSf matrix is also a matrix operator with components Sab (). For N  3, the structure of the scattering matrix is essentially more complicated than for N = 2. This is discussed in some detail in the next section.

Resolvent Equations for Three-Particle Systems Let the Hamiltonian H be defined by eqns [2]–[4], where N = 3, and let the configuration space of each particle be R d , d  3. The operator H acts in the space H = L2 (Xcm ), where the subspace Xcm  R 3d is distinguished by condition [5]. Let R0 (z) = (H0  z)1 , R(z) = (H  z)1 . Since V(x) does not tend to 0 as jxj ! 1, x 2 Xcm , in the three-particle case, the resolvent equation RðzÞ ¼ R0 ðzÞ  R0 ðzÞVRðzÞ

½16

is not Fredholm even for Im z 6¼ 0. To overcome this difficulty, Faddeev (1965) derived a system of equations for components of the resolvent. The entries of this system are constructed in terms of three Hamiltonians H ¼ H0 þ V   = (12), (13), (23), containing only one pair interaction each, and their resolvents R (z) = (H  z)1 . Let us write down the resolvent equation for each pair H , H X RðzÞ ¼ R ðzÞ  R ðzÞ V  RðzÞ 6¼

We multiply it by jV  j1=2 and set r 0 ðzÞ ¼ jV  j1=2 R ðzÞ; t ; ðzÞ ¼ 0;

r  ðzÞ ¼ jV  j1=2 RðzÞ

t ; ðzÞ ¼ jV  j1=2 R ðzÞðV  Þ1=2

where (V  )1=2 = V  jV  j1=2 . This yields a system of equations X r  ðzÞ ¼ r 0 ðzÞ  t ; ðzÞr  ðzÞ ½17 6¼

for the operators r  (z). Note that the resolvent R(z) can be recovered from its components r  (z) by the formula X RðzÞ ¼ R0 ðzÞ  R0 ðzÞ ðV  Þ1=2 r  ðzÞ 

It is convenient to rewrite eqn [17] in the matrix notation rðzÞ ¼ r 0 ðzÞ  tðzÞrðzÞ

½18

where r 0 (z) = {r 0 (z)}, r(z) = {r  (z)} are the ‘‘vector’’ cm operators in the three-component space L(3) 2 (X ) and t(z) = {t ,  (z)} is the ‘‘matrix’’ operator in this space. The advantage of eqn [17] compared to [16] is that the operators t ,  (z) are compact for Im z 6¼ 0. This can be deduced from the fact that the product V  (x )V  (x ), where  6¼  tends to 0 as

N-Particle Quantum Scattering

jxj ! 1, x 2 Xcm , provided that V  (x ) ! 0 as jx j ! 1 for all . Moreover, the homogeneous equation [17] has only a trivial solution. Indeed, if for some z with Im z 6¼ 0 X f ¼  t ; ðzÞf ½19

589

satisfies the equation u = R0 (z)Vu. Since the operator H is self-adjoint, this implies that u = 0 and hence f = 0 for all . According to the Fredholm alternative, eqns [17] for r  (z) or [18] for r(z) can be solved if Im z 6¼ 0, that is,

points  2 (0, 1) is closed and has Lebesgue measure zero. In particular, the operators hx il , l > 1, are H-smooth on any compact subinterval of  = (0, 1)nN . Therefore, the smooth method of scattering theory can be directly applied. It yields the existence and completeness of the wave operators W (H, H0 ). In this case, three-particles are necessarily asymptotically free. ‘‘Two-particle’’ channels of scattering arise if the operators H  have negative eigenvalues. To simplify notation, we assume that every H  has exactly one eigenvalue  < 0. Moreover, it is supposed that the corresponding eigenfunction  (x ) tends to zero sufficiently rapidly as jx j ! 1. Analytically, the appearance of new channels is due to new singularities of the resolvents. Indeed, in this case

rðzÞ ¼ ðI þ tðzÞÞ1 r 0 ðzÞ

^  ðzÞ R ðzÞ ¼ ð  zÞ1 P þ R

6¼

then the function u¼

X

ðV  Þ1=2 f



½20

This equation allows one to deduce the existence of necessary boundary values of the ‘‘sandwiched’’ resolvent R(z) from similar results for the resolvents R (z) of the ‘‘two-particle’’ operators H . In its turn, R (z) can be expressed in terms of the resolvent R (z) of the operator H  acting in the space L2 (R d ). Indeed, in the ‘‘mixed’’ representation ( , x ), where the Fourier transform in the variable x is performed and the variable  is dual to x , we have ðR ðzÞf Þð ; x Þ ¼ ðR ðz  ð2m Þ1 j j2 Þf Þ ð ; x Þ

½21

The passage to the limit Im z ! 0 requires that assumption [11] be satisfied for  > 2. Moreover, we have to suppose that the operators H  do not have the so-called zero-energy resonances as well as eigenvalues embedded in the continuous spectrum. Then the operator functions hx il R (z)hx il , l > 1, hx i = (1 þ jx j2 )1=2 , are analytic in the complex plane cut along [0, 1), they have poles only at the points , n , and are continuous up to the cut, the point z = 0 included. In particular, it follows from eqn [21] that, if the operators H  do not have negative eigenvalues, then the operator functions hx il R (z)hx il , l > 1, are also analytic in the complex plane cut along [0, 1) and are continuous up to the cut. The next result is of genuinely three-particle nature and is crucial for the study of the operator t(z). The operator functions hx il R0 (z)hx il ,  6¼ , l > 1, are continuous in norm up to the cut along [0, 1). Now it follows from eqn [20] that the operatorvalued functions r  (z)jV  j1=2 are continuous up to the cut (0, 1) except points  2 (0, 1), where the homogeneous equation [19] for z =   i0 has a nontrivial solution. The set N = N þ [ N  of such

^  (z) is analytic and continuous where the function R up to the cut in the complex plane cut along [0, 1). It follows from eqn [21] that in this case the resolvent R (z) contains the additional term ðð2m Þ1 j j2 þ   zÞ1  P which is analytic only in the complex plane cut along [ , 1). To take these terms into account, system [17] should be further rearranged. This yields the following result. Let us set X G0 ¼ hx il ðI  P Þ; G1 ¼ hx il ðJ Þ

V  ½22 6¼

Then, for all , , i, j = 0, 1, a suitable l > 1 and 0 = min { }, the operator functions Gi R(z)G j are norm continuous as z approaches the cut (0 , 1) at the points of  = (0 , 1)nN , where N is again a closed set of measure zero. In particular, the operators G0 and G1 are H-smooth on any compact subinterval of . In the multichannel case, to fit scattering for the Hamiltonian H into the framework of smooth theory, it is convenient to reformulate the result in terms of scattering theory in a couple of spaces. Let ^ and the identification ^J ^ the operator H, the space H, be defined by eqns [12], [13], and [14], respectively, where the index a takes four values a = 0,  and  = (12), (13), (23). One, further, needs to introduce auxiliary identifications X J0 ¼ I  P 

and ^J ¼ J 0

M 

J

590 N-Particle Quantum Scattering

^ smoothness of operators [22] imply The H- (and H-) that the wave operators ^ ^JÞ W  ðH; H;

and

^ H; ^J Þ W  ðH;

exist. ^ ^J) are isometric because The operators W  (H, H; s-lim P expðiH0 tÞ ¼ 0 jtj!1

½23

and the operators P P are compact for  6¼ . Using that the operator X ^J^J  I ¼ P  P 6¼

is compact (whereas ^J^J  I is not), we see that the ^ H; ^J ) are also isometric. Finally, operators W  (H, we remark that, by eqn [23], ^ ^JÞ ¼ W  ðH; H; ^ ^JÞ W  ðH; H; This implies the asymptotic completeness. Let us discuss properties of the scattering matrix in the one-channel case where the pair operators H  do not have negative eigenvalues. The scattering matrix S() : L2 (S2d1 ) ! L2 (S2d1 ),  > 0, is of course a unitary operator, but in contrast to the two-particle case the operator S()  I is not compact because its kernel contains the Dirac functions (  0 ). Nevertheless, the structure of its singularities can be explicitly described. Actually, let S () be the ‘‘two-particle’’ scattering matrix for the pair H0 , H . Then SðÞ SðÞ ¼ S12 ðÞS23 ðÞS13 ðÞ~ where the operator ~ S()  I is compact. The approach described briefly in this section relies on a kind of an advanced perturbation theory where the free problem is determined by the set of all sub-Hamiltonians. Its generalization to the case of an arbitrary number of particles meets with numerous difficulties. A different, nonperturbative, approach which works well for any number of particles will be discussed in the next section. A purely time-dependent method in three-particle scattering is exposed in Enss (1983).

Nonperturbative Approach Now N and d are arbitrary. In the nonperturbative approach (see Graf (1990), Sigal and Soffer (1989), and Yafaev (1993)) the operators H and H0 as well as the Hamiltonians of all subsystems are treated on an equal basis. It is supposed that all pair potentials satisfy condition [11]. No assumptions on subsystems are required.

The starting point of this approach is the limitingabsorption principle, which claims that the operator hxil , x 2 Xcm , for l > 1=2 is H-smooth on any compact interval  not containing the thresholds and eigenvalues of H. Its proof relies on the Mourre commutator method (see Cycon et al. (1987)). To be more precise, it is deduced from the following estimate: ið½H; Af ; f Þ  ckf k2 ;

c ¼ cðÞ > 0

f 2 Eð ÞH

½24

for the commutator of H with the generator of translations X A ¼ i ðxj @j þ @j xj Þ j

Here xj are coordinates of x 2 Xcm in some orthonormal (with respect to scalar product [6]) basis in Xcm ,  is neither a threshold nor an eigenvalue of the operator H and  is a sufficiently small interval. Very roughly speaking, the Mourre estimate [24] means that, similarly to the two-particle case, the observable ðAeiHt f ; eiHt f Þ is a strictly increasing function of t for all f 2 H(ac) . The limiting-absorption principle implies that the singular continuous spectrum of the operator H is empty, but it is not sufficient for scattering theory. If the limiting-absorption principle were true for the critical value l = 1=2, then it would imply asymptotic completeness. Unfortunately, the operator hxi1=2 is definitely not smooth even with respect to the free operator H0 . However, by introducing an auxiliary differential operator we can fix this problem. This leads to the radiation estimates. These estimates look differently in different regions of the configuration space. Choose any cluster decomposition a = (C1 , . . . , Cn ). The radiation estimate morally implies that the motion of a system is asymptotically free in the variable xa (describing the relative motion of clusters) in the region where particles from each cluster Cl , l = 1, . . . , n, are close to each other compared to distances between different clusters. On the contrary, this motion is very complicated in the variable xa pertaining to bound states of different clusters. In particular, the radiation estimate is the same as for the two-particle case in the ‘‘free’’ region where all particles are far from each other. To be more precise, let ra = rxa be the gradient in the variable xa and let r? a,  ?  ra u ðxÞ ¼ ðra uÞðxÞ  jxa j2 hðra uÞðxÞ; xa ixa be its orthogonal projection in Xa on the subspace orthogonal to the vector xa . Let a be the

N-Particle Quantum Scattering

characteristic function of a closed cone Y a  Xcm satisfying the condition Y a \ Xb = ; for all b such that Xa 6 Xb . Then the operator Ga ¼ a hxi1=2 r? a is H-smooth on . A proof of the radiation estimates is based on the consideration of the commutator of H with P some differential operator M = i (m(j) @j þ @j m(j) ), where m(j) = @m=@xj . Here m (it depends on a) is a specially constructed function satisfying the following properties: 1. m(x) is homogeneous (for jxj  1) of order 1; 2. for any b it does not depend on xb in some conical neighborhood of the subspace Xb ; 3. m(x) is convex; and 4. m(x) = a jxa j, a  1, on support of the function a . Note that we can set m(x) = jxj in the case of the operator H0 . Due to properties (1) and (2) the commutator [V, M] is a short-range function (estimated by hxi1" for " > 0). Due to properties (3) and (4) the commutator [H0 , M]  cG a Ga , c > 0, up to short-range terms. The estimate ½H; M  cG a Ga  c1 hxi1" implies that the operator Ga is H-smooth on . The main difficulty in the N-particle problem is that pair potentials V  (x ) do not tend to zero as jxj ! 1. The idea of the proof of asymptotic completeness is to introduce auxiliary wave operators such that ‘‘effective’’ perturbations are decaying functions. This requires a suitable smooth partition of unity. Moreover, it is convenient to choose auxiliary identifications as first-order differential operators rather than operators of multiplication. Unfortunately, although such identifications allow one to ‘‘kill’’ directions where the potentials V  (x ) do not tend to zero, their commutators with the operator H0 have coefficients decaying at infinity only as jxj1 . Thus, we introduce differential operators  X ðjÞ mðjÞ Ma ¼ i a @j þ @j ma (j)

with coefficients ma = @ma =@xj . The functions ma satisfy properties (1), (2) formulated above and 5. ma (x) = 0 in some conical neighborhoods of the subspaces Xb such that Xa 6 Xb . To put it differently, ma (x) = 0 in some conical neighborhood of the subspace where xi = xj for some i, j belonging to different clusters C1 , . . . , Cn .

591

Let the operator Ha be defined by eqn [9]. Given the limiting-absorption principle and the radiation estimates, we first check the existence of auxiliary wave operators W  ðH; Ha ; Ma Ea ðÞÞ and W  ðHa ; H; Ma EðÞÞ

½25

Here we use that according to (5) coefficients of the differential operator (V  V a )Ma are, under assumption [11], short-range (in the configuration space Xcm ). By property (2), the function [V a , Ma ] is also short-range. Thus, the operator VMa  Ma V a can be taken into account by the limiting-absorption principle. The commutator [H0 , Ma ] factorizes into a product of Ha - and H-smooth operators according to the radiation estimates. P Similar P arguments show that, for a ma = m and M = a Ma (the sums here are taken over all possible breakups of the N-particle system), the wave operator (observable) W  ðH; H; MEðÞÞ

½26

also exists. Moreover, it can be easily achieved that m(x)  1. Then it follows from the Mourre estimate that operator [26] is positive definite on the subspace E()H and hence its range coincides with this subspace. It means that for all f 2 E()H lim k expðiHtÞf  M expðiHtÞg k ¼ 0

t!1

½27

if f = W  (H, H; ME())g . The existence of wave operators [25] implies that   for any g = E()g and g a = W (Ha , H; Ma E())g lim kM expðiHtÞg

t!1



X

expðiHa tÞg ak¼ 0

½28

a

Combining eqns [27] and [28], we see that exp (iHt)f decomposes asymptotically into simpler evolutions exp (iHa t)g a . This is one of the equivalent formulations of asymptotic completeness and leads to eqn [15]. Finally, we note that eqn [15] can be rewritten as expðiHtÞf ¼

X

expðia ðxa ; tÞÞð2itÞda =2

a

^fa ðxa =ð2tÞÞ

a

ðxa Þ þ oð1Þ

½29

592 Nuclear Magnetic Resonance

where t ! 1, da = dim Xa , ^fa is the Fourier transform of fa and a ðxa ; tÞ ¼ x2a ð4tÞ1  a t

½30

bound state. In these additional channels, the bound state of a couple of particles depends on a position of the third particle, and it is destroyed asymptotically. See also: Quantum Mechanical Scattering Theory; Schro¨dinger Operators.

Long-Range Interactions: New Channels The multiparticle problem acquires a long-range character if pair potentials decay as Coulomb potentials or slower. Similarly to the two-particle problem, for long-range potentials the definition of wave operators should be naturally modified. As in the short-range case, only the asymptotic completeness is a really difficult mathematical problem. Assume that pair potentials satisfy condition pffiffiffi jð@  V  Þðx Þj  Cð1 þ jx jÞjj ;  > 3  1 for all jj  0 and sufficiently large 0 . Then only phase factors in eqn [29] should be modified. Actually, instead of eqn [30] we should set Z 1 a ðxa ; tÞ ¼ x2a ð4tÞ1  a t  t Va ðsxa ; 0Þ ds 0

a

where Va (x) = V(x)  V (x) and as usual x = (xa , xa ). As shown in Derezin´ski (1993), with this definition of wave operators, the asymptotic completeness holds. On the contrary, if pair potentials decay slower than jxj1=2 , then the traditional picture of scattering breaks down (see Yafaev (1996)). Actually, a three-particle system might have additional scattering channels intermediary between the channel where three particles are asymptotically free and the channels where a couple of particles form a

Further Reading Cycon H, Froese R, Kirsh W, and Simon B (1987) Schro¨dinger Operators, Texts and Monographs in Physics. Berlin: Springer. Derezin´ski J (1993) Asymptotic completeness of long-range quantum systems. Annals of Mathematics 138: 427–473. Derezin´ski J and Ge´rard C (1997) Scattering Theory of Classical and Quantum N Particle Systems. Berlin: Springer. Enss V (1983) Completeness of three-body quantum scattering. In: Blanchard P and Streit L (eds.) Dynamics and Processes, Springer Lecture Notes in Mathematics, vol. 1031, pp. 62–88. Faddeev LD (1965) Mathematical Aspects of the Three Body Problem in Quantum Scattering Theory, Israel Program of Sci. Transl. Graf GM (1990) Asymptotic completeness for N-body shortrange quantum systems: a new proof. Communications in Mathematical Physics 132: 73–101. Reed M and Simon B (1979) Methods of Modern Mathematical Physics III. New York: Academic Press. Sigal IM and Soffer A (1987) The N-particle scattering problem: asymptotic completeness for short-range systems. Annals of Mathematics 126: 35–108. Yafaev DR (1993) Radiation conditions and scattering theory for N-particle Hamiltonians, Communications in Mathematical Physics 154: 523–554. Yafaev DR (1996) New channels of scattering for three-body quantum systems with long-range potentials, Duke Mathematical Journal 82: 553–584. Yafaev DR (2000) Scattering Theory: Some Old and New Problems, Springer Lecture Notes in Mathematics, vol. 1735.

Nuclear Magnetic Resonance P T Callaghan, Victoria University of Wellington, Wellington, New Zealand ª 2006 Elsevier Ltd. All rights reserved.

Introduction The existence of nuclear spin and its associated magnetism was first suggested by Wolfgang Pauli in 1924, a conjecture based on the fine details of atomic spectra, the so-called hyperfine structure. The interaction of this nuclear magnetism with an external magnetic field was predicted to result in a finite number of discrete energy levels known as the Zeeman structure. However, the first direct

excitation of transitions between nuclear Zeeman levels was by Isador Rabi in 1933, using radiofrequency (RF) waves in an atomic beam apparatus. In 1945, Felix Bloch and co-workers at Stanford, and Edward Purcell and co-workers at MIT, performed the first nuclear magnetic resonance (NMR) experiments in condensed matter, with the RF response of the hydrogen nucleus (proton) being directly detected. The early prospects for this new technique were limited to precise measurements of magnetic fields and nuclear magnetic moments. However, three transformational discoveries intervened to set NMR on a course that would result in initially unimaginable contributions to physics, chemistry,

Nuclear Magnetic Resonance

engineering, medicine, geology, food science, and biochemistry. In 1950, it was found that atomic nuclei at different sites of a molecular orbital had slightly different resonant frequencies, a phenomenon known as ‘‘chemical shift.’’ In the a same year, Erwin Hahn discovered the spin echo, thus opening the possibility that multiple RF pulse trains could be used to remove unwanted nuclear spin interactions while being used to manipulate spin coherences with exquisite resolution. In addition, in 1951, using this spin echo, Herbert Gutowsky and Charles Slichter revealed a hitherto unobserved scalar spin–spin interaction between nuclei, mediated by the molecular orbital electrons. The discovery of the chemical shift and the scalar coupling would immediately revolutionize chemistry. Further discoveries of nuclear quadrupole interactions and through-space dipolar interactions would add to the capacity of NMR to provide insight regarding structure and order in the solid and liquid crystalline state. But the spin echo would provide a platform for new advances in science in every one of the six decades following the discovery of NMR in 1945. These were successively diffusion and flow NMR, multidimensional NMR, magnetic resonance imaging, protein structure NMR, ex situ NMR, and quantum computing NMR.

Resonant Excitation and Detection In quantum-mechanical language, the Zeeman Hamiltonian H for a nuclear spin experiencing a magnetic field B0 along the laboratory z-axis may be written as H ¼ B0 Iz

½1

 being the (nuclear) gyromagnetic ratio while Iz is the operator for the z-component of angular momentum, with eigenvalues m h, m lying in the range I, I þ 1, . . . , I. I is the angular momentum quantum number, being either integer or half-integer. From the Schro¨dinger equation, it can be seen that the eigenkets of H precess about the z-axis at a rate B0 , the frequency corresponding to the energy difference between the 2I þ 1 Zeeman levels. For convenience, we shall take the eigenvalues of Iz to be simply m, dropping the factor  h, and leading to a Hamiltonian expressed in frequency rather than in energy units. Resonant excitation between the Zeeman levels is achieved by the application of an RF (!) magnetic field of amplitude 2B1 linearly polarized normal to B0 such that the total Hamiltonian becomes H ¼ B0 Iz  2B1 cos !tIx

½2

593

This excitation is easily applied by means of a transversely oriented antenna coil, the same coil generally being used to detect the nuclear spin response. In the frame of reference rotating about B0 at !, the Hamiltonian transforms to   ! H ¼   B0  Iz  B1 Ix   B1 expði2!tIz ÞIx expði2!tIz Þ

½3

At resonance, ! = !0 = B0 . The last term in eqn [3] averages to zero and may be neglected (the Heisenberg condition) provided !  B1 , that is, B0  B1 . Given B0 of the order of tesla and B1 of the order of millitesla, this condition is easily satisfied. Hence, from the perspective of the rotating frame, the spins at resonance see only the static magnetic interaction B1 Ix , so that application of this resonant RF field causes spins to nutate about the rotating frame x-axis at a rate B1 . Thus, by application of RF pulses of different duration, and phases, one may produce arbitrary reorientation of the spins about various axes in the rotating frame. With the spin system disturbed from equilibrium, the NMR ‘‘signal’’ is detected via the subsequent free precession, and usually via the same antenna coil used for resonant excitation, Semiclassically, the phenomenon may be pictured as follows. RF excitation nutates an initial z-magnetization into the transverse plane of the rotating frame. Such transverse magnetization corresponds the laboratory frame to a magnetization precessing at the Larmor frequency, thus inducing an oscillating emf in the receiver coil. In the next section, we see how to describe this phenomenon in the language of quantum mechanics. Typically, NMR is performed using the nuclei of common atoms in organic molecules, (1 H, 2 H, 13 C, 15 N, 19 F, 31 P) although for inorganic matter a wider class of nuclei are available. Of all these, the proton is most abundant and most sensitive, having the highest gyromagnetic ratio, , of all stable nuclei.

The Quantum Statistics of the Spin Ensemble The nuclear Zeeman energy in typically available laboratory magnetic fields, B0 h, is many orders of magnitude smaller than the Boltzmann energy, kB T, except at millikelvin temperatures. At room temperature in thermal equilibrium, the fractional difference in populations between the Zeeman levels

594 Nuclear Magnetic Resonance

is normally very small, for example, for protons, about 105 . Of course, the total number of spins available may be very large, for example, on the order of 1020 . The signal in magnetic resonance is detected as a collective effect of the large ensemble of nuclear spins. The natural language of quantum statistics is that of the density matrix, ; the time-dependent expectation value for any observable represented by an operator O is then, tr(O(t)), the diagonal sum of the product of O and . The time evolution of the density matrix is given by the Liouville equation i

@ ¼ ½H;  @t

½4

where [ , ] is a commutator. For a constant Hamiltonian, this equation gives ðtÞ ¼ expðiHtÞð0Þ expðiHtÞ

½5

Physical solutions to the density matrix (Liouville space) are (2I þ 1)2 (square) matrices formed in the (2I þ 1)-dimensional angular momentum eigenbasis. Generally, we may write the density matrix in a representation of irreducible tensor operators. One very convenient representation is the set formed by taking products of spin operators. For example, in the case of spin-1/2 where Liouville space is 22 -dimensional, we may write ðtÞ ¼ 12 I þ ax Ix þ ay Iy þ az Iz

½6

where I is the identity operator. The operators Ix and Iy provide the off-diagonal elements of  and define the degree of phase coherence in the ensemble, while the operator Iz defines the degree to which the diagonal elements differ, thus defining the polarization. ax and ay give the amount of ‘‘onequantum coherence’’ in the ensemble while az gives the polarization. In thermal equilibrium ax = ay = 0, and the spin ensemble exists in a state of pure longitudinal polarization given, in the hightemperature approximation, B0  h t2 ). In solids, the proton T2 relaxation may take place in a few tens of microseconds. In the fast motion limit < !2 >1=2 c 1, the decay of the detected magnetization is exponential, and given by a factor exp( c t). Liquid state T2 values approach T1 under extreme narrowing conditions.

HCS ¼  i B0 Iiz  12 ð3 cos2   1Þ

The Details of the Nuclear Spin Hamiltonian

ð 33  i ÞB0 Iiz

Atomic nuclei interact with their environment, with surrounding electrons, and with other nuclear spins. It is precisely this feature that provides such a sensitive probe of material structure and dynamics. For a material immersed in a steady magnetic field B0 along the laboratory z-axis, the Hamiltonian for the ith nuclear spin can be written X :B þ J I i :I j H ¼ B0 Iiz  I i :S ¼ 0 j

þ

X j

I i :D :I þ I i :Q:I i ¼ j ¼

superconducting magnets, this interaction can be as large as 1000 MHz, although in earth field applications it can be as small as 2.5 kHz. Given that the sensitivity and resolution of NMR generally improve with increasing magnetic field, the range of 100–1000 MHz is typically the operating regime of choice. All other terms in the nuclear spin Hamiltonian are smaller and thus act as first-order perturbations only, projecting their quantum operators into the zeroth-order Zeeman eigenbasis, the quantum frame of the operator Iz . Because several of the terms in H depend on the orientation of the local nuclear environment (e.g., the molecular orbital) with respect to the magnetic field, these terms will fluctuate in the presence of reorientational motions. By the Heisenberg uncertainty principle, fluctuations faster in frequency than the size of the Hamiltonian contribution, expressed in frequency units, will result in an averaging to the mean, a phenomenon known as ‘‘motional averaging.’’ The term I i .S .B is the chemical shift that occurs ¼ 0 for nuclei in molecular atoms, or the knight shift for nuclei in metals. It is typically a few ppm to several 100 ppm (i.e., 100’s Hz to 10 kHz), depending on the nucleus. ¼S = 

¼ is a tensor whose principal axes (1, 2, 3) are associated with the local symmetry axis of the molecular orbital (bond) in the vicinity of the nucleus. For a liquid state molecule tumbling rapidly and isotropically, only the averaged trace of ¼ , i = (1=3)( 11 þ 22 þ 33 ) survives under motional averaging, giving a fixed frequency shift  i B0 Iiz . However, in a solid-state environment, the remaining terms also contribute to the anisotropic chemical shift

½11

It is the variety of the terms in the nuclear spin Hamiltonian that imparts power to NMR. The first is the nuclear Zeeman interaction with the applied magnetic field. In modern laboratory

½12

where  is the polar angle between the magnetic field and the principal axis (thePaxis ‘‘3’’). The scalar coupling term, j JI i .I j causes each (ith spin) energy level to be sensitive to the quantum states of the neighboring j-spins, the coupling constant J being typically tens to hundreds of hertz for nearby spins, but reducing rapidly with greater distance P in the molecular orbital. Note that the operator j JI i .I j is nondiagonal in the zeroth-order representation, but provided that the chemical shift between the I and j spins is larger than the coupling frequency (known in chemistryPas an AX spin system), the operator reduces to j JIiz Ijz the effect being to split the i-spin resonance in to a multiplet, depending on the state of the nearby j-spin. For m identical nearby j-spins, the multiplet bears a simple

Nuclear Magnetic Resonance

binomial relationship to m, allowing one to ‘‘read’’ this number directly. The combination of chemical shift and scalar coupling information is of profound importance in identifying molecular structure in chemistry. P The terms j I i .D .I and I i .Q .I i are, respectively, ¼ j ¼ the through-space dipolar interaction, HD , and the nuclear quadrupole interaction, HQ , the latter being nonzero only for nuclear spin quantum numbers I  1=2, for example, 2 H. These interactions, projected into the zeroth-order Zeeman frame, for the dipole– dipole interaction, are  0  h X i j 1  1  3 cos2 ij HD ¼ 3 4 j>i rij 2   3Iiz Ijz  I i :I j

½13

where rij is the internuclear distance and ij is the angle made by the internuclear vector with the magnetic field direction; while, for the quadrupole interaction HQ ¼

3eVZZ Q 1 ð1  3 cos ZZ Þ 4Ið2I  1Þh 2   3Iz2  IðI þ 1Þ

½14

where Q is the nuclear quadrupole moment, VZZ is the electric field gradient (assuming axial symmetry) and ZZ is the angle made by the principal axis of that gradient with the magnetic field direction. For protons in organic matter, the internuclear dipole interaction strength is on the order of 100 kHz, a similar strength being found for the quadrupole interaction of deuterons. However, in the liquid state, these orientation-dependent interactions fluctuate so rapidly that they are typically motionally averaged to zero. Nonetheless, their fluctuations do contribute to the relaxation process. Liquid-state NMR can result in exceptionally high-resolution (sub-Hz) spectra, if care is taken to adjust the magnetic field harmonics (shims) to produce a highly uniform Zeeman field across the sample. The last contribution of residual inhomogeneities to line broadening can often be removed by gently spinning the sample about its axis at a rate of a few tens of hertz.

The Evolution Domain, Multiple RF Pulses, and Multidimensional NMR Having seen the complexity of the spin Hamiltonian, one may envisage experiments where the spin coherences evolve in a much more complicated manner. To this end, consider the case of a

597

molecular liquid two-spin (AX) system coupled via the scalar spin–spin interaction. In first-order perturbation theory, we may represent the simple twospin Hamiltonian (in the rotating frame of the averaged Larmor frequency) as Hrot ¼  1 B0 I1z  2 B0 I2z þ J I1z I2z ¼ !1 I1z  !2 I2z þ J I1z I2z

½15

We now write down the density matrix in the rotating frame following a single 90x RF pulse (Ix ), ðtÞ ¼ expði!1 tI1z þ i!2 tI2z þ iJ I1z I2z tÞ      exp i Ix aeqbm ðI1z þ I2z Þexp i Ix 2 2 expði!1 tI1z  i!2 tI2z  iJ I1z I2z tÞ ¼ expði!1 tI1z þ i!2 tI2z þ iJ I1z I2z tÞaeqbm ðI1y þ I2y Þ expði!1 tI1z  i!2 tI2z  iJ I1z I2z tÞ ¼ expði!1 tI1z þ i!2 tI2z Þaeqbm     I1y þ I2y cos 12 Jt þ 2ðI1z I2x þ I1x I2z Þ   sin 12 Jt expði!1 tI1z  i!2 tI2z Þ 0 1 I1y cos!1 t þ I2y cos !2 t B C B þ I sin ! t þ I sin ! t cos 1 Jt C 1x 1 2x 2 B C 2 B C  B C ¼ aeqbm B þ 2 I1z I2x cos !2 t  I1z I2y sin !2 t C B C B C B þ I1x I2z cos !1 t  I1y I2z sin !1 tÞ C @ A 1  sin 2 Jt

½16

Detection in the rotating frame with Ix þ iIy gives a signal   SðtÞ aeqbm ðexpði!1 tÞ þ expði!2 tÞÞ cos 12 Jt ½17 Fourier transformation with respect to t yields a spectrum corresponding to two spectral lines at !1 and !2 , each split into a doublet of two sidebands separated by J. Notice that it is easier to follow the evolution of the density matrix by simply writing down a time sequence of behaviors under the influence of the successive Hamiltonians. Where simultaneous terms in the Hamiltonians commute, the order of their operation may be set at will. Thus, the above example becomes     JI1z I2z t  2Ix I1z þ I2z ! I1y þ I2y ! I1y þ I2y cos 12 Jt   þ 2ðI1z I2x þ I1x I2z Þ sin 12 Jt !1 tI1z þ!2 tI2z   ! I1y cos !1 t þ I2y cos !2 t   þ iI1x sin !1 t þ iI2x sin !2 tÞ cos 12 Jt  þ 2 I1z I2x cos !2 t  iI1z I2y sin !2 t    þI1x I2z cos !1 t  iI1y I2z sin !1 t sin 12 Jt

½18

598 Nuclear Magnetic Resonance

Indirect (scalar) spin–spin coupling via electron Local chemical shift A diamagnetic shielding A A

A

A

ppm

ppm 6.5

3.7

3.6

6.4

3.0

ppm 1.4

6.3

ppm

1.4

1.2

1.0

1.2

1.0

A

σ

J coupling σ

ppm 6

5

4

3

2

1

Figure 3 The proton NMR spectrum of ethanol showing three major peaks, separated by chemical shift, each split into multiplets arising from nearby protons via the scalar coupling.

Now consider a two RF pulse scheme as shown in Figure 4, each RF pulse being 90 x . The evolution is  2Ix

^

I1z þ I2z ! I1y þ I2y !1 t1 I1z þ!2 t1 I2z JI1z I2z t1  I1y cos !1 t1 þ I2y cos !2 t1 þ I1x sin !1 t1   þ I2x sin !2 t1 Þ cos 12 Jt1 þ 2ðI1z I2x cos !2 t1 I1z I2y sin !2 t1 þ I1x I2z cos !1 t1    I1y I2z sin !1 t1 sin 12 Jt1  2Ix

!ðI1z cos !1 t1  I2z cos !2 t1 þ I1x sin !1 t1   þ I2x sin !2 t1 Þ cos 12 Jt1  þ 2 I1y I2x cos !2 t1 þ I1y I2z sin !2 t1    þ I1x I2y cos !1 t1 þ I1z I2y sin !1 t1 sin 12 Jt1

^

!1 t2 I1z þ!2 t2 I2z þJI1z I2z t2

Keeping only observable magnetization

ðI1x sin !1 t1 cos !1 t2 þ I2x sin !2 t1 cos !2 t2 Þ     cos 12 Jt1 cos 12 Jt2 þ ðI1x sin !2 t1 sin !1 t2     þ I2x sin !1 t1 sin !2 t2 Þ sin 12 Jt1 sin 12 Jt2 ½19 If the idealized experiment is performed with two independent time dimensions t1 and t2 , then detection in the rotating frame over the t2 period with

Ix þ iIy gives a signal (restricting our attention to the quadrant of positive frequencies) Sðt1 ; t2 Þ aeqbm ðexpði!1 t1 Þ expði!1 t2 Þ þ expði!2 t1 Þ     expði!2 t2 ÞÞ cos 12 Jt1 cos 12 Jt2 þ aeqbm ðexpði!2 t1 Þ expði!1 t2 Þ þ expði!1 t1 Þ expði!2 t2 ÞÞ     sin 12 Jt1 sin 12 Jt2

½20

When Fourier transformed in two dimensions with respect to t1 and t2 , the pattern shown in Figure 5 results. Remarkably, while the diagonal spectrum is the same pair of doublets seen in the figure, this two-dimensional spectrum contains off-diagonal antiphase peaks for scalar-coupled sites where magnetization transfer has occurred. The idea of performing NMR in two or more dimensions was first proposed by Jean Jeener in 1971. The example outlined above, correlation spectroscopy (COSY), is just one of an array of coherence transfer experiments using multiple RF pulse trains and time domain evolution of the spin ensemble. Notice that in the COSY experiment, t1 is an evolution dimension during which no detection of NMR signal occurs, while t2 is the detection domain.

Nuclear Magnetic Resonance

t1 90°x

t2 90°x

rot

= – ω1I1z – ω2I2z + JI1zI2z

Figure 4 RF pulse scheme used for COSY experiment.

Figure 5 Schematic COSY (modulus) spectrum for an AX spin system. Not that the (antiphase) off-diagonal peaks indicate J-couplings between chemical-shift-separated spins.

The effect of the evolution is indelibly imprinted in the spin system density matrix allowing later recall of vital information concerning the interactions present in the spin Hamiltonian. The COSY experiment allows one to determine which spins are coupled via their molecular orbital electrons. Other multidimensional methods that rely on dipole–dipole relaxation effects, such as NOESY, determine which spin sites have ‘‘through-space’’ proximity. The use of two- and higher-dimensional methods has allowed the NMR spectra of biological macromolecules to be unraveled, with COSY methods used for spectral assignment of amino acid units, and NOESY methods used to determine any close proximities of amino acids otherwise well separated in the sequence. Such distance information has allowed the reconstruction of protein conformations by NMR. The second RF pulse of Figure 4 also generates a state of the density matrix, I1y I2x known as a double quantum coherence, and, in the simple COSY experiment, lost to observable magnetization. Other RF pulse schemes can take advantage of this state, converting it via suitable ‘‘coherence pathways’’ into an observable. For a detailed summary of these various NMR phenomena, readers are referred to the book by Ernst et al. (1987).

599

Solid-State NMR As with J couplings, dipolar interactions and quadrupole interactions (I > 1=2) are bilinear in the spin operators and can be used to generate various higher-order coherence pathways in NMR experiments. Unlike the simple spin–spin coupling, they have an angular dependence. In solids, these interactions may broaden the NMR resonance line by tens to hundreds of kilohertz. In the case where a probe nucleus is located at a known site in the material (often achieved by deuteron labeling), these Hamiltonian terms may contribute important information about structure, and especially orientational anisotropy. For example, the quadrupole interaction for the spin-1 deuteron (see eqns [11] and [14]) depends as P2 ( cos ZZ ) = (1=2)(3 cos ZZ  1) on the angle between the external magnetic field and the electric field gradient (generally associated with the local molecular orbital or bond direction, and taken here to be axially symmetric). Note that the first-order contribution of the quadrupole interaction leads to an unequal separation of the m = 1, 0, 1 Zeeman energy levels, resulting in a doublet NMR spectrum, for any particular orientation, ZZ . Such a unique orientation might be found in a single crystal, or in a nematic liquid crystalline state. For a polycrystalline material, however, the NMR spectrum has a contribution from all orientations, leading to a characteristic powder pattern. The details of 2 H spectral distributions may be used to characterize the degree of orientational order in solids and soft, anisotropic matter. For 1 H, 13 C, and other spin-1/2 nuclei, dipolar interactions (with a wide distribution of spin spacings and internuclear vector orientation) may severely broaden the NMR spectrum in the solid state (see eqns [11] and [13]). Such interactions, along with quadrupole interactions for nuclei with I > 1=2, may be significantly reduced by modulating the effective dipolar Hamiltonian at a rate faster than its strength in frequency units. Two methods are available, one (magic angle spinning or MAS) relying on the angular terms in eqns [13] and [14], and the other (multiple pulse line narrowing) on the spin terms. The MAS technique relies on spinning the sample rapidly about at angle oriented at 54.4 to the magnetic field, such that the average value of P2 (cos ij ) becomes its projection along this spinning axis, while the projection of the spinning axis residual is P2 (cos 54.4 )  0. Multiple pulse methods rely on a successive reorientation of the spin system such that the effective dipolar Hamiltonian that results from the application of the nested evolution operators is rendered close to zero.

600 Number Theory in Physics

In practice, MAS techniques work best with 13 C NMR where the moderate 1 H–13 C dipolar interactions may be removed with achievable spinning speeds (a few tens of kilohertz). Furthermore, the larger proton magnetization (1 H =13 C  4) can be transferred to the 13 C nuclei via Hartman–Hahn cross-polarization thus significantly enhancing sensitivity. Such methodology is referred to as CPMAS NMR. The real art of solid-state NMR is in removing the unwanted dipolar or quadrupolar interactions, but leaving specific interactions of interest. This may be achieved by including in the MAS experiment, specific combinations of pulses which recouple selected spins. Some of the most sophisticated experiments in modern NMR are to be found in this domain of application.

Conclusion NMR provides exceptional structural information concerning molecules, biomolecules as well as molecular assemblies, liquid crystals, soft solids, and solids. In addition, the method provides unique information concerning molecular dynamics, through both relaxation methods and the direct measurement of diffusion or flow. One spectacular application of NMR concerns its use in imaging, achieved by giving the Larmor frequency a spatial tag through the use of deliberately inhomogeneous

magnetic fields. This topic is covered in the article on Magnetic Resonance Imaging. See also: Magnetic Resonance Imaging.

Further Reading Abragam A (1961) Principles of Nuclear Magnetism. Oxford: Clarendon. Bloch F, Hansen WW, and Packard M (1946) Nuclear induction. Physical Review 70: 474. Ernst RR, Bodenhausen G, and Wokaun A (1987) Principles of Nuclear Magnetic Resonance in One and Two Dimensions. Oxford: Clarendon. Goldman M (1991) Quantum Description of High-Resolution NMR in Liquids. New York: Oxford University Press. Hahn EL (1950) Spin echoes. Physical Review 80: 580. Jeener J (1971) Ampere International Summer School. Yugoslavia: Basko Polje. McNeil EB, Slichter CP, and Gutowsky HS (1951) Slow beats in 19F nuclear spin echoes. Physical Review 84: 1245. Mehring M (1983) Principles of High Resolution NMR in Solids. Berlin: Springer. Purcell EM, Torrey HC, and Pound RV (1946) Resonance absorption by nuclear magnetic moments in a solid. In: Physical Review 69: 37. Rabi II and Cohen VW (1933) Measurement of nuclear spin by the method of molecular beams: the nuclear spin of sodium. Physical Review 46: 707. Schmidt-Rohr K and Spiess H-W (1994) Multi-Dimensional Solid State NMR and Polymers. London: Academic Press. Slichter CP (1963) Principles of Magnetic Resonance. New York: Harper and Row.

Number Theory in Physics M Marcolli, Max-Planck-Institut fu¨r Mathematik, Bonn, Germany ª 2006 Elsevier Ltd. All rights reserved.

Several fields of mathematics have closely been associated to physics: this has always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras, and functional analysis, or group theory also developed a close relation to physics. In the 1990s, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics,

mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other, we will assume a somewhat extensive definition of number theory that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopedia. The choice of topics represented here inevitably reflects the limited knowledge, particular interests, and bias of the author. Very useful references, collecting a lot of material on number theory and physics, are the proceedings of the Les Houches conference in 2003 (Beilinson and Manin 1986), as well as the two volumes of a previous Les Houches conference on number theory and physics, which took place in 1989, published by Springer in 1990 and 1992. A number theory and physics database is presently maintained online by M R Watkins.

Number Theory in Physics

In the following, we have organized the material by topics in number theory that have so far made an appearance in physics, and for each we briefly describe the relevant context and results. This singles out many themes. We first discuss a class of functions that occur in physics and their special values that are of great number-theoretic importance. This includes the dilogarithm, the polylogarithms and multiple polylogarithms, and the multiple zeta values. We also discuss the most important symmetry groups of number theory, the Galois groups, and occurrences in physics of some forms of Galois theory. We then discuss how techniques from the arithmetic geometry of algebraic varieties, especially Arakelov geometry, play a role in string theory. Finally, we discuss briefly the theory of motives and outline its possible relation to quantum physics. From the physics point of view, it seems that the most promising directions in which number-theoretic tools have come to play a crucial role are to be found mostly in the realm of rational conformal field theories and of noncommutative geometry, as well as in certain aspects of string theory. Among the topics that are very relevant to this theme, but that will not be touched upon in this article, there are important subjects such as the theory of ‘‘arithmetic quantum chaos,’’ the use of methods of random matrix theory applied to the study of zeros of zeta functions, or mirror symmetry and its connection to modular forms. The interested reader can find such topics treated in other articles of this encyclopedia and in the references mentioned above (see Quantum Ergodicity and Mixing of Eigenfunctions; Random Matrix Theory in Physics; Mirror Symmetry: a Geometric Survey).

Dilogarithm, Multiple Polylogarithms, Multiple Zeta Values The dilogarithm is defined as Li2 ðzÞ ¼

Z z

0

1 n X logð1  tÞ z dt ¼ 2 t n n¼1

It satisfies the functional equation Li2 (z) þ Li2 (1  z) = Li2 (1)  log (z) log (1  z), where Li2 (1) = (2), for (s) the Riemann zeta function. A variant is given by the Rogers dilogarithm L(x) = Li2 (x) þ (1=2) log (x) log (1  x). For more details, see Zagier’s paper (Julia et al. 2005, vol. II). The polylogarithms are similarly defined by the P series Lik (z) = n1 zn =nk . In quantum electrodynamics, there are corrections to the value of the

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gyromagnetic ratio, in powers of the fine structure constant. The correction terms that are known exactly involve special values of the zeta function such as (3), (5) and values of polylogarithms such series defining the polylogarithm as Li4 (1=2). The P function Lis (z) = n1 zn =ns converges absolutely for all s 2 C and jzj < 1 and has analytic continuation to z 2 C n [1, 1). The Fermi–Dirac and Bose–Einstein distributions are expressed in terms of the polylogarithm function as Z 1 xs dx ¼ ðs þ 1ÞLi1þs ð e Þ x e  1 0 The multiple polylogarithms are functions defined by the expressions Lis1 ;...;sr ðz1 ; z2 ; . . . ; zr Þ X zn11 zn22    znr r ¼ ns1 ns2    nsrr n >n >>n >0 1 2 1

2

½1

r

By analytic continuation, the functions Lis1 ,..., sr (z1 , z2 , . . . , zr ) are defined for all complex si and for zi in the complement of the cut [1, 1) in the complex plane. Multiple zeta values of weight k and depth r are given by the expressions X 1 ½2 ðk1 ; . . . ; kr Þ ¼ k1 kr n1 >n2 >>nr >0 n1    nr with ki 2 N and k1  2. These satisfy many combinatorial identities and nontrivial relations over Q. For an informative overview on the subject, see Cartier (2002). Notice that, for the sums in [1] and [2], a different summation convention can also be found in the literature. Conformal Field Theories and the Dilogarithm

There is a relation between the torsion elements in the algebraic K-theory group K3 (C) and rational conformally invariant quantum field theories in two dimensions (see Nahm (2005)). There is, in fact, a map, given by the dilogarithm, from torsion elements in the Bloch group (closely related to the algebraic K-theory) to the central charges and scaling dimensions of the conformal field theories. This correspondence arises by considering sums of the form X qQðmÞ ðqÞm m2N r

½3

where (q)m = (q)m1    (q)mr , (q)mi = (1  q)(1  q2 )    (1  qmi ) and Q(m) = mt Am=2 þ bm þ h has rational coefficients. Such sums are naturally obtained from considerations involving the partition function of a bosonic rational conformal field theory (CFT). In

602 Number Theory in Physics

particular, [3] can define a modular function only if all the solutions of the equation X Aij logðxj Þ ¼ logð1  xi Þ ½4 j

determine elements of finite order in an extension ^ B(C) of the Bloch group, which accounts for the fact that the logarithm is multivalued. The Rogers dilogarithm gives a natural group homomorphism ^ (2i)2 L : B(C) ! C=Z, which takes values in Q=Z on the torsion elements. These values give the conformal dimensions of the fields in the theory. Feynman Graphs

Multiple zeta values appear in perturbative quantum field theory. D Kreimer (2000) developed a connection between knot theory and a class of transcendental numbers, such as multiple zeta values, obtained by quantum field-theoretic calculations as counterterms generated by corresponding Feynman graphs. Broadhurst and Kreimer (1997) identified Feynman diagrams with up to nine loops whose corresponding counterterms give multiple zeta values up to weight 15. Recently, Kreimer showed some deep analogies between residues of quantum fields and variations of mixed Hodge–Tate structures associated to polylogarithms. Testing predictions about the standard model of elementary particles, in the hope of detecting new physics, requires developing effective computational methods handling the huge number of terms involved in any such calculation, that is, efficient algorithms for the expansion of higher transcendental functions to a very high order. The interesting fact is that abstract number-theoretic objects, such as multiple zeta values and multiple polylogarithms, appear naturally in this context (cf., e.g., Moch et al. (2002)). The explicit recursive algorithms are based on Hopf algebras and produce expansions of nested finite or infinite sums involving ratios of gamma functions and Z-sums (Euler–Zagier sums), which naturally generalize multiple polylogarithms and multiple zeta values. Such sums typically arise in the calculation of multiscale multiloop integrals. The algorithms are designed to recursively reduce the Z-sums involved to simpler ones with lower weight or depth.

Galois Theory Given a number field K, which is an algebraic extension of Q of some degree [K : Q] = n, there is an associated fundamental symmetry group, given   is by the absolute Galois group Gal(K=K), where K an algebraic closure of K. Even in the case of Q, the

 absolute Galois group Gal(Q=Q) is a very complicated object, far from being fully understood. One can consider an easier symmetry group, which is the abelianization of the absolute Galois group. This corresponds to considering the field Kab , the ‘‘maximal abelian extension’’ of K, which has the property that ab  GalðKab =KÞ ¼ GalðK=KÞ

The Kronecker–Weber theorem shows that for K = Q the maximal abelian extension can be identified with the cyclotomic field (generated by all roots of unity), Q ab = Q cycl , and the Galois ^ , where group is identified with Gal(Q ab =Q) ffi Z ^ = A =Q . In general, for other number fields, Z f þ one has the ‘‘class field theory isomorphism’’ ’

 : GalðKab =KÞ!CK =DK where CK = A K =K is the group of idele classes and DK the connected component of the identity in CK . In general, however, one does not have an explicit description of the generators of the maximal abelian extension Kab and the action of the Galois group. This is the content of the explicit class field theory problem, Hilbert’s 12th problem. In addition to the Kronecker– Weber case, a complete answer is known pffiffiffiffiffiffiffi in the case of imaginary quadratic fields K = Q( d), with d > 1 a positive integer. In this case generators are obtained by evaluating modular functions at a point  in the upper-half plane such that K = Q() and the Galois action is described explicitly through the group of automorphisms of the modular field, through Shimura reciprocity. For a survey of the explicit class field theory problem and the case of imaginary quadratic fields, see Stevenhagen (2001). As we mentioned above, understanding the  structure of the absolute Galois group Gal(Q=Q) is a fundamental question in number theory. Grothendieck described, in his famous proposal ‘‘Esquisse d’un programme,’’ how to obtain an action of  Gal(Q=Q) on an essentially combinatorial object, the set of ‘‘dessins d’enfants.’’ These are connected graphs (on a surface) such that the complement of the graph is a union of open cells and the vertices have two different markings, with the properties that adjacent vertices have opposite markings. Such objects arise by considering the projective line P1 minus three points. Any finite cover of P1 branched only over {0, 1, 1} gives an algebraic curve defined  The dessin is the inverse image under the over Q. covering map of the segment [0, 1] in P1 . The  absolute Galois group Gal(Q=Q) acts on the data of the curve and the covering map, hence on the set of

Number Theory in Physics

dessins. A theorem of Bielyi shows that, in fact, all  are obtained as algebraic curves defined over Q coverings of the projective line ramified only over the points {0, 1, 1}. This has the effect of realizing the absolute Galois group as a subgroup of outer automorphisms of the profinite fundamental group of the projective line minus three points. For a general reference on the subject, see Schneps (1994). A different type of Galois symmetry of great arithmetic significance is ‘‘motivic’’ Galois theory. This will be discussed later in the section dedicated to motives, where we discuss a surprising occurrence in the context of perturbative quantum field theory and renormalization. Quantum Statistical Mechanics and Class Field Theory

In quantum statistical mechanics, one considers an algebra of observables, which is a unital C -algebra A with a time evolution t . States are given by linear functionals ’ : A ! C satisfying ’(1) = 1 and positivity ’(x x)  0. Equilibrium states ’ at inverse temperature satisfy the Kubo–Martin–Schwinger (KMS) condition, namely, for all x, y 2 A there exists a bounded holomorphic function Fx, y (z) on the strip 0 < =(z) < , which extends continuously to the boundary, such that for all t 2 R Fx;y ðtÞ ¼ ’ðxt ðyÞÞ

603

temperature, the evaluation of KMS1 states on elements of a rational subalgebra intertwines the ^ by automorphisms of (A, t ) with the action of Z action of Gal(Q ab =Q) on the values of the states. This recovers the explicit class field theory of Q from a physical perspective. Noncommutative space of adele classes The algebra A of the Bost–Connes system is the noncommutative ^ and r 2 Q algebra of functions f (r, ), for  2 Z ^ with the convolution product such that r 2 Z, X f1 ðrs1 ; sÞf2 ðs; Þ ½6 f1 f2 ðr; Þ ¼ ^ s2Q :s2Z

and the adjoint f (r, ) = f (r1 , r). According to the general philosophy of Connes style noncommutative geometry, it is the algebra of coordinates of the noncommutative space defined by the ‘‘bad quotient’’ GL1 (Q) n (Af {1}) – a noncommutative version of the zero-dimensional Shimura variety Sh(GL1 , {1}) = GL1 (Q) n (GL1 (Af ) {1}). Its ‘‘dual system’’ (in the sense of Connes’s duality of type III and type II factors) is obtained by taking the crossed product by the time evolution. It gives the algebra of coordinates of the noncommutative space defined by the quotient A=Q . This is the noncommutative space of ‘‘adele classes’’ used by Connes in his spectral realization of the zeros of the Riemann zeta function.

and Fx;y ðt þ i Þ ¼ ’ðt ðyÞxÞ

½5

Cases of number-theoretic interest arise when one considers the noncommutative space of commensurability classes of Q-lattices up to scaling as algebra of observables, with a natural time evolution determined by the covolume, as shown in the paper Quantum Statistical Mechanics of Q-Lattices of Connes–Marcolli (Julia et al. 2005, vol. I). A Q-lattice in R n consists of a pair (, ) of a lattice  R n together with a homomorphism of abelian groups : Q n =Zn ! Q=. Two Q-lattices are commensurable, (1 , 1 )  (2 , 2 ), iff Q1 = Q2 and 1 = 2 mod 1 þ 2 . The Bost–Connes system The quantum statistical mechanical system considered by Bost and Connes (1995) corresponds to the case of one-dimensional Q-lattices. The partition function of the system is the Riemann zeta function ( ). The system has spontaneous symmetry breaking at = 1, with a single KMS state for all 0 < 1. For > 1, the extremal equilibrium states are parametrized by the embeddings of Q cycl in C with a free transitive ^ . At zero action of the idele class group CQ =DQ = Z

The GL2 -system A generalization of the Bost– Connes system was introduced by Connes and Marcolli in the paper Quantum Statistical Mechanics of Q-Lattices (Julia et al. 2005). This corresponds to the case of two-dimensional Q-lattices. The partition function is the product ( )(  1). The system in this case has two phase transitions, with no KMS states for 1. For > 2, the extremal KMS states are parametrized by the invertible Q-lattices, namely, those for which is an isomorphism. The algebra A has an arithmetic structure given by a rational algebra of unbounded multipliers. This rational algebra contains modular functions and Hecke operators. At zero temperature, extremal KMS states can be evaluated on these multipliers. Symmetries of (A, t ) are realized in part by endomorphisms (as in the theory of superselection sectors) and the symmetry group acting on low-temperature KMS states is the group of automorphisms of the modular field GL2 (Af )=Q . For a generic set of extremal KMS1 states, evaluation at the rational algebra intertwines this action with the action on the values of an embedding of the modular field as a subfield of C.

604 Number Theory in Physics

The complex multiplication system In the case of an imaginary quadratic field K = Q(), an analogous construction is possible. A one-dimensional K-lattice is a pair (, ) of a finitely generated O-submodule  of C, with K = K, and a homomorphism of O-modules

: K=O ! K=. Two K-lattices are commensurable iff K1 = K2 and 1 = 2 mod 1 þ 2 . Connes et al. (Preprint 2005) constructed a quantum statistical mechanical system describing the noncommutative space of commensurability classes of one-dimensional K-lattices up to scale. The partition function is the Dedekind zeta function K ( ). The system has a phase transition at = 1 with a unique KMS state for higher temperatures and extremal KMS states parametrized by the invertible K-lattices at lower temperatures. There is a rational subalgebra induced by the rational structure of the GL2 -system (one-dimensional K-lattices are also two-dimensional Q-lattices with compatible notions of commensurability). The symmetries of the system are given by the idele class group A K, f =K . The action is partly realized by endomorphisms corresponding to the possible presence of a nontrivial class group (for class number > 1). The values of extremal KMS1 states on the rational subalgebra intertwine the action of the idele class group with the Galois action on the values. This fully recovers the explicit class field theory for imaginary quadratic fields.

outer automorphisms on the profinite completion of the tower. The basic building blocks of the tower are provided by ‘‘pairs of pants,’’ that is, by projective lines minus three points. This leads to a conjectural relation between the Moore–Seiberg equations and this Grothendieck– Teichmu¨ller setting (cf. Degiovanni 1994) according to which solutions of the Moore–Seiberg equations provide projective representations of the Teichmu¨ller tower, and the action of the absolute Galois group  Gal(Q=Q) corresponds to the action on the coefficients of the Moore–Seiberg matrices. Rational conformal field theories are, in general, one of the most promising sources of interactions between number theory and physics, involving interesting Galois actions, modular forms, Brauer groups, and complex multiplication. Some fundamental work in this direction was done by, for example, Borcherds and Gannon.

Arithmetic Algebraic Geometry In this section we describe occurrences in physics of various aspects of the arithmetic geometry of algebraic varieties. Arithmetic Calabi–Yau

Conformal Field Theory and the Absolute Galois Group

Moore and Seiberg considered data associated to any rational conformal field theory, consisting of matrices, obtained as monodromies of some holomorphic multivalued functions on the relevant moduli spaces, satisfying polynomial equations. Under reasonable hypotheses, the coefficients of the Moore–Seiberg matrices are algebraic numbers. This allows for the presence of interesting arithmetic phenomena. Through the Chern–Simons/Wess–Zumino–Witten correspondence, it is possible to construct three-dimensional topological field theories from solutions to the Moore– Seiberg equations. On the arithmetic side, Grothendieck proposed in his ‘‘Esquisse d’un programme’’ the existence of a Teichmu¨ller tower given by the moduli spaces Mg, n of Riemann surfaces of arbitrary genus g and number of marked points n, with maps defined by operations such as cutting and pasting of surfaces and forgetting marked points, all encoded in a family of fundamental groupoids. He conjectured that the whole tower can be reconstructed from the first two levels, providing, respectively, generators and relations. He called this a ‘‘game of Lego–Teichmu¨ller.’’ He also conjectured that the absolute Galois group acts by

In the context of type II string theory, compactified on Calabi–Yau 3-folds, Greg Moore considered certain black hole solutions and a resulting dynamical system given by a differential equation in the corresponding moduli. The fixed points of these equations determine certain ‘‘black hole attractor varieties.’’ In the case of varieties obtained from a product of elliptic curves or of a K3 surface and an elliptic curve, the attractor equation singles out an arithmetic property: the elliptic curves have complex multiplication. The class number of the corresponding imaginary quadratic field counts U-duality classes of black holes with the same area. Other results point to a relation between the arithmetic properties of Calabi–Yau 3-folds and conformal field theory. For instance, it was shown by Schimmrigk that, in certain cases, the algebraic number field defined via the fusion rules of a conformal field theory as the field defined by the eigenvalues of the integer-valued fusion matrices

i j ¼ ðNi Þkj k can be recovered from the Hasse–Weil L-function of the Calabi–Yau. An interesting case is provided by the Gepner model associated with the Fermat quintic Calabi–Yau 3-fold.

Number Theory in Physics Arakelov Geometry

For K a number field and OK its ring of integers, a smooth proper algebraic curve X over K determines a smooth minimal model XOK , which defines an arithmetic surface X OK over Spec(OK ). The closed fiber X} of X OK over a prime } 2 OK is given by the reduction mod }. When Spec(OK ) is ‘‘compactified’’ by adding the Archimedean primes, one can correspondingly enlarge the group of divisors on the arithmetic surface by adding formal real linear combinations of irreducible ‘‘closed vertical fibers at infinity.’’ Such fibers are only treated as formal objects. The main idea of Arakelov geometry is that it is sufficient to work with ‘‘infinitesimal neighborhood’’ X (C) of these fibers, given by the Riemann surfaces obtained from the equation defining X over K under the embeddings : K ,! C that constitute the Archimedean primes. Arakelov developed a consistent intersection theory on arithmetic surfaces, by computing the contribution of the Archimedean primes to the intersection indices using Hermitian metrics on these Riemann surfaces and the Green function of the Laplacian. A general introduction to the subject of Arakelov geometry can be found in Lang (1988). Manin (1991) showed that these Green functions can be computed in terms of geodesics in a hyperbolic 3-manifold that has the Riemann surface X (C) as its conformal boundary at infinity. The Polyakov measure A first application to physics of methods of Arakelov geometry was an explicit formula obtained by Beilinson and Manin (1986) for the Polyakov bosonic string measure in terms of Faltings’s height function at algebraic points of the moduli space of curves. The partition function for the closed Pbosonic string has a perturbative expansion Z = g0 Zg , with Z ð22gÞ Zg ¼ e eSðx;Þ DxD ½7 

written in terms of a compact Riemann surface  of genus g, maps x :  ! Rd , and metrics  on . The classical action is of the form Z pffiffiffiffiffiffi Sðx; Þ ¼ d2 z jj ab @a x @b x ½8 

Using the invariance of the classical action with respect to the semidirect product of diffeomorphisms of  and the conformal group, the integral is reduced (in the critical dimension d = 26 where the conformal anomaly cancels) to a zeta regularized

605

determinant of the Laplacian for the metric on  and an integration over the moduli space Mg of genus g algebraic curves. Beilinson and Manin gave an explicit formula for the resulting Polyakov measure on Mg using results of Faltings on Arakelov geometry of arithmetic surfaces. In particular, their argument uses essentially the properties of the Faltings metrics on the invertible sheaves d(L) given by the ‘‘multiplicative Euler characteristics’’ of sheaves L of relative 1-forms. For a suitable choice of bases { j } and {wj } of differentials and quadratic differentials, the formula for the Polyakov measure is then of the form (up to a multiplicative constant)  1 ^ ^ dg ¼ jdet Bj18 ðdet =Þ13 W1 ^ W  W3g3 ^ W3g3 ½9 R with  in the Siegel upper-half space, Bij = ai j , and the Wj given by the images of the basis wj under the Kodaira–Spencer isomorphism. Holography In the case of the elliptic curve Xq (C) = C =qZ , a formula of Alvarez-Gaume, Moore, and Vafa gives the operator product expansion of the path integral for bosonic field theory as gðz; 1Þ ¼ log jqjB2 ðlog jzj= log jqjÞ=2 j1  zj

1 Y

! n

n 1

j1  q zj j1  q z j

½10

n¼1

where B2 is the second Bernoulli polynomial. Expression [8] is in fact the Arakelov Green function on Xq (C) (cf. Lang (1988)). Using this and analogous results for higher genus Riemann surfaces, Manin and Marcolli (2001) showed that the result of Manin (1991) on Arakelov and hyperbolic geometry can be rephrased in terms of the AdS/CFT correspondence, or holography principle. Expression [8] can then be written as a combination of terms involving geodesic lengths in the Euclidean BTZ black hole. In the case of higher genus curves, the Arakelov Green function on a compact Riemann surface, which is related to the two-point correlation function for bosonic field theory, can be expressed in terms of the semiclassical limit of gravity (the geodesic propagator) on the bulk space of Euclidean versions of asymptotically AdS2þ1 black holes introduced by K Krasnov.

Motives There are several cohomology theories for algebraic varieties: de Rham, Betti, e´tale cohomology. de Rham

606 Number Theory in Physics

and Betti are related by the period isomorphism, and comparison isomorphisms relate E´tale and Betti cohomology. In the smooth projective case, they have the expected properties of Poincare´ duality, Ku¨nneth isomorphisms, etc. Moreover, E´tale cohomology provides interesting ‘-adic representations of  Gal(k=k). In order to understand what type of information, such as maps or operations can be transferred from one to another cohomology, Grothendieck introduced the idea of the existence of a ‘‘universal cohomology theory’’ with realization functors to all the known cohomology theories for algebraic varieties. He called this the theory of ‘‘motives.’’ Properties that can be transferred between different cohomology theories are those that exist at the motivic level. A short introduction to motives can be found in Serre (1992). The first constructions of a category of motives proposed by Grothendieck covers the case of smooth projective varieties. The corresponding motives form a Q-linear abelian category of ‘‘pure motives.’’ Roughly, objects are varieties and morphisms are ‘‘correspondences’’ given by algebraic cycles in the product, modulo a suitable equivalence relation. The category also contains Tate objects generated by Q(1), which is the inverse of the pure motive H 2 (P1 ). Grothendieck’s standard conjectures imply that the category of pure motives is equivalent to the category of representations RepG of a ‘‘motivic Galois group,’’ which in the case of pure motives is proreductive. The subcategory of pure Tate motives has as motivic Galois group the multiplicative group Gm . The situation is more complicated for ‘‘mixed motives,’’ for which constructions were only very recently proposed (e.g., in the work of Voevodsky). These provide a universal cohomology theory for more general classes of algebraic varieties. Mixed Tate motives are the subcategory generated by the Tate objects. There is again a motivic Galois group. For mixed motives it is an extension of a proreductive group by a prounipotent group, with the proreductive part coming from pure motives and the prounipotent part from the presence of a weight filtration on mixed motives. The multiple zeta values appear as periods of mixed Tate motives. Renormalization and Motivic Galois Theory

A manifestation of motivic Galois groups in physics arises in the context of the Connes–Kreimer theory of perturbative renormalization (for an introduction to this topic, see Hopf Algebra Structure of Renormalizable Quantum Field Theory). In fact, according to the Connes–Kreimer theory, the Bogoliubov–Parasiuk– Hepp–Zimmerman (BPHZ) renormalization scheme

with dimensional regularization and minimal subtraction can be formulated mathematically in terms of the Birkhoff factorization ðzÞ ¼  ðzÞ1 þ ðzÞ

½11

of loops in a prounipotent Lie group G, which is the group of characters of the Hopf algebra of Feynman graphs. Here, the loop  is defined on a small punctured disk around the critical dimension D, þ is holomorphic in a neighborhood of D, and  is holomorphic in the complement of D in P1 (C). The renormalized value is given by þ (D) and the counterterms by  (z). The paper of Connes and Marcolli Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory in volume II of Julia et al. (2005) shows that the data of the Birkhoff factorization are equivalently described in terms of solutions to a certain class of differential systems with irregular singularities. This is obtained by writing the terms in the Birkhoff factorization as time-ordered exponentials, and then using the fact that Rb 1 Z X ðtÞ dt Te a :¼ 1 þ ðs1 Þ    ðsn Þ ds1    dsn n¼1

a s1  sn b

is the value g(b) at b of the unique solution g(t) 2 G with value g(a) = 1 of the differential equation dg(t) = g(t) (t) dt. The singularity types are specified by physical conditions, such as the independence of the counterterms on the mass scale. These conditions are expressed geometrically through the notion of G-valued ‘‘equisingular connections’’ on a principal C -bundle B over a disk , where G is the prounipotent Lie group of characters of the Connes–Kreimer Hopf algebra of Feynman graphs. The ‘‘equisingularity’’ condition is the property that such a connection ! is C -invariant and that its restrictions to sections of the principal bundle that agree at 0 2  are mutually equivalent, in the sense that they are related by a gauge transformation by a G-valued C -invariant map regular in B; hence, they have the same type of (irregular) singularity at the origin. The classification of equivalence classes of these differential systems via the Riemann–Hilbert correspondence and differential Galois theory yields a Galois group U = Uo Gm , where U is prounipotent, with Lie algebra the free graded Lie algebra with one generator en in each degree n 2 N. The group U is identified with the motivic Galois group of mixed Tate motives over the cyclotomic ring Z[e2i=N ], for N = 3 or N = 4, localized at N.

Number Theory in Physics Speculations on Arithmetical Physics

In a lecture written for the 25th Arbeitstagung in Bonn, Y Manin presented intriguing connections between arithmetic geometry (especially Arakelov geometry) and physics. The theme is also discussed in Manin (1989). These considerations are based on a philosophical viewpoint according to which fundamental physics might, like adeles, have Archimedean (real or complex) as well as non-Archimedean (p-adic) manifestations. Since adelic objects are more fundamental and often simpler than their Archimedean components, one can hope to use this point of view in order to carry over some computation of physical relevance to the non-Archimedean side where one can employ number-theoretic methods. Adelic physics? Some of the results mentioned in the previous sections seem to lend themselves well to this adelic interpretation. The quantum statistical mechanics of Q-lattices relies fundamentally on adeles and it admits generalizations to systems associated to other algebraic varieties (Shimura varieties) that have an adelic description and adelic groups of symmetries. The result on the Polyakov measure also has an adelic flavor, as it uses essentially the Archimedean component of the Faltings height function. The latter is in fact a product of contributions from all the Archimedean and non-Archimedean places of the field of definition of algebraic points in the moduli space, so that one can expect that there would be an adelic Polyakov measure, of which one normally sees the Archimedean side only. The Freund–Witten adelic product formula for the Veneziano string amplitude fits in the same context, with p-adic amplitudes Z 1 Bp ð ; Þ ¼ jxj 1 dx p j1  xjp Qp

and B1 ( , ) (2004)).

1

=

Q

p

Bp ( , )

(cf.

Varadarajan

Adelic physics and motives A similar adelic philosophy was taken up by other authors, who proposed ways of introducing non-Archimedean and adelic geometries in quantum physics. A recent survey is given in Varadarajan (2004). For instance, Volovich (1995) proposed spacetime models based on cohomological realizations of motives, with e´tale topology ‘‘interpolating’’ between a proposed nonArchimedean geometry at the Planck scale and Euclidean geometry at the macroscopic scale. In this viewpoint, motivic L-functions appear as partition functions and actions of motivic Galois groups govern the dynamics.

607

See also: Hopf Algebra Structure of Renormalizable Quantum Field Theory; Mirror Symmetry: A Geometric Survey; Quantum Ergodicity and Mixing of Eigenfunctions; Random Matrix Theory in Physics; Regularization for Dynamical Zeta Functions.

Further Reading Beilinson A and Manin Yu (1986) The Mumford form and the Polyakov measure in string theory. Communications in Mathematical Physics 107(3): 359–376. Bost JB and Connes A (1995) Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Mathematica (N.S.) 1(3): 411–457. Broadhurst DJ and Kreimer D (1997) Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Physics Letters 393(3–4): 403–412. Cartier P (2002) Fonctions polylogarithmes, nombres polyzetas et groupes pro-unipotents, Se´minaire Bourbaki, Vol. 2000/2001. Aste´risque No. 282 (2002), Exp. No. 885, viii, 137–173. Connes A, Marcolli M, and Ramachandran N (2005) KMS states and complex multiplication. Preprint (to appear in Selecta Mathematica), arXiv math.OA/0501424. Degiovanni P (1994) Moore and Seiberg equations, topological field theories and Galois theory. In: Leila S (ed.) The Grothendieck Theory of Dessins D’enfants. Cambridge: Cambridge University Press. Julia BL, Moussa P, and Vanhove P (eds.) (2005) Frontiers in Number Theory, Physics and Geometry, vols. I, II, Papers from the Meeting held in Les Houches, March 9–21, 2003. Berlin: Springer. Kreimer D (2000) Knots and Feynman Diagrams. Cambridge Lecture Notes in Physics, vol. 13. Cambridge: Cambridge University Press. Lang S (1988) Introduction to Arakelov Theory. Berlin: Springer. Manin Yu (1989) Reflections on arithmetical physics. In: Dita P and Georgescu V (eds.) Conformal Invariance and String Theory, pp. 293–303. Boston: Academic Press. Manin Yu (1991) Three-dimensional hyperbolic geometry as 1-adic Arakelov geometry. Inventiones Mathematicae 104(2): 223–243. Manin Yu and Marcolli M (2001) Holography principle and arithmetic of algebraic curves. Advanced Theoretical and Mathematical Physics 5(3): 617–650. Moch S, Uwer P, and Weinzierl S (2002) Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. Journal of Mathematical Physics 43(6): 3363–3386. Moore G (1998) Arithmetic and attractors, hep-th/9807087. Nahm W (2005) Conformal field theory and torsion elements of the bloch group. In: Julia BL, Moussa P, and Vanhove P (eds.) Frontiers in Number Theory, Physics and Geometry, vol. I. Berlin: Springer. Schneps L (ed.) (1994) The Grothendieck Theory of Dessins d’enfants. Cambridge: Cambridge University Press. Serre JP (1992) Motifs, in Journe´es Arithme´tiques, 1989 (Luminy, 1989). Aste´risque No. 198–200, (1991), 11, 333–349. Stevenhagen P (2001) Hilbert’s 12th problem, complex multiplication and Shimura reciprocity. In: Class Field Theory – Its Centenary and Prospect (Tokyo, 1998), 161–176, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001. Varadarajan VS (2004) Arithmetic quantum physics: why, what and whether. Proceedings of the Steklov Institute of Mathematics 245: 258–265. Volovich IV (1995) From p-adic strings to e´tale strings. Proceedings of the Steklov Institute of Mathematics 203(3): 37–42.

O Operads J Stasheff, Lansdale, PA, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction An operad is an abstraction of a family of composable functions of n variables for various n, useful for the ‘‘bookkeeping’’ and applications of such families. Operads are particularly important and useful in categories with a good notion of ‘‘homotopy,’’ where they play a key role in organizing hierarchies of higher homotopies, reflecting their original use as a tool in homotopy theory, especially for studying (iterated) loop spaces. For several years now, operads have become increasingly important in mathematical physics, especially in string field theory, where they organize the terms of higher order in perturbed actions, and in deformation quantization. The major focus of this article will be on operads as they are relevant to mathematical physics, but will also include some background material from homotopy theory, where they originated. A borderland where homotopy theory and cohomological physics overlap is the world of differential graded vector spaces, including those of differential forms, ghosts, anti-ghosts, etc., sometimes lumped together as BRST theory. Here, as elsewhere in contemporary mathematical physics, the flow has been in both directions – sometimes physicists have discovered or reinvented known mathematics but finding new applications, at other times physics has suggested new concepts for mathematicians to develop further. In the case of operads, they have provided general structure for varieties of algebras, some of which are novel types contributed by physicists. For a reasonably up-to-date introduction and survey, consider Markl et al. (2002), although there have been many developments since then. Two particularly important original works are Boardman and Vogt (1973) and May (1972).

Definitions and Examples The term ‘‘operad’’ is due to May, building on work of Stasheff and of Boardman–Vogt. The most

fundamental example of an operad is the endomorphism operad EndX := {Map(Xn , X)}n1 where, for a set or topological space X, {Map(Xn , X)} means the set or space of functions or continuous functions from the n-fold product of X with itself to X, together with the operations i : MapðXn ; XÞ  MapðXm ; XÞ ! MapðXnþm1 ; XÞ given, for 1  i  n, by ðf i gÞðx1 ; . . . ; xmþn1 Þ ¼ f ðx1 ; . . . ; xi1 ; gðxi ; . . . ; xiþm1 Þ; xiþm ; . . .Þ In the endomorphism operad EndX , there are easily discovered relations involving iterated i operations and the symmetric group n actions on the Xn s. For example, ðf i gÞ j h ¼ f j ðg jiþ1 hÞ for i  j  i þ m  1 if g is a function of m variables, since only the name of the position for the insertion is changed. An operad (O, i ) consists of a collection {O(n)}n1 of objects and maps i : O(n)  O(m) ! O(n þ m  1) for m, n  1 and i  n satisfying the relations manifest in the example EndX . May’s original definition corresponds to simultaneous insertions into all possible positions of inputs into f 2 Map(Xn , X). In most examples, the structures are ‘‘manifest’’ without appeal to the technical definitions. It helps to see graphic examples of operads, particularly ones relevant for physics. Two kinds that are particularly important are the tree operads and the little disks (or cubes) operads. Let T (n) be the set of planar trees with one root and n leaves labeled (arbitrarily) 1 through n. The collection T = {T (n)}n1 of sets of trees forms an operad by grafting the root of g to the leaf of f labeled i, as in Figure 1, where the leaves are assumed labeled in order from left to right. Figure 1 can be interpreted as portraying the 4 result of inserting a 3-linear operation into a 5-linear one. The little n-disks operad Dn = {Dn (j)}j1 where Dn (j) consists of an ordered collection of j n-disks

610 Operads

(ab)(cd )

a(b(cd )) =

4

a((bc)d ) Figure 1 Grafting with the leaves numbered from left to right.

3 2

2 3

2

((ab)c)d

1

2

4

=

1

1

Figure 2 The little 2-disks operad.

embedded in the standard n-dimensional unit disk Dn with disjoint interiors, the embedding being of the form az þ b with 0 < a 2 R. The operations are given as indicated in Figure 2. Just as group theory without representations is rather sterile, so are operads best appreciated by their representations known as (varieties of) algebras, especially algebras with higher homotopies. An algebra A over an operad P ‘‘is’’ a map of operad P ! EndA . This is just a compact way of saying that an algebra A has a coherent system of maps P (n)  An ! A. Much of this article will speak in terms of such algebras with the corresponding operad being understood.

(a(bc))d

Figure 3 The associahedron K4 :

of n letters. The edges correspond to a single application of an associating homotopy. More generally, the cellular structure of the associahedra is well described by planar rooted trees, the vertices corresponding to binary trees and so forth (see Figure 4). For K5 , see Figure 5 or a rotatable image available at http://igd.univ-lyon1.fr/ chapoton/stasheff.html. The facets are all products of two associahedra of lower dimension and specific imbeddings can be given to play the role of the i operations as in an operad. An A1 -space is a space Y which admits a coherent family of maps mn : Kn  Y n ! Y so that they make Y an algebra over the operad (without n -actions) K = {Kn }n1 . The main result by Stasheff is: A connected space Y (of the homotopy type of a CW-complex) has the homotopy type of a based loop space X for some X if and only if Y is an A1 -space. Homotopy characterization of iterated loop spaces n Xn for some space Xn required the full power of the theory of operads with the symmetries.

Operads in Homotopy Theory A major motivation for the development of operads was the desire to have a homotopy-invariant characterization of based loop spaces and iterated loop spaces. Precisely such coherent systems of higher homotopies provided the answers. For based loop spaces, the operad in question K = {Kn }n1 consists of the polytopes known as ‘‘associahedra.’’ The usual product of based loops is only homotopy associative. If we fix a specific associating homotopy and consider the five ways of parenthesizing the product of four loops, there results a pentagon whose edges correspond to a path of loops (Figure 3). From the leftmost vertex to the rightmost, consider the two paths of loops across the top or around the bottom. By further adjustment of parameters, the pentagon can be filled in by a family of such paths. The associahedron Kn can be described as a convex polytope with one vertex for each way of associating n ordered variables, that is, ways of inserting parentheses in a meaningful way in a word

Figure 4 K4 with vertices labeled by trees.

Figure 5 The associahedron K5 :

Operads 611

An early motivation for the invention of a theory of operads was the consideration of infinite loop spaces, that is, a sequence of spaces Xn such that each Xn is homotopy equivalent to Xnþ1 . Although introduced originally in the category of topological spaces, operads were available almost immediately for differential graded (dg) vector spaces, also known as chain complexes. In physics, the differential is often called a BRST operator, a term that should be reserved for a special kind of dg algebra, see below.

(A, m ) is called an A1 -algebra when the multilinear maps mk satisfy the following relations: X

p X

mp i mq ¼ 0

½4

pþq¼nþ1 i¼1

with an appropriate set of signs for n  1. A weak A1 -algebra consists of a collection of degree 1 multilinear maps m :¼ fmk : Ak ! Agk0 satisfying the above relations, but for n  0 and in particular with k, l  0.

Operads in Algebra The i notation first appeared in Gerstenhaber’s study of the algebraic structure of the Hochschild cohomology of an associative algebra, about the same time as the construction of the associahedra where the operations were given in a less convenient notation. Recall the Hochschild cohomology of an associative algebra A is the homology of the complex Hom(An , A) with the coboundary given as follows (all signs below are indicated as , any of the standard references will specify conventions and signs): for f 2 Hom(An , A) and g 2 Hom(Am , A) let f  g ¼ n1 f i g

½2

where m : A  A ! A is the multiplication. Moreover, the associativity of m is equivalent to ½m; m ¼ 0

½3

A1 -Algebras

In the setting of graded vector spaces V = r2Z V r , there are two conventions for defining A1 -algebras, which differ by a shift in grading. We adopt the physics convention so that A here is the suspension of that considered in the original papers. The cellular chains of the associahedra form the A1 operad, providing the following definition. Definition 1 A1 -algebra (Strong homotopy associative algebra). Let A be a Z-graded vector space A = r2Z Ar and suppose that there exists a collection of degree 1 multilinear maps m :¼ fmk : Ak ! Agk1

m1 m1 ¼ m2 ðm0  1Þ m2 ð1  m0 Þ

½5

Just as associativity was captured by the equation [m, m] = 0, so the defining relations of the definition of an A1 -algebra are captured by ½m ; m ¼ 0

½1

Gerstenhaber then defines his bracket as [f , g] = f  g g  f . With hindsight, he realized that the Hochschild coboundary can be written as h ¼ ½m; h

Remark 1 The ‘‘weak’’ version is fairly new, inspired by physics, where m0 : C ! A, regarded as an element m0 (1) 2 A, is related to what physicists refer to as a ‘‘background.’’ The augmented relation then implies that m0 (1) is a cycle, but m1 m1 need no longer be 0, rather

½6

Decades later it was realized that considering T c A = An as a coalgebra with ða1   an Þ ¼ pþq ða1   ap Þ  ðapþ1   an Þ we then have an isomorphism HomðAn ; AÞ ’ CoderðT c AÞ Here Coder is the space of all coderivations of T c A. The Gerstenhaber bracket is indeed the ‘‘intrinsic’’ commutator bracket of coderivations via the above isomorphism. As such, it satisfies a graded version of the Jacobi identity; after a shift in grading from the original one of Hochschild, the Hochschild cochain complex forms a dg Lie algebra. L1 -Algebras

Since an ordinary Lie algebra g is regarded as ungraded, the defining bracket is regarded as skewsymmetric. For dg Lie algebras and L1 -algebras, we need graded symmetry, which refers to symmetry with signs determined by the grading. The basic operation is  : x  y 7! ð1Þjxjjyj y  x

½7

Also we adopt the convention that tensor products of graded functions or operators have the signs built

612 Operads

in; for example, (f  g)(x  y) = (1)jgjjxj f (x)  g(y). By decomposing each permutation as a product of transpositions, there is then defined the sign of a permutation of n graded elements, for example, for any ci 2 V, 1  i  n, and any  2 S n , the permutation of n graded elements is defined by ðc1 ; . . . ; cn Þ ¼ ð1ÞðÞ ðcð1Þ ; . . . ; cðnÞ Þ

½8

The sign (1)() is often referred to as the Koszul sign of the permutation. Definition 2 (Graded symmetry). A graded symmetric multilinear map of a graded vector space V to itself is a linear map f : V n ! V such that for any ci 2 V, 1  i  n, and any  2 S n (the permutation group of n elements), the relation f ðc1 ; . . . ; cn Þ ¼ ð1ÞðÞ f ðcð1Þ ; . . . ; cðnÞ Þ

½9

The operad of Lie algebras was defined rather late, although it was earlier implicit in the work of Fred Cohen. It is defined as the homology Hn1 (Config(R2 , n)) for n  1, where Config(R2 , n) denotes the configuration space of ordered n-tuples of distinct points in R2 . Equivalently, the configurations can be thought of as the centers of the little 2-disks. The open disks being contractible to their centers, this is a suboperad of the full homology H (D2 ). Just as a Lie algebra is obtained from an associative algebra using the commutator as bracket and, inversely, a Lie algebra gives rise to its universal enveloping associative algebra, an L1 -algebra can be obtained from an A1 -algebra by n-variable analogs of commutators and there is a universal enveloping A1 -algebra of a given L1 -algebra.

holds.

Open–Closed Homotopy Algebras

Definition 3 By a (k, l)-unshuffle of c1 , . . . , cn with n = k þ l is meant a permutation  such that for i < j  k, we have (i) < (j) and similarly for k < i < j  k þ l. We denote the subset of (k, l)-unshuffles in S kþl by S k, l and by S kþl = n , the union of the subsets S k, l with k þ l = n. Similarly, a (k1 , . . . , ki )-unshuffle means a permutation  2 S n with n = k1 þ þ ki such that the order is preserved within each block of length k1 , . . . , ki . The subset of S n consisting of all such unshuffles we denote by S k1 ,..., ki .

Open–closed string field theory suggests interaction between an L1 -algebra Hc and an A1 -algebra Ho including a strong homotopy representation of Hc on Ho by strong homotopy derivations. Here is the formal definition:

Definition 4 L1 -algebra (Strong homotopy Lie algebra). Let L be a graded vector space and suppose that a collection of degree 1 graded symmetric linear maps l := {lk : Lk ! L}k1 is given. (L, l) is called an L1 -algebra iff the maps satisfy the following relations: X ð1ÞðÞ l1þl ðlk ðcð1Þ ; . . . ; cðkÞ Þ;

each of which is graded symmetric on (Hc )l . We denote the collection also by n . We call (H, n , l) a (partial) open-closed homotopy algebra (OCHA) when n satisfies the following relations (up to some factorial coefficients):

2S

kþl¼n

½cðkþ1Þ ; . . . ; cðnÞ Þ ¼ 0

½10

for n  1. A weak L1 -algebra consists of a collection of degree 1 graded symmetric linear maps l:= {lk : Lk ! L}l0 satisfying the above relations, but for n  0 and with k, l  0. Remark 2 summation unshuffles, coefficients

The alternate definition in which the is over all permutations, rather than just requires the inclusion of appropriate involving factorials.

Just as an A1 -algebra can be described as a coderivation of T c A, similarly an L1 -algebra L can be described as a coderivation on Sc L, the symmetric subcoalgebra of T c A.

Definition 5 Let H = Ho  Hc be a graded vector space and (Hc , l) be a weak L1 -algebra. Consider a collection of multilinear maps n :¼ fnk;l : ðHo Þk  ðHc Þl ! Ho gk;l0

0¼

k X Xm X k;l0 p¼0 2S

nmþ1k;nl ðo1 ; . . . ; op ;

n

nk;l ðopþ1 ; . . . ; opþk ; cð1Þ ; . . . ; cðlÞ Þ; opþkþ1 ; . . . ; om ; cðlþ1Þ ; . . . ; cðnÞ Þ n XX þ nm;nþ1l ðo1 ; . . . ; om ; 2S

n

l¼1

ll ðcð1Þ ; . . . ; cðlÞ Þ; cðlþ1Þ ; . . . ; cðnÞ Þ

½11

Other Algebras of Interest

The Hochschild complex also has a graded product (without invoking the shift) known as the cup product. Except for the signs and the grading, the bracket and the product satisfy the Leibniz rule of a Poisson algebra on the cohomology; the result is

Operads 613

axiomatized as a ‘‘Gerstenhaber algebra.’’ However, on the cochain complex, the Lie bracket and the associative product are compatible only up to homotopy. This naturally raises the issue of an operad for strong homotopy Gerstenhaber algebras. The operad G for Gerstenhaber algebras is the homology of the little disks operad, H (D2 ). But now we have choices: in addition to relaxing the Leibniz rule up to homotopy, the bracket could be relaxed to be part of an L1 -algebra and/or the product could be relaxed to be part of an A1 -algebra. The choice which is now known as the G1 -operad is defined in terms of a procedure which works for what are known as quadratic operads, indicating they have generators in O(2) and relations in O(3): the corresponding O1 has ‘‘dual’’ relations. For example, this gives the classical Koszul duality between Lie and commutative associative algebras. The G1 operad can also be described as the ‘‘minimal model’’ of G in the sense of Markl. Another alternative is to consider just the ‘‘brace’’ operations, originally introduced by Kadeishvili and later independently by Getzler, but described in the Hochschild complex setting by Gerstenhaber– Voronov. Together with the cup product, these determine an operad denoted HG which acts on the Hochschild complex; there is an operad map from G1 to HG, hence G1 also acts on the Hochschild complex. Finally, Tamarkin showed that G1 is quasi-isomorphic to the dg operad of singular chains on the little disks operad, thus providing one of several proofs of what had been a conjecture by Deligne. Algebras with invariant inner products < ,  > are of considerable importance in mathematics and especially in mathematical physics; invariance means < a, bc > = < ab, c > or < a, [b, c] > = < [a, b], c > in, respectively, the associative or the Lie case (with appropriate signs in the graded case). Using the inner product, n-ary operations An ! A can be converted to operations Anþ1 ! C of which we can require cyclic symmetry. To handle such algebras, there is a notion of ‘‘cyclic operad.’’ In terms of trees, the transition is to take a rooted tree and then regard the root edge as just another leaf. This point of view corresponds to an essential symmetry for particle interactions.

These operadic structures were directly related to the moduli spaces of Riemann surfaces with punctures or boundaries (or other decorations) in these physical theories. Two special ‘‘higher-homotopy algebras’’ have been emphasized because they are particularly important in mathematical physics: A1 for openstring field theory and L1 for closed-string field theory and for deformation quantization. Open– closed string field theory combines A1 -algebra and L1 -algebra in a particular way known as an OCHA. The operad for L1 -algebras is given a very nice and physically relevant geometric interpretation in terms of a real compactification of the moduli space of Riemann spheres with punctures, while for OCHAs, there is a real compactification of the moduli space of Riemann disks with punctures on the boundary or in the interior (bulk). Thus, this operad can be regarded as obtained from a moduli space of configurations of points (punctures) in the disk by compactifying the moduli spaces by adding boundary strata where two (or more) points (punctures) collide. Points on the boundary strata can be visualized as ‘‘bubble trees’’ of disks and/or spheres, see Figure 6. Alternatively, the little disks operad can be regarded as being obtained by ‘‘decorating’’ the points with little disks, while for OCHAs there is also a basic half-disk decorated with little disks in the bulk and little half-disks for the boundary points. The corresponding colored operad is Voronov’s ‘‘Swiss-cheese operad.’’ ‘‘Colored’’ refers to the fact that disks can be inserted into half-disks but not vice versa. Compare trees with two ‘‘colors’’ of edges with grafts restricted to ones which match colors. On-Shell versus Off-Shell

In cohomological physics, the ‘‘on-shell’’ states or observables are usually given by the cohomology with respect to an internal differential, which in physics is called the BRST differential or BRST operator, though originally this meant the Chevalley–Eilenberg differential associated to the action of the Lie algebra of

Operads in Mathematical Physics One reason for the explosive development of operad theory in the 1990s was the introduction of operadic structures in field theories, for example, conformal field theories (CFTs) and string field theories (SFTs).

Figure 6 Bubble tree for circle configurations.

614 Operads

gauge symmetries of a physical theory. The generators of the Chevalley–Eilenberg cochain complex are known as ‘‘ghosts’’. On-shell subspaces of algebras which are not closed under the product of the larger ‘‘off-shell’’ algebra are called ‘‘open’’ algebras by physicists. Quite generally, this situation gives rise to an algebra over an appropriate operad. A special case involves a differential graded algebra A and a linear imbedding H(A)  A. The (co)homology is in turn a graded algebra (with 0 as differential), but inherits a higher-homotopy structure so that cohomology and original algebra are equivalent. In the associative case, the inheritance is a result of Kadeishvili: Let (A, d) be a differential graded associative or A1 -algebra, then the homology H(A) inherits the structure of an A1 -algebra. Even if the original algebra A is strictly associative, the inherited A1 -structure generally has nontrivial operations mi . Analogous results hold for L1 -algebras and others. It is the L1 -version that is relevant for closed-string field theory (CSFT). Zwiebach showed the quantum theory of covariant closed strings has an action defined in terms of an infinite chain of string field products. The genus-0 (tree level) string field algebra is an L1 -algebra inherited from the offshell state space modeled by the Batalin–Vilkovisky (BV) construction. The higher-order brackets provide higher-order correlation or n-point functions which play a crucial role in the extended Lagrangian of the theory. Batalin–Fradkin–Vilkovisky and Batalin–Vilkovisky Constructions

The constructions of Batalin–Fradkin–Vilkovisky (BFV) for constrained Hamiltonian systems and of Batalin–Vilkovisky (BV) for Lagrangians with symmetries are important examples of L1 -structures derived from ‘‘open’’ algebra settings, though the L1 -structures were recognized quite a while after the constructions. The BFV setting is that of a symplectic manifold W with a family of constraints, that is, a family of functions  2 C1 (W). The constraints are called ‘‘first class’’ if the ideal they generate is closed under the Poisson bracket. The vector space spanned by the constraints will in general be an open algebra; the structure of the bracket is given by structure functions, rather than structure constants. The zero locus of all the constraints forms the constraint surface V. In the first-class case, the constraints are in involution and determine a foliation F of V. If the space of leaves V=F is a manifold, it would be

considered the true physical space and the physical observables would be functions in C1 (V=F ). BFV construct a differential graded Poisson algebra such that the cohomology in degree 0 agrees with C1 (V=F ) when that makes sense and, in the regular case, the rest of the cohomology is that of the differential forms along the leaves of the foliation. The BFV differential is a deformation of the Chevalley–Eilenberg/BRST differential and can be constructed most efficiently by the same techniques used in proving Kadeishivili’s inheritance theorem. Crucially, it is an inner derivation with respect to the Poisson bracket. After the fact, an L1 -structure can be observed in the extended algebra. For a Lagrangian with symmetries, BV develop a similar construction, the main difference being that there is no Poisson bracket initially, but one is constructed by adjoining ‘‘anti-fields’’ as conjugate to the fields but of ghost degree 1 and the differential of an anti-field being the Euler–Lagrange expression for the corresponding field. Then, as in the Hamiltonian case, ghosts and anti-ghosts, etc. are adjoined and the construction proceeds in a parallel fashion. Deformation Quantization

Once algebras over an operad P are considered, it is natural to consider also morphisms of such algebras over a fixed P . From a homotopy point of view, the appropriate maps need not respect the operad structure strictly but only up to higher homotopy; indeed, there is a related operad to define such maps. For L1 algebras, such L1 -maps play a key role in deformation quantization. That refers to deformation of the commutative multiplication of a Poisson algebra in the direction of the Poisson bracket; that is, to first order, the deformation is given by the bracket. More generally, for any associative algebra A with multiplication m, one considers formal deformations a ? b ¼ mða; bÞ þ tm1 ða; bÞ þ t 2 m2 ða; bÞ þ ½12

where each mi 2 Hom(A  A, A). The associativity of ? provides a sequence of constraints on the mi . In particular, m1 must be a Hochschild cocycle and the obstruction to the existence of m2 is a class in the Hochschild cohomology of degree 3. In fact, the primary obstruction is represented by [m1 , m1 ]. If it is cohomologous to zero, that fact identifies candidates for m2 , that is, ½m1 ; m1 ¼ 2½m; m2

½13

or, using the notation d = [m,], dm2  1=2½m1 ; m1 ¼ 0

½14

Operads 615

once known as the integrability equation but now, more frequently, as a Maurer–Cartan equation. For a Poisson algebra, the Poisson bracket is a Hochschild cocycle but in general a full deformation need not exist. However, for the algebra A of smooth functions on a Poisson (e.g., symplectic) manifold M, Kontsevich showed that such a full formal deformation does exist. The guiding philosophy is that deformations are controlled by a dg Lie or L1 -algebra L, unique up to L1 -homotopy equivalence. Therefore, the obstructions can be computed in any of the equivalent dg Lie algebras. Moreover, the structure of the obstructions is known sufficiently so that if there is an equivalent dg Lie algebra with d in fact zero, then all the obstructions to deformation quantization vanish. The key to Kontsevich’s proof was the construction of an L1 -map, inducing an isomorphism in cohomology, from the Lie algebra of polyvector fields on Rd with the Schouten bracket and d = 0 to the Lie algebra of multidifferential operators on A = C1 (Rd ) regarded as a subalgebra of the Hochschild cochain complex for A with the Gerstenhaber bracket. BV Algebras

In addition to their construction of a differential graded Gerstenhaber algebra (a differential graded commutative algebra with a compatible Poisson bracket of degree 1), BV introduced a new mathematical structure, adding a second-order differential operator  relating the commutative product and the bracket. The operator  is a derivation of the bracket and of square zero. Moreover, ½a; b ¼ ðabÞ  ðaÞb aðbÞ

½15

so that the failure of  to be a derivation of the product is given by the bracket. The definition of a BV algebra is then a Gerstenhaber algebra with such an operator, though alternative definitions exist in which  and the product are primary and the bracket is defined by the above equation. From the operadic/higher-homotopy point of view, one can then go on to consider BV1 algebras. Recall that A1 -algebras and L1 -algebras (among others) can be characterized by an ‘‘inner’’ coderivation d = [m, ] of square zero on an appropriate ‘‘standard’’ construction. In the context of BV algebras, where the bracket is more commonly written as {,}, the classical action is an element S0 such that {S0 , S0 } = 0 or, equivalently, d = {S0 , } is of square zero. The quantum analog S is a perturbation of S0 and satisfies instead fS; Sg ¼ S

½16

This was originally called the ‘‘master equation,’’ but now is increasingly referred to as a ‘‘Maurer– Cartan’’ equation. Insertion Operads

There is another class of operads illustrated by trees (and more generally graphs) with a very different sort of ‘‘composition,’’ namely insertion of one graph into another. The most directly relevant to physics is the kind of insertion used by Connes and Kreimer in their Hopf algebra constructed for renormalization of Feynman diagrams. For example, consider all finite graphs with exactly two external edges and internal numbered edges. Given two graphs 1 , 2 , define 1 i 2 by cutting edge i of 1 and identifying the dangling edges with the two external edges of 2 . For planar trees, yet another insertion operad is obtained by Chapoton, isolating a part of a structure due to Kontsevich, in which a small neighborhood of a vertex of the second planar tree is removed and the dangling edges are attached to a vertex of the first tree by entering through the angles between the edges at that vertex (Figure 7). Inside the HG-operad is the operad Brace for an abstract brace algebra (forgetting the cup product), first described as such by Chapoton using the insertion operations of Kontsevich and Soibelman. A1 -Categories

Also of importance for applications to mathematical physics is the notion of an A1 -category, first made explicit by Fukaya and now playing a major role in string D-brane theory and homological mirror symmetry. The D-branes are the objects of the A1 category and the open strings with boundaries on two (possibly equal) D-branes B1 , B2 are the morphisms from B1 to B2 . The operations mi are defined only on tuples (a1 , . . . , ai ) of ‘‘composable’’ morphisms (e.g., strings). PROPs

While an operad is an abstraction of a family of composable functions of n variables for various n, a PROP is an abstraction of a family of functions in

Figure 7 Angles determined by edges with leaves extended to the semicircle.

616 Operator Product Expansion in Quantum Field Theory

Hom(Ap , Aq ) for all p and q. Now the relevant images are graphs with p input legs and q output legs with composition being defined by grafting output legs of one graph to inputs of another. Feynman diagrams are the obvious example in physics or, in conformal field theory, tubular neighborhoods of such graphs, which is to say, Riemann surfaces with boundary circles: p as inputs and q as outputs. See also: Algebraic Approach to Quantum Field Theory; Batalin–Vilkovisky Quantization; Constrained Systems; Deformations of the Poisson Bracket on a Symplectic Manifold; Deformation Quantization; Deformation Theory; Hopf Algebra Structure of Renormalizable Quantum Field Theory; String Field Theory.

Further Reading Batalin IA and Fradkin ES (1983) A generalized canonical formalism and quantization of reducible gauge theories. Physics Letters B 122: 157–164. Batalin IA and Vilkovisky GS (1981) Gauge algebra and quantization. Physics Letters B 102: 27–31. Boardman JM and Vogt RM (1973) Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Mathematics, vol. 347. Berlin–New York: Springer. Connes A and Kreimer D (2000) Renormalization in quantum field theory and the Riemann–Hilbert problem, I: the Hopf algebra structure of graphs and the main theorem. Communications in Mathematical Physics 210: 249–273, (hep-th/ 9912092). Fradkin ES and Vilkovisky GS (1975) Quantization of relativistic systems with constraints. Physics Letters B 55: 224–226. Fukaya K (1997) Floer homology, A1 -categories and topological field theory. In: Andersen J, Dupont J, Pertersen H, and Swan A (eds.) Geometry and Physics, Lecture Notes in Pure and

Applied Mathematics, Notes by Seidel P, vol. 184, pp. 9–32. New York: Marcel Dekker, Inc. Gerstenhaber M (1963) The cohomology structure of an associative ring. Annals of Mathematics 78: 267–288. Getzler E (1995) Operads and moduli spaces of genus 0 Riemann surfaces. In: Dijkgraaf R, Faber C, and van der Geer (eds.) The Moduli Space of Curves, Progr. Math., vol. 129, pp. 199–230. Boston: Birkha¨user. Getzler E and Jones JDS (1993) n-algebras and Batalin– Vilkovisky algebras. Preprint. Getzler E and Jones JDS (1994) Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint, Department of Mathematics, MIT; Department of Mathematics Northwestern University, March 1994, hep-th/9403055. Getzler E and Kapranov M (1994) Modular operads. Compositio Mathematica 110: 65–126 (dg-ga/9408003). Hinich V and Schechtman V (1993) Homotopy Lie algebras. Advanced Studies in Soviet Mathematics 16: 1–18. Lada T and Markl M (1995) Strongly homotopy Lie algebras. Communications in Algebra 2147–2161 (hep-th/9406095). Lada T and Stasheff JD (1993) Introduction to sh Lie algebras for physicists. International Journal of Theoretical Physics 32: 1087–1103. Markl M, Shnider S, and Stasheff J (2002) Operads in Algebra, Topology and Physics, Mathematical Surveys and Monographs, vol. 96, MR 2003f: 18011. Providence, RI: American Mathematical Society. May JP (1972) The Geometry of Iterated Loop Spaces, Lecture Notes in Mathematics, vol. 271. Springer. Stasheff J (1963) Homotopy associativity of H-spaces, I, II. Transactions of the American Mathematical Society 108: 293–312, 313–327. Voronov AA (1999) The Swiss Cheese Operad, Contemp. Math. vol. 239, pp. 365–373. Providence, RI: American Mathematical Society. Zwiebach B (1993) Closed string field theory: quantum action and the Batalin–Vilkovisky master equation. Nuclear Physics B 390: 33–152. Zwiebach B (1998) Oriented open–closed string theory revisited. Annals of Physics 267: 193–248 (hep-th/9705241).

Operator Product Expansion in Quantum Field Theory H Osborn, University of Cambridge, Cambridge, UK ª 2006 Elsevier Ltd. All rights reserved.

Introduction The operator product expansion (OPE) provides an algebraic structure in quantum field theory. In a sense it supercedes or rather transcends the equaltime commutation relations, which provide the traditional starting point for the canonical quantization of any quantum field theory. The essential idea is that for any two local operator quantum fields at spacetime points x1 , x2 their product may be expressed in terms of a series of other local quantum fields at a point x, which may be identified with x1

or x2 , times c-number coefficient functions which depend on x1  x2 . The set of operators which may appear depends on the particular quantum field theory and must of course be in accord with any requirements of conserved quantum numbers. The coefficient functions depend on x1  x2 in a fashion which depends on the dimensions of the various operators involved, at least up to renormalization group corrections. The most singular contributions are those for the operators appearing in the OPE with lowest scale dimension. From a phenomenological point of view, only the first few terms in the OPE are of relevance. However, theoretically, especially for conformal field theories, it is desirable to know the full expansion to all orders in powers of x1  x2 in such a way that the operator product may

Operator Product Expansion in Quantum Field Theory

be replaced by the full expansion in appropriate correlation functions. We first discuss the OPE for free theories and then the interacting case.

Free Field Theory The OPE is most straightforward in free field theory when it almost reduces to a Taylor series expansion. For a simple free massless scalar field (x) then in four dimensions we may write ðxÞð0Þ ¼

C þ : ðxÞð0Þ: x2

½1

where : : denotes normal ordering (moving all annihilation operators to the right of creation operators) and C is just a normalization numerical constant (for canonical normalization C = 1=42 ). The 1=x2 term proportional to the identity operator reflects the leading singular behavior at short distances of (x)(0), the power being determined by  having dimension 1. For the normal-ordered term we may expand in terms of an infinite set of local operators by using the Taylor expansion :ðxÞð0Þ: ¼

1 X 1

n! n¼0

x1    xn :@1    @n ð0Þð0Þ:

½2

where the operator appearing in the nth term has dimension n þ 2. Manifestly at short distances only the leading terms are relevant. Equation [1] also provides a point splitting definition of the local composite operator :2 (0): in terms of limit of (x)(0) as x ! 0 after subtraction of the singular C=x2 term. The OPE can be easily generalized to composite operators defined by normal ordering. For :2 : we have, by applying Wick’s theorem, : 2 ðxÞ:: 2 ð0Þ: ¼

2C2 4C þ 2 : ðxÞð0Þ: x x4 2 þ :  ðxÞ2 ð0Þ:

½3

where Taylor series expansion may be applied to both :(x)(0): and also :2 (x)2 (0): to give an infinite sequence of local operators of increasing dimensions. The expansion in terms of local operators may be reordered. For instance, from [1] we may write, using @ 2  = 0, ðxÞð0Þ ¼

C x2   1 2 2 þ 1 þ 12 x @ þ 14 x x @ @ þ 16 x @ : 2 ð0Þ:  12 x x T þ Oðx3 Þ

½4

where T ¼ : @ @  : 14  : @  @ :

½5

617

is the energy–momentum tensor. In [4], and also in a similar context subsequently, we define @ :2 (0): = @y :2 (y): jy = 0 . The expansion [4] provides a point splitting definition of T and also demonstrates that many operators appearing in the OPE are expressible in terms of overall derivatives of lowerdimension operators. We may also note that without further input there is an ambiguity in the definition of T of the form T  T þ að@ @  14  @ 2 Þ : 2 :

½6

In a conformal theory, however, we require a = 1=6.

Interacting Theories The OPE becomes an essential tool in the context of interacting quantum field theories. For renormalizable quantum field theories various results can be proved to all orders in the standard perturbative expansion and are naturally assumed to be properties of the complete theory. In interacting theories we may no longer use normal ordering to define composite operators which, in general, have anomalous dimensions. The coefficient functions appearing in the OPE also gain perturbative corrections but these are constrained by renormalization group (RG) Callan–Symanzik equations. Again if we consider the simplest case of a massless scalar theory as above but now with a renormalized coupling constant g the leading terms in the expansion of (x)(0) are of the form (here we assume a Z2 symmetry under  ! , otherwise the operator  would be expected to appear in the OPE) ðxÞð0Þ ¼

Cðg; 2 x2 Þ þ Dðg; 2 x2 Þ2 ð0Þ þ    x2

½7

where  is an arbitrary renormalization scale. This arbitrariness is reflected in the RG equation   @ @ þ ðgÞ þ 2 ðgÞ Cðg; 2 x2 Þ ¼ 0 ½8  @ @g At a fixed point (g ) = 0 this equation may be solved with an arbitrary choice of normalization to give C(g , 2 x2 ) = (2 x2 ) (g ) , which corresponds to the fields  having a modified scale dimension 1 þ  (g ). In a similar fashion the coefficient D(g, 2 x2 ) in [7] satisfies   @ @ þ ðgÞ þ 2 ðgÞ  2 ðgÞ  @ @g  Dðg; 2 x2 Þ ¼ 0

½9

where it is necessary to introduce a new anomalous dimension function 2 (g) related to the composite

618 Operator Product Expansion in Quantum Field Theory

operator 2 . Although it is natural to label the operator as 2 its definition in terms of the elementary field  is essentially only as given in terms of the OPE [9]. At a fixed point again D(g , 2 x2 ) = k(2 x2 ) (g )þ(1=2)2 (g ) , where the coefficient k is determined by the scale of the three-point function h(x)(y)2 (0)i. In asymptotically free theories the RG equations show that at short distances the coefficient functions tend to those of free field theory but with calculable logarithmic corrections. More generally, for a set of operators {Oi } the OPE has the form 1 X Cijk ðg; 2 x2 ÞOk ð0Þ ½10 Oi ðxÞOj ð0Þ  2 p ðx Þ k where p is determined by the free scale dimensions of the Oi and   @ @ þ ðgÞ  Cijk ðg; 2 x2 Þ @ @g X kn ðgÞCijn ðg; 2 x2 Þ  in ðgÞCnjk ðg; 2 x2 Þ ¼ n

 jn ðgÞCink ðg; 2 x2 Þ



½11

with in (g) the anomalous dimension matrix arising from the mixing of composite operators. An important aspect of the OPE is that the coefficient functions may be calculated perturbatively, essentially by applying the OPE in some suitable correlation function. Essentially the OPE provides a factorization between short-distance UV singularities and nonperturbative effects. In a Feynman graph the short distances in an operator product correspond to the large-momentum behavior and power-counting theorems allow a factorization up to calculable logarithmic corrections. A detailed analysis depends on the detailed technicalities of the proofs of renormalization to all orders of perturbation theory. The coefficient functions in the OPE should be independent of any infrared or nonperturbative longdistance effects (such as confinement in QCD). However, the operators which appear in the OPE, such as 2 above, may have nonzero expectation values which are absent to all orders in perturbation theory.

With a mass term the operator 2 mixes with the identity operator so that ðD þ 2 ðgÞÞh2 ð0Þi ¼ 2 I ðgÞm2 @ @ @ þ ðgÞ þ 2 ðgÞm2 D¼ @ @g @m2

where 2 I reflects the mixing. At one loop order we have ðgÞ ¼

3g2 ; 162

2 ðgÞ ¼

g ; 162

2 I ðgÞ ¼

1 82

½13

and we may also set  (g) = 0. In this case in the operator product expansion (7) the coefficient C also depends on m2 x2 and the RG equations [8] and [9] are now modified to include the effects of mixing DCðg; m2 x2 ; 2 x2 Þ ¼ m2 x2 2 I ðgÞDðg; 2 x2 Þ   D  2 ðgÞ Dðg; 2 x2 Þ ¼ 0

½14

From lowest order perturbation theory with [13], and using [14] to include all orders in g ln 2 x2 , we have in this approximation Cðg; m2 x2 ; 2 x2 Þ ¼

 2=3 1 2m2 x2 3g 2 2 1 þ  ln  x 42 322 g  1=3 ! 3g 2 2  1þ ln  x 1 322  1=3 3g 2 2 Dðg; 2 x2 Þ ¼ 1 þ ln  x 322

½15

The operator product expansion then reproduces the small x behavior of the two point function h(x)(0)i at one loop, expanding C, D to first order in g, if we take h2 ð0Þi ¼ 

m2  ln þ OðgÞ 82 m

½16

which is in accord with [12]. If m2 < 0 the symmetry  $  is broken and it is necessary to shift the field  =  þ f , with  2 = 6m2 =g and the field f has a mass mf with m2f = 2m2 . The operator product expansion [7] with the same coefficient functions as in [15] remains valid. The two point function h(x)(0)i, which includes a nonperturbative term  2 , is again reproduced for small x at one loop now if

Perturbative Example The general considerations can be illustrated by considering a scalar field theory to lowest order in a perturbative expansion. We consider a four dimensional theory with a single scalar field and a 1 potential V() = 12 m2 2 þ 24 g4 . Using dimensional 2 regularization m , as well as g, is treated as a coupling with an associated -function 2 (g)m2 .

½12

h2 ð0Þi ¼ 

6m2 m2   2 ln þ OðgÞ mf g 2

½17

but in this case it is necessary to expand D(g, 2 x2 ) to O(g2 ) as a consequence of the leading 1=g term in [17]. Note that both [16] and [17] contain the nonperturbative dependance on ln m and ln mf which is present in the two point function.

Operator Product Expansion in Quantum Field Theory

Conformal Field Theories When the -function vanishes and a quantum field theory enjoys conformal invariance the operator product expansion is a potentially convergent expansion. It is natural to restrict to conformal quasiprimary operators which do not mix with lower scale dimensions under conformal transformations. If we consider, for instance, two scalar operators  with scale dimension  then the OPE has the generic form X 1 1 COI ðxÞð0Þ ¼ 2 þ 1=2 x ðx2 Þ ð2  I þ ‘Þ I ð‘Þ

 CI ðx; @Þ1 ;...;‘ OI1 ;...;‘ ð0Þ

T ðxÞOð0Þ  A ðxÞOð0Þ þ B ðxÞ@ Oð0Þ þ   

If the theory has a symmetry with corresponding conserved currents then there are Ward identities which constrain the OPE of fields with the conserved current. For a current Ja then we have, in d dimensions, the singular contribution in the OPE is given by x 1 ta Oð0Þ Sd ðx2 Þð1=2Þd

½19

where ta are a set of matrix generators corresponding to the symmetry acting on the fields O and Sd is the volume of the unit (d  1)-dimensional sphere, S4 = 22 . For a conserved current there are no anomalous dimensions and the coefficient in [19], which depends on the normalization for the current Ja , is chosen so that [Qa , O(0)] = ta O(0) with Qa the charge formed from Ja . For the energy– momentum tensor the operator there is an analogous result. We consider the simpler case of a

½20

where A (x) = O(xd ) and B (x) = O(xdþ1 ). As a distribution A (x) is ambiguous up to terms proportional to d (x). If  is the scale dimension of O and s are the Lorentz spin generators acting on O the Ward identities then give     þ C þ 12 s @ d ðxÞ @  A ðxÞ ¼ d A ðxÞ ¼ C d ðxÞ @  B ðxÞ ¼   d ðxÞ

½21

where C is a constant tensor reflecting the arbitrariness in A , it is immaterial as far as Ward identities are concerned. We may choose   þ C ¼ 0 d

½22

(If desired, we might also take A0 (x) = A (x) þ (1=2)s d (x) in which case @ A0 (x) = 0, A0[] (x) = (1=2)s d (x) but such an antisymmetric piece seems unnatural). In general there is no unique form for A (x), as a consequence of the freedom of choice for C in [21]. However, for a scalar field O we must have, for x 6¼ 0, A ðxÞ ¼

x x   1 1   d 2 2 x d  1 Sd ðx Þð1=2Þd

¼

Ward Identities

Ja ðxÞOð0Þ  

conformal theory when the energy–momentum tensor is both conserved and traceless and

½18

where there is a sum over quasiprimary operators OI1 ,..., ‘ with scale dimension I and spin ‘, so they are symmetric traceless tensors of rank ‘. In the first term in [18] the coefficient is chosen to be 1 by a choice of normalization. The coefficients COI , with a standard normalization for OI , are then determined by the coefficients of the corresponding three-point functions involving  and OI . In [18] C(‘) are differential operators which sum up the I contributions of all derivatives or descendants of the quasiprimary operator OI . They can be explicitly given in terms of an integral representation, for any spacetime dimension, where the scale is fixed by requiring for the leading term C(‘) (x, 0)1 ,..., ‘ = I x1    x‘ – traces. The spectrum of operators which appear is obviously a property of the particular conformal field theory.

619

 1 1 @ @ ðd  1Þðd  2Þ Sd ðx2 Þð1=2Þd1

½23

with the overall scale determined by [21].

For the operator product of the current Ja with itself there is an additional term proportional to the identity operator of the form  x x  1 ½24 Ja ðxÞJb ð0Þ  CJ ab   2 2 x x2ðd1Þ where the coefficient CJ , which determines the scale of the two-point function for Ja , is well defined since the normalization of the current is determined through the Ward identity. A similar result also holds for the operator product of the energy– momentum tensor with itself, with an overall coefficient CT . In general, we may also write for the operator product of two scalar fields O: OðxÞOð0Þ  CO

1 CO d 1  2 ð1=2Þdþ1 x CT Sd d  1 ðx2 Þ  x x T ð0Þ ½25

620 Optical Caustics

neglecting other contributions. The contribution of the energy–momentum tensor does not therefore introduce any new coefficient.

Two Dimensions In two dimensions the OPE plays an essential role in the discussion of conformal field theories. For a Euclidean metric it is natural to use complex variables z and z. The energy–momentum tensor in this case reduces to a chiral field T(z) and its  z). For the operator product with a conjugate T( chiral field (z) with scale dimension , TðzÞð0Þ 

 1 ð0Þ þ 0 ð0Þ z2 z

½26

and, for the operator product of T with itself, TðzÞTð0Þ 

c 2 1 þ Tð0Þ þ T 0 ð0Þ 2z4 z2 z

½27

Here c is the Virasoro central charge, which plays a critical role in the discussion of two-dimensional conformal field theories, it is given by the two-point function which follows from [27], hT(z)T(0)i = (1=2)cz4 . In simple rational conformal field theories the operators are organized into conformal blocks by the infinite-dimensional extended conformal symmetry in two dimensions. This allows the full spectrum of operators and their dimensions to be determined and in consequence complete results for the OPE to be found in many cases.

Further Remarks The OPE reflects the locality properties of quantum field theories and can be extended without difficulty to curved space backgrounds. For a product (x)(0), the separation x2 may be replaced by a biscalar at x and 0 but it is necessary to include in the OPE contributions involving the background

Riemann tensor as well as the operator fields present in flat space. There is also a generalization of the OPE for superfields on superspace. At a fundamental level although the OPE can be derived to all orders in perturbation theory the contribution of nonperturbative effects such as instantons to the coefficients is not entirely clear. Issues of associativity have yet to be fully analyzed. There are also important applications to the phemenonological analysis of QCD when assumptions about the OPE and saturation of sum rules can lead to results for the vacuum expectation value of gauge-invariant operators such as F F . See also: Boundary Conformal Field Theory; Effective Field Theories; Quantum Chromodynamics; Renormalization: General Theory; Renormalization: Statistical Mechanics and Condensed Matter; Two-Dimensional Models.

Further Reading Cardy J (1987) Anisotropic corrections to correlation functions in finite size systems. Nuclear Physics B 290: 355. Collins JC (1984) Renormalization: An Introduction to Renormalization, The Renormalization Group and the OperatorProduct Expansion. Cambridge: Cambridge University Press. David F (1984) On the ambiguity of composite operators, I.R. renormalons and the status of the operator product expansion. Nuclear Physics B 234: 237. Di Francesco P, Mathieu P, and Se´ne´chal D (1997) Conformal Field Theory. New York: Springer. Erdmenger J and Osborn H (1996) Conserved currents and the energy–momentum tensor in conformally invariant theories for general dimensions. Nuclear Physics B 483: 431. Kadanoff LP (1969) Operator algebra and the determination of critical indices. Physical Review Letters 23: 1430. Novikov VA, Shifman MA, Voinshtein AJ, and Zakharov VI (1985) Wilson’s Operator product expansion: can it fail? Nuclear Physics B249: 445. Wilson KG (1961) Non-Lagrangian models of current algebra. Physical Review 179: 1499. Wilson KG (1971) Renormalization group and strong interactions. Physical Review D 3: 1818.

Optical Caustics A Joets, Universite´ Paris-Sud, Orsay, France ª 2006 Elsevier Ltd. All rights reserved.

Introduction Optical caustics are the bright forms created by the focalization, natural or artificial, of light (Figure 1). Special caustic points, called focuses, are produced by stigmatic optical systems in order to visualize objects. However, there are no special conditions for

producing usual caustics. Every congruence of rays always generates a caustic, more or less intricate. Caustics have been observed and described since a long time, tracing back to antiquity. The name itself was coined after the Greek root ‘‘kausticos’’ meaning burning and expressing that a high energy density is produced by ray focalization at a caustic point. Conceptually, they appeared in the literature as ‘‘evolutes,’’ ‘‘envelopes,’’ ‘‘centers of curvature,’’ ‘‘focals,’’ etc. However, these different approaches, often too restricted, were unable to clarify the

Optical Caustics

621

Figure 1 Optical caustics may be produced by reflection (on window glasses) or by refraction (through the wavy surface of a swimming pool). Here the light source, the Sun, has some angular extension and the caustic appears somewhat blurred.

general properties of caustics, for instance, their classification in generic types. This difficult question was solved only recently in the framework of the singularity theory which appeared in the second half of the twentieth century (Whitney 1955, Thom 1956). Caustics are now understood as physical realizations of Lagrangian singularities, and they are often called optical singularities or optical catastrophes. The aim of this introductory article is to show in which sense caustics can be understood as singularities, and to present their main properties.

The Physical Phenomenon Caustics are usually observed by interposing a screen on the ray trajectories and their trace in the screen forms a set of bright curves called ‘‘fold’’ (A2 ). Across the fold, the number of rays passing through a given point jumps by 2. Two fold curves may join at some point forming there a tip called cusp (A3 ). A simple example is provided by the nephroid that one sees in a cup of coffee when the light is reflected off the cylindrical sides. In the threedimensional (3D) space, the folds form surfaces and the cusps form curves (Figure 2). For particular

positions of the screen, three other types of caustics may be observed: the swallowtail (A4 ), the meeting point of two cusp lines; the elliptic umbilic (D 4 ), the meeting point of three cusp lines; and the hyperbolic umbilic (Dþ 4 ) where a cusp line tangentially meets a fold surface (Figure 2). These five caustic types are generic in the sense that any other type of caustic point is unstable and decomposes into these generic caustic points under small perturbations. The perfect focus is an example of a nongeneric caustic point, obtained by imposing a special symmetry. The natural focusing of light, as in gravitational optics, produces only generic caustics. A caustic point is then a generalized focus. The caustic surface is a complex surface in the 3D physical space, generally self-intersecting and possessing singular lines A3 þ ending at singular points A4 , D 4 , or D4 . At the scale of the wavelength of the light, the caustics have a more complex structure. Instead of well-defined surfaces, lines and points, one observes a system of interference fringes concentrated in the vicinity of the geometrical caustic. Each type of caustic point has its own diffraction pattern (also called diffraction catastrophe) (Figure 3). These interference systems are easily produced, for instance, by focusing a coherent laser beam by a

Fold A2

Cusp A3

Swallow tail A4

Figure 2 The five generic types of caustics of the 3D space.

_

Elliptic umbilic D4

Hyperbolic umbilic D4+

622 Optical Caustics

A2

A3

+

D4

A4

–

D4

Figure 3 Interference fringes produced by the five generic caustics of the 3D space (numerical simulation).

corrugated glass or by a water droplet. An important feature is revealed by Gouy’s experiment, in which bright and dark fringes are inverted when the rays are forced to pass through a focus (Guillemin and Sternberg 1977). The experiment shows that the wave undergoes a phase shift of =2 when the associated ray passes through a caustic point. So, caustics are fundamental objects of both the geometrical optics and the wave optics.

Modeling Caustics Because of the presence of a caustic, a congruence of rays generally presents intersecting rays. At the points of intersection, the coordinates q1 , q2 , q3 of the physical space R3 are unable to distinguish the various intersecting rays and they do not constitute a convenient system of coordinates. It is then interesting to construct an abstract space in which the rays are represented by nonintersecting curves. The initial congruence is recovered by projecting the abstract space into the physical one. All the models use this type of construction in which the properties of the caustics are deduced from those of the projection. Caustics as Envelopes of Rays

In this geometrical modeling, each ray is labeled by two parameters r1 , r2 , for instance, the coordinates on the initial wave front W. A third coordinate r3 specifies the points along the ray, for instance, by assigning their distance to W. Taken together, these three coordinates represent the congruence of rays, and define a 3D space, the source space M = {r1 , r2 , r3 }. By construction, the rays in M do not intersect. The coordinates (q1 , q2 , q3 ) of the current point P 2 R3 along each ray depend

differentiably on the coordinates (r1 , r2 , r3 ) and define a ‘‘projection’’ f : (r1 , r2 , r3 ) 7! (q1 , q2 , q3 ) from the source space M into the physical space R3 . The caustic points correspond to the envelope of the rays. At a caustic point P, the energy density flowing along the rays becomes infinite, since the small volume delimited by neighboring rays is shrunk into a small surface at P. This behavior may be simply expressed with the help of the projection f: the rank rk of the derivative Df is equal to 2 at the point representing P in M. This motivates the following definition. Given a map f : M ! N, a point x 2 M is said to be critical (or singular) if the rank of the derivative Df is less than the maximal possible value min(dim M, dim N). Here, dim M = dim N = 3, and a critical point is a point where rk < 3. The set   M of the critical points is called the singular set. The caustic C is the image of the singular set: C = f (). One also says that the caustic points are the critical values of f. In practice, the derivative Df is expressed by the Jacobian matrix J = @(q1 , q2 , q3 )=@(r1 , r2 , r3 ) and the singular set  is defined by solving the equation detðJÞ ¼ 0

½1

If this equation permits one to express explicitly one coordinate, say r3 , as a function of the other two, the caustic surface C is found in parametric form: q1 = q1 (r1 , r2 , r3 (r1 , r2 )), etc. For a homogeneous medium, equation [1] is of second degree in r3 and the caustic is composed of two sheets which meet at the umbilic points D4 . Equation [1] gives all caustic points independently of their nature, that is, it does not distinguish þ between A2 , A3 , A4 , D 4 , and D4 . A refinement allows one to recognize different types of caustic points. One defines the Thom–Boardman class i as the points in M where Df has a kernel of dimension i. Then one defines inductively the class i, ..., j, k as the class k of the restriction of f to i, ... , j . Thus, 0 represents the regular points (noncaustic points), 1,0 the fold points A2 , 1,1,0 the cusp points A3 , 1,1,1,0 the swallow-tail points A4 , and 2,0 the umbilics D4 (hyperbolic or elliptic). Altogether, the classes I , I 6¼ 0, form the singular set . The Thom–Boardman classes constitute a simple and powerful tool for computing the structure of a caustic. Each class is obtained by canceling some functional determinants associated with the map f or with its restriction to some class. However, the method presents the weakness of ignoring the special nature of a set of rays: its Lagrangian character. As a consequence, it is unable, for þ instance, to distinguish between D 4 and D4 .

Optical Caustics Caustics as Lagrangian Singularities

As for mechanics, the natural framework for geometrical optics is a phase space: the cotangent space T  R3 = {pi , qi } of the configuration space R3 = {qi }. The phase space is characterized by its symplectic P structure, that is, the differential 2-form ! = i dpi ^ dqi , which is nondegenerate and closed (d! = 0). A set of rays in the phase space is defined by specifying the wave vector (or momentum) p at each point q of the congruence. In the simple case where only one ray passes through each point, one R has p = rS, where S is the optical length n ds and n the refractive index. In other words, p is the differential of the optical length. The wave vector p is tangent to the ray and orthogonal to the (geometrical) wave front S = const. The eikonal equation shows that its modulus is n. As a direct consequence of the relation p = rS, the symplectic form annihilates identically for these p. However, in general, because of the presence of the caustics, one must not expect to have p = rS for some function S. Nevertheless, it is possible to keep the more general property to annihilate !. This motivates the definition of a Lagrangian submanifold: a submanifold L  T  R3 of dimension 3 (that is, half of the dimension of the phase space) on which the symplectic form vanishes: !jL = 0. Every congruence of rays is described by a Lagrangian submanifold. The Lagrangian submanifold plays the same role as the source space in the preceding section. The role of the projection f is played by the natural projection  from the phase space into the configuration space (p, q) = q, or more precisely to its restriction to L: f = jL . It is called a Lagrangian map (or Lagrangian projection) and it is again a map between two spaces of the same dimension (here 3). When L is given by an embedding  : L ! T  R3 , one has f =   . A caustic is then defined as the set of critical values of a Lagrangian map. There exist two remarkable results showing that a Lagrangian submanifold may be described in terms of functions or of families of functions. As a consequence, caustics are not directly related to the singularities of maps but, more particularly, to the singularities of functions. ",1,5,3,0,0pc,0pc,0pc,0pc>Generating function of a Lagrangian submanifold The 3D Lagrangian submanifold L  {pi , qi } is locally defined by three coordinates p ( 2 A) and q ( 2 B) depending on the three other ones p and q : p = p (q , p ), q = q (q , p ). One can show that this may be done in such a way that each

623

conjugate pair (qi , pi ) gives exactly one independent variable and one dependent variable. Formally: A [ B = {1, 2, 3}, A \ B = ;. R In fact, introducing the function S(q , p ) = hp, dqi  hq , p i(h,i denotes the scalar product), the local equation for L takes a more simple form: q ¼ 

@S ; @p

p ¼

@S @q

½2

The function S is well defined, since, R by the definition of a Lagrangian submanifold h p, dqi is locally path independent: it depends only on its end points. S is called a (local) generating function. Formula [2] generalizes p = rS, to which it reduces when B = ;, that is, for nonintersecting rays. ",1,5,3,0,0pc,0pc,0pc,0pc>Generating family and optical catastrophes Formula [2] may be rewritten in an interesting way. Taking the jBj variables p as internal parameters x and q = (q , q ) as external parameters, we construct a function F of x parametrized by q: F(x, q) = S(q , x) þ hq , x i. Now the Lagrangian submanifold L is defined by   @F @F ¼ 0; p ¼ L ¼ ðq; pÞ : 9x : @x @q F is called the generating family. The first equation @F=@x = 0 determines the rays passing through the fixed external parameter q 2 R3 . The second one distinguishes these rays according to their wave vector p. Each ray corresponds to a critical point (i.e., an extremum) of F considered as a function of x. At a caustic point, two infinitely close rays are converging and F then presents a degenerate critical point. So the generating-family technique links the caustics to the theory of singularities of functions depending on some parameters, that is, to the catastrophe theory (Thom 1969). Caustics are also called optical catastrophes. The generating families are not uniquely defined, even locally. In optics, one may always take for F the equivalent family ‘‘optical length’’ d, considered as a function defined on the initial wave front W (this is discussed in the following). Caustics as the Locus of Wave Front Singularities

There exists a remarkable duality linking rays and wave fronts. As a consequence, the caustic points (i.e., Lagrangian singularities) are related to singularities of wave fronts (i.e., Legendrian singularities). A typical wave front W may possess only two types of singularities: cuspidal curves and swallow-tail points. During the motion of W, governed by the eikonal equation, the cuspidal curves generate surfaces, and

624 Optical Caustics

swallow tails generate curves. These surfaces are exactly the fold surfaces of the caustic C and the curves are the cusp lines of C. The point singularities of the caustic, that is, the swallow tails and the umbilics, correspond to bifurcations of the instantaneous wave front, at certain moments of its motion. Caustics as Short Wave Asymptotic

The fine observation of the optical caustics shows that they never appear as the well-defined surfaces given by the geometrical optics, but rather as diffraction patterns concentrated around these surfaces. So wave optics is the natural framework to account for this fundamental feature. One exploits the fact that the wave number k = 2= ( : wavelength of the light) is a large parameter. This short-wave approximation permits the use of powerful expansion techniques and clarifies the relation with the geometrical optics viewpoint, formally obtained for k tending to infinity.

1 where 1 1 and 2 are the two principal radii of curvature at Q 2 W, and ] the number of caustic points (also called focal points) along the ray PQ. In the stationary-phase approach, the caustic C, locus of centers of curvature of W, appears as an obstacle in constructing asymptotics, since formula [4] diverges when di ! 1, that is, when P tends to C. It is, nevertheless, remarkable that C also appears explicitly when [4] is valid, via the i ’s and ]. In particular, the term e]i=2 , applied in the case of a focus (] = 2), accounts for the phase shift of  observed in Gouy’s experiment.

Asymptotics on caustics Uniform asymptotic formulas, valid also on the caustic, need a more complex theoretical framework, for instance, Maslov’s theory, presented here in a necessarily simplified version (see Maslov and Fedoriuk (1981) for more detail). The starting point is the equation of wave optics, that is, the Helmholtz equation ð þ k2 n2 ÞU ¼ 0

The stationary phase In the most simple model, the Huygens–Fresnel principle, the amplitude U(P) of the optical field may be evaluated by adding the secondary disturbances emitted from the points Q of some initial wave front W: Z Z ikd e UðPÞ ¼ c G ds ½3 W d where d is the distance QP. G is the inclination factor, a smooth function defined on W and c some prefactor. For simplicity, G and n (the refractive index) are assumed to be constant. Defining a = cG=d, formula [3] appears as an integral of the R form a(y)eik(y) dy. This type of integral may be evaluated for large k by the method of stationary phase. The principal contributions are due to points where the phase  is stationary: r = 0. For wave optics,  is the length PQ, considered as a function of Q and parametrized by P. The stationary condition means that PQ is normal to W, that is, it represents a ray of geometrical optics. The function PQ is a generating family in the sense of the discussion earlier. If no stationary points exist, that is, if P is in the shadow, the integral is O(kN ) for any N. Otherwise, and if the critical points are not degenerate, the phase stationary method gives (Guillemin and Sternberg 1977): 2 X ð1]Þi=2 UðPÞ ¼ e k rays PQ

aðQÞeikd 1=2

jð1  1 dÞð1  2 dÞj

þ Oðk2 Þ ½4

½5

where the refractive index n generally varies from point to point. For k ! 1, one looks for an asymptotic solution in the (tentatively) form: UðPÞ ¼ eikSðq1 ;q2 ;q3 Þ

1 X

ðikÞj ’j ðq1 ; q2 ; q3 Þ

½6

j¼0

Inserting this form in eqn [5] one obtains the eikonal equation (or characteristic equation) for the phase S: ðrSÞ2 ¼ n2 and an infinite series of equations for the amplitudes ’j , called the transport equations. One knows that the Cauchy problem for the eikonal equation may be reduced to the integration of the corresponding Cauchy problem for the Hamilton system (or bicharacteristic system): dq @H ¼ ¼ 2p; dt @p

dp @H ¼ ¼ rn2 dt @q

where H = h p, p i  n2 . Its solutions, the bicharacteristics q(t, ), p(t, ) are parametrized by the ‘‘time’’ t and the 2D parameter parametrizing the points on the initial wave front W. The bicharacteristics form a 3D Lagrangian submanifold L in the phase space {pi , qi } and one recovers the preceding situation. Assuming L to be simply connected, one defines aR global phase function S on L by formula S(t, ) = h p, dqi. In a domain j  L not containing the singular set and in which the coordinates t, are in a one-to-one correspondence with the physical coordinates, S

Optical Caustics

becomes a function of qi . Using the transport equation, one finds the leading term of the asymptotic solution (with accuracy to k1 ) in the following form: UðPÞ ¼ ðKðj Þ’ÞðqÞ sffiffiffiffiffiffiffiffiffiffi    d  ¼  eikSðq1 ;q2 ;q3 Þ ’ðq1 ; q2 ; q3 Þ dqi

½7

where d and dqi , respectively, represent the measures on the Lagrangian submanifold and on the physical space. The amplitude ’ depends on the initial conditions. Formula [7] defines a precanonical operator K(j ). It has the same form as [4], with the same drawback to diverge near the singular set , where dqi = 0. In a domain j containing the singular set, L is locally parametrized by mixed coordinates q , p . The basic idea is then, roughly speaking, to carry out a Fourier transform Fk with respect to these p (in fact, a variant of the usual Fourier transform, in which the parameter k appears in the prefactor and in the phase term). This leads one to consider, instead of L =  þ k2 n2 , the operator L^= Fk LFk1 , and instead of U, the unknown function V = Fk U. In this Fourier space, V may be found in the same way as U was found in the preceding case, with S replaced here by the local generating function Sj (q , p ) = S  hq , p i. Coming back to the real space by Fk1 , one obtains (with the same accuracy): UðPÞ ¼ ðKðj Þ’ÞðqÞ "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #  ffi   d   ikSj ðq ;p Þ ’ðq ; p Þ ½8 ¼ Fk1 dp dq e   There is no divergence in this local solution. So local short-wave asymptotics may be found everywhere, even on the caustic where they have a more complex form than the form [6] or [7]. Global asymptotics and Maslov’s index The global asymptotic solution is obtained by formally gluing the local solutions by a partition of unity ej = 1 subordinate to a covering {j } of L. However there is a difficulty. The representations of the same precanonical operator in different local coordinates q , p , even not containing the singular set, agree only up to a constant multiplier eim=2 , where the integer m is the number of negative eigenvalues of some matrix. One is led to multiply every precanonical operator by a convenient phase factor ei =2 , where 2 Z4 is called Maslov’s index. The coherency of the phase factor in different domains is realized by using the important property of  to be co-oriented. Thus, counts the number of passages

625

of an oriented path on L from the negative side of  to its positive side, minus the number of passages in the opposite sense. Maslov’s index is locally constant and jumps by 1 only across the singular set . The global canonical operator is now formally defined as K = j ei j =2 K(j )ej . Finally, the canonical operator K is well defined only if it is independent of the {j } and ej used for its definition. This possibility is expressed (in the case of a simply connected L) by the following property, intrinsically attached to L: the Maslov index cancels on every closed loop. So the only obstruction for global asymptotics is the nontriviality of the characteristic class defined by Maslov’s index and not the caustic. The central object of the caustic modeling is then the projection of the submanifold representing the rays (M or L) into the physical space. The possibility to reduce this projection to some normal form is the key result for the local classification of caustics.

Local Classification of Caustics Equivalence, Stability, and Genericity

In order to distinguish different types of singularities, one has to define an equivalence relation. Two Lagrangian maps fi : T  Mi Li ! Mi (i = 1, 2), are said to be Lagrange equivalent if there is a diffeomorphism h : T  M1 ! T  M2 preserving both the symplectic and the fiber structures, and sending L1 to L2 . In fact, only the local problem of classification makes sense, and one considers, instead of Lagrangian maps, germs of Lagrangian maps. A map germ is a map locally defined, that is, defined in an infinitely small neighborhood around a point (depending on the germ). The notion of Lagrange equivalence is extended to the germs. A Lagrangian singularity is then the Lagrange equivalence class of a germ at a critical point. Each equivalence class represents a type of Lagrangian singularity, that is, a type of caustic point. The example of the perfect focus point shows that there exist singularities which are totally unstable. In this sense, they correspond to idealized situations not physically realizable, and they have to be disregarded. Conversely, stable singularities resist under the action of small perturbations. They correspond to Lagrangian germs for which all neighboring germs are Lagrange equivalent (not necessarily at the same point, but near the point considered). Now the important question is: do the stable germs represent the generality? In the best case, stable germs form a dense open set. This means that every germ may be approximated by stable germs. In this case, one says that the stable germs are generic.

626 Optical Caustics

Stability and genericity are disctinct notions. It turns out that they coincide for low values of the dimension n of the ‘‘physical space’’ (n < 6), but they may disagree at higher dimensions. Classification of Stable Caustics

The fundamental result of the theory is the local classification of Lagrangian singularities (Arnol’d 1972). With the help of the generating families, the study of Lagrangian singularities is reduced to the study of singularities of families of functions. More precisely, at a singular point, every stable Lagragian map is equivalent to one of the following maps, given by their generating function S and by their generating family F: A2 :

S ¼ p31 F ¼ x3 þ q1 x

A3 :

S ¼ p41 þ q2 p21 F ¼ x4 þ q1 x2 þ q2 x

A4 :

S ¼ p51 þ q2 p31 þ q3 p21 F ¼ x5 þ q1 x3 þ q2 x2 þ q3 x

D 4 :

S ¼ p31  p1 p22 þ q3 p21 F¼

x21 x2



x32

þ

q1 x22

number of nondegenerate critical points of F, that is, the number of rays coinciding at the singularity. In the 3D space, one has  = c þ 1: (A2 ) = 2, (A3 ) = 3, (A4 ) = (D 4 ) = 4. Short-wave asymptotics near the caustic present remarkable scaling properties (Berry and Upstill 1980). In particular, the amplitude jU(P)j increases like k as k ! 1. The number depends only on the type of the singularity and it is called the singularity index. The more ‘‘degenerate’’ the singularities, the larger the index, and then the brighter the caustic point: (A2 ) = 1=6 < (A3 ) = 1=4 < (A4 ) = 3=10 <

(D 4 ) = 1=3.

Global Organization of Caustics The global properties of caustics are less understood than the local ones. There is, nevertheless, an interesting result concerning specifically the caustics in the 3D space (Chekanov 1986). Given a Lagrangian map f : L ! R3 , the Euler characteristic () of the singular set   L and the number ]D4 (1=2) of umbilics of index 1=2 are related by the formula ðÞ þ 2]D4 ð1=2Þ ¼ 0

þ q2 x2 þ q3 x1

These polynomial functions are called normal forms. The stable singularities are generic. In other words, every other type of singularity is destroyed by infinitely small perturbations and gives a set of singularities belonging to the list. The five generic caustics have been observed and experimentally studied in detail (Berry and Upstill 1980, Nye 1999). By inserting the normal forms S in a short-wave asymptotic, one obtains the diffraction patterns associated with the five caustic types (Figure 3). They generalize the Airy function which corresponds to the fold singularity. The normal forms describe at once the geometry of the caustics and the interference systems around them. Codimension, Corank, Multiplicity, and Index

Lagrangian singularities are also characterized by some numbers. They have a codimension c equal to the difference between the dimension of the physical space and their dimension: c(A2 ) = 1, c(A3 ) = 2, c(A4 ) = c(D 4 ) = 3. They have a corank ck, equal to the difference between the dimension of the space and the rank of the Lagrangian map: ck(A2 ) = ck(A3 ) = ck(A4 ) = 1, ck(D 4 ) = 2. The corank is the number of internal parameters of the generating family F. They also have a multiplicity , which is the

½9

At an umbilic point T,  is locally a cone with vertex at T. The index is defined according to the relative positions of the following elements: the 2D plane  = ker f , the cusp lines A3   passing through T, and the characteristic line l which represents the ray at T. If l and A3 are separated by , the index is equal to þ1=2, and to 1=2 in the other case. The index of an elliptic umbilic is always equal to 1=2. The validity of Chekanov’s formula [9] requires that L lies on a hypersurface E of the phase space, convex with respect to the wave vectors. The characteristics are the orthocomplements of E. In this framework, the singularities are called optical singularities, because such an E is always defined in geometrical optics by the eikonal equation. All Lagrangian singularities can be realized as optical singularities. Chekanov’s formula has been experimentally checked (Joets and Ribotta 1996). The Chekanov relation has an important consequence on the caustic bifurcations (also called metamorphoses or perestroikas), that is, the generic transformations modifying the topology of a caustic depending on one parameter. Among the 11 possible caustic bifurcations, considered as bifurcations of general Lagrangian singularities, four of them cannot be realized as bifurcations of optical Lagrangian singularities. So Chekanov’s relation reduces the number of optical metamorphoses to seven.

Optical Caustics

Extensions Caustics in Spaces of Higher Dimension

The local classification of Lagrangian singularities has been extended in spaces of higher dimension. For n = 4, in addition to the preceding ones, two new singularities appear: the butterfly A5 and the parabolic umbilic D5 . For n = 5, in addition to A6 and D 6 , one has a new type of umbilic: E6 . However, in higher dimensions, the classification becomes more complex. In addition to stable singularities, like those of the series Ai , Di , Ei , one encounters unstable generic singularities which depend on arbitrary parameters (moduli). Despite this difficulty, there exists a classification of generic Lagrangian singularities up to the dimension n = 10. The Maslov index has been extended in spaces of higher dimension and has led to the discovery of invariants associated with particular types of singularities (Vassilyev 1988). These invariants control the number of some types of singularities. For instance, in dimension n = 4, the number of A5 (taking account of sign) is equal to zero. Symmetrical Caustics

Another extension consists in imposing some constraint, for instance, a symmetry (Janeczko and Roberts 1993). Symmetrical caustics are not merely the symmetrized usual caustics. Many of them result from the stabilization of unstable singularities of higher codimension by the symmetry. For example, in the 3D space, the butterfly A5 is unstable, but the symmetrical butterfly is a generic singularity in the class of Lagrangian singularities having the mirror symmetry. Nonoptical Caustics

Caustics, as locus of focalization, are not restricted to the usual optics. They are also observed in electronic optics or in gravitational optics and the preceding results apply to these waves. They also appear in nonelectromagnetic waves, for instance, acoustic waves, seismic waves, etc. Propagation always generates caustics. Optical caustics are now understood as Lagrangian singularities and, as singularities, their interest is not restricted to optics. They became indispensable for understanding other domains of mathematical physics, for instance, the variational calculus,

627

the classical mechanics, the Hamilton–Jacobi equations, the control theory, the field theory, etc. See also: Billiards in Bounded Convex Domains; Normal Forms and Semiclassical Approximation; Stationary Phase Approximation; Singularity and Bifurcation Theory.

Further Reading Arnol’d VI (1972) Normal forms for functions near degenerate critical points, the Weyl groups Ak ; Dk ; Ek and Lagrangian singularities. Functional Analysis and its Applications 6: 254–272. Arnol’d VI (1983) Singularities of systems of rays. Uspekhi Matematicheskykh Nauk 38: 77–147. Arnol’d VI (1990) Singularities of Caustics and Wave Fronts, Math. Appl. (Soviet Series), vol. 62. Dordrecht: Kluwer Academic. Arnol’d VI, Gusein-Zade SM, and Varchenko AN (1985) Singularities of Differentiable Maps, Volume I, Monographs in Mathematics, vol. 82. Boston: Birkha¨user. Bennequin D (1986) Caustique Mystique, Sie´minaire Bourbaki, 37e anne´e, no. 634, S.M.F., Aste´risque 133–134, pp. 19–56. Berry MV and Upstill C (1980) Catastrophe optics: morphologies of caustics and their diffraction patterns. In: Wolf E (ed.) Progress in Optics XVIII, pp. 257–346. Amsterdam: NorthHolland Publishing Company. Chekanov Yu V (1986) Caustics in geometrical optics. Functional Analysis and its Applications 6: 223–226. Guillemin V and Sternberg S (1977) Geometric Asymptotics, Mathematical Surveys and Monographs, vol. 14. Providence: American Mathematical Society. Janeczko S and Roberts M (1993) Classification of symmetric caustics II: caustic equivalence. Journal of the London Mathematical Society 48: 178–192. Joets A and Ribotta R (1995) Structure of caustics studied using the global theory of singularities. Europhysics Letters 29: 593–598. Joets A and Ribotta R (1996) Experimental determination of a topological invariant in a pattern of optical singularities. Physical Review Letters 77: 1755–1758. Kravtsov Yu A and Orlov Yu I (1993) Caustics, Catastrophes and Wave Fields, Wave Phenomena, vol. 15. Berlin: Springer. Maslov VP and Fedoriuk MV (1981) Semi-classical approximation in quantum mechanics. In: Mathematical Physics and Applied Mathematics, vol. 7. Dordrecht: D Reidel Publishing Company. Nye JF (1999) Natural Focusing and Fine Structure of Light. Bristol: Institute of Physics Publishing. Thom R (1956) Les singularite´s des applications diffe´rentiables. Annales de l’Institut Fourier 6: 43–87. Thom R (1969) Topological models in biology. Topology 8: 313–335. Vassilyev VA (1988) Lagrange and Legendre characteristic classes. Advanced Studies in Contemporary Mathematics, vol. 3. New York: Gordon and Breach Science Publishers. Whitney H (1955) On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane. Annals of Mathematics 62: 374–410.

628 Optimal Cloning of Quantum States

Optimal Cloning of Quantum States while the Schro¨dinger picture representation is given in terms of the (pre-)dual of T, that is,

M Keyl, Universita` di Pavia, Pavia, Italy ª 2006 Elsevier Ltd. All rights reserved.

T : B ðHN Þ ! B ðHM Þ

Introduction According to the well-known ‘‘no-cloning theorem’’ (Wootters and Zurek 1982) perfect copying of quantum information is impossible, that is, there is no machine which takes a quantum system in an unknown state as input and produces two systems of the same kind, such that none of them is distinguishable from the input by a statistical experiment. In this qualitative form, however, the theorem is not very useful, because in the presence of noise classical information cannot be copied perfectly as well. Therefore, the crucial point is that even under ideal conditions the errors produced in the clones cannot be made arbitrarily small. The best we can hope for is to find an optimal cloning device which makes these errors as small as possible. More generally, we can consider cloning devices, which take as input a certain number, N, of identically prepared systems, and produce a larger number, M, of systems as output. Again, the cloning task is to make the output state resemble as much as possible a state of M systems all prepared in the same state as the inputs. This variant of the problem is of interest as a ‘‘quantum amplifier.’’ It also has a better chance of reasonable success than a cloning device operating on singleinput systems: in the limit of many-input systems, the device can make a good statistical estimate of the input density matrix and hence produce arbitrarily good clones. Figures of Merit

To get a precise mathematical description of the problem, let us consider a one-particle Hilbert space H (which is assumed to be finite dimensional, H = Cd , if nothing else is explicitly stated) and the algebras B(HN ), B(HM ) of (bounded) operators on the N-fold, respectively M-fold, tensor product of H. A quantum operation which takes N particles as input and produces M output particles is then described, in the Heisenberg picture, by a completely positive, unital map (a completely positive, unital and normal map if H is infinite dimensional): T : BðHM Þ ! BðHN Þ

½1

½2

where B (  ) denotes the space of trace-class operators. Hence, if T operates on input systems in the (joint) state N , the output systems (i.e., the ‘‘clones’’) are in the state T (N ). We will call each such T a cloning map. Now our aim is to find an operation T such that the output state T (N ) approximates the product state M as well as possible. The quality of the approximation is measured by a distance function  on the convex set S(HM )  B (HM ) of density operators on HM and, since it is impossible to minimize (T (N ), M ) for all  simultaneously, we are looking only for the worst case. Hence, the quality of a cloning map T is measured by a figure of merit of the form   X; ðTÞ ¼ sup  T ðN Þ;  M ½3 2X

Here X  S(H) is a set of ‘‘preferred’’ density operators whose role will be explained in the next section. An optimal cloning device is described by a ^ which minimizes X,  , that is, cloning map T ^  X; ðTÞ X; ðTÞ

½4

should hold for each cloning map T. The Preferred Set of States

The set X  S(H) of density operators introduced in the last equation describe a priori knowledge about the one-particle input state ; for example, if we want to clone only signal states 1 , . . . , k used to transmit classical information through a quantum channel, the choice for X is {1 , . . . , k }. Other possibilities include: X = S(H) if nothing is known about , the set of pure states, the states in the ‘‘equatorial plane’’ of the Bloch sphere, or Gaussian states if H is infinite dimensional. Each different choice for X leads to a different variant of the cloning problem, and we will summarize the most relevant cases treated in the literature in the section ‘‘Examples.’’ A different kind of a priori knowledge is a priori measures, that is, instead of knowing that all possible input states lie in a special set X, we know for each measurable set X  S(H) the probability (X) for  2 X. Such a situation typically arises when we are trying to clone states of systems which

Optimal Cloning of Quantum States

originate from a source with known characteristics. In this case, we can use mean errors, Z    ; ðTÞ ¼ ½5   T  ðN Þ; M ðdÞ SðHÞ

as a figure of merit. Sometimes these are easier to compute than maximal errors as in eqn [3]. Often,  however,  leads to stronger results than , therefore we will concentrate our discussion on maximal rather than mean errors. The Distance Measure

The remaining freedom in eqn [3] is the distance measure  and there are mainly two physically different choices: we can either check the quality of each clone separately or we can test, in addition, the correlations between output systems. The most common choice for a figure of merit for the first ^ j denotes partial trace over type is given by (where tr all but the jth tensor factor)    ^ j T ðN Þ;   ½6 1 ðTÞ ¼ sup 1  F tr 2X;j

Here F (, ) denotes the (quadratic) fidelity of  and , that is,  1=2 2 1=2 1=2 F ð; Þ ¼ tr   ½7 and the supremum is taken over all  2 X and j = 1, . . . , N. 1 measures the worst one-particle error of the output state T  (N ), and we will refer to it in the following as the local error. If we are interested in correlations too, we have to choose    all ðTÞ ¼ sup1  F T ðN Þ; M  ½8 2X

all measures again a ‘‘worst-case’’ error, but now of the full output with respect to M uncorrelated copies of the input . We will call it the global error. Alternative figures of merit arise if we replace the fidelity in eqns [6] and [8] by other distance measures like the trace norm, the Hilbert–Schmidt norm, or the relative entropy. If X consists only of pure states, the operations T which minimize 1 or all are usually not altered by such different choices. If X is a set of mixed states, however, the correct choice is unclear and might depend on the precise physical context (there is, in particular, no reason to prefer fidelities).

General Properties Before we consider more special examples in the next section, let us discuss some general properties

629

of the figure of merit X,  from eqn [3] and the corresponding optimization problem. Existence of Solutions

If the distance measure  is continuous in the first argument, the optimization problem [4] has a solution, that is, optimal cloning machines exist: the set T of cloning maps [1] is compact and the quantity X,  is – as a supremum over continuous functions – lower-semicontinuous. Hence, the statement follows from the fact that a lowersemicontinuous function on a compact set always admits a minimizer. This argument can be generalized to the infinitedimensional case, if we choose the set T of allowed cloning maps more carefully (the restriction to normal channels proposed above is most probably not sufficient for this purpose) and if we equip it with an appropriate topology. The latter should be weak enough for T to be compact, and strong enough for X,  to be lower-semicontinuous. A typical choice is the weak-topology arising from an embedding of T into the dual of a Banach space (such that we can apply the Banach–Alaoglu Theorem). Detailed studies in this direction are, however, not yet available. Covariant Cloning Maps

To solve the optimization problem [4] is a difficult and, in many cases, impossible task. However, it can be simplified significantly if X and  admit a nontrivial symmetry group. Hence, consider again a distance  which is continuous and convex in its first argument and a closed subgroup G of the group U(d) of unitary operators on H = Cd , such that   UXU  X;  UM UM ; UM U M ¼ ð; Þ ½9 hold for all U 2 G and ,  2 S(HM ). Then X,  is invariant under the induced G action on the set T of cloning maps, that is, X; ðU TÞ ¼ X; ðTÞ with ðU TÞðAÞ ¼ UN TðUM AUM ÞUN

½10

holds for all U 2 G and all T 2 T . Convexity of X,  in T implies (with the Haar measure H on G)   X; ðTÞ; X; ðTÞ Z  with T ¼ U ðTÞH ðdUÞ

½11

G

for all T. Hence, we can replace each cloning map  without sacrificing the by its group average T

630 Optimal Cloning of Quantum States

 is optimal quality of the clones. This implies that T  is G-covariant, if T is, and, since T  ¼T  U ðTÞ

8U 2 G

can be found on the web at http://www.imaph. tu-bs.de/qi/problems/problems-html.

½12

we can conclude, together with the arguments from the last section, that the optimization problem [4] always admits covariant solutions. Similarly, we can show that permutation invariant (sometimes called ‘‘symmetric’’) solutions exist, that is, cloners which do not prefer a particular clone or a particular input system. This is a very useful result, because the set of covariant and permutation-invariant T is much smaller than the set of all cloning maps, and it can be parametrized in terms of irreducible representations of G and the permutation group. In particular, the case G = U(d) (such a T is often called ‘‘universal’’ because it does not prefer any direction in the Hilbert space H) leads to quite general solutions. Relationships with Quantum State Estimation

If a procedure to estimate the input state  from a measurement on the N-fold system in the joint state N is given, there is a simple way to produce a cloning machine: we just have to take the estimate ^ for the density matrix  and prepare M > N systems in the state ^M . If X is finite and estimation (which in this case is called hypothesis testing) is done in terms of a positive operator valued measure (E )2X , E 2 B(HN ), the probability to get the estimate  2 X when the input is in the state N is given by tr(E N ). Hence, the cloning map derived from this estimation scheme is given by X   ~  ðN Þ ¼ E tr E N M ½13 2X

A generalization to arbitrary X is straightforward, but requires the use of measure theory. It is easy to ~ from eqn [13] is in see that the cloning map E general not optimal, in particular if M is only ~ has the interestslightly bigger than N. However, E ~ ing feature that X,  (E) depends only on the number of input systems, N, but not on the number of clones, M, we want to produce. This observation ~ becomes leads immediately to the conjecture that E optimal in the limit M ! 1. A general proof is currently not available, in those cases, however, where optimal cloner and estimater can be explicitly calculated for all N and M (i.e., the cases treated in the sections ‘‘Universal pure-state cloning’’ and ‘‘Phase-covariant pure-state cloning’’) the conjecture is true. A more detailed discussion of this problem together with information about its current status

Examples In this section, we will discuss concrete examples that arise from different choices of the distance measure  and the set X of preferred states. Universal Pure-State Cloning

The most frequently discussed case arises if X is the set of pure states, that is, the input states are pure, but otherwise unknown. Under this condition, it is sufficient to consider the symmetric part HN þ of the tensor product HN , and only cloning maps T : B(HM ) ! B(HN þ ), because only this part affects the local or the global error. A complete solution for arbitrary N, M and all finitedimensional Hilbert spaces is available for all in Werner (1998) and for 1 in Keyl and Werner (1999). Both cases admit the same (surprisingly simple) unique solution   ^  ðÞ ¼ d½N SM   1ðMNÞ SM T d½M

½14

where SM is the projection onto the symmetric tensor product HM and d[M] denotes the dimenþ sion of HM . To derive these results, the groupþ theoretic methods sketched in the section ‘‘Covariant cloning maps’’ are used. The fact that global and local figures of merit are minimized by the same cloning map is surprising and a special feature of pure-state cloning. It implies that correlations and entanglement between the clones does not matter at all. Phase-Covariant Pure-State Cloning

Consider a fixed basis jji, j = 0, . . . , d  1, in H and let X be the set of states given by ¼ j0i þ

d1 X

eij jji

½15

j¼1

where the j denote arbitrary phases. Obviously, this set is invariant under the set of all unitaries which are diagonal in the given basis (i.e., a maximal torus in U(d)). Using the methods outlined in the section ‘‘Covariant cloning maps,’’ the corresponding cloning problem is (almost) completely solved in Buscemi et al. (2005). For arbitrary d = dim H, N and all M = N þ dk, with k 2 N a

Optimal Cloning of Quantum States

cloning map which minimizes global as well as local errors is given in terms of the unitary ^ 0 ; . . . ; nd i ^ : HN ! HM ; Ujn U þ þ ¼ jn0 þ k; . . . ; nd þ ki

½16

where jn1 , . . . , nd i, nj 2 N, denotes the number basis of HN associated with the distinguished basis jji of H. Cloning Finitely Many States

If X is a finite set of pure states, a general solution is not available, but there are several important partial results. The easiest situation arises if the elements of X are mutually orthogonal pure states. In this case, ideal cloning is possible in terms of an appropriately chosen unitary. If the states are linearly independent but nonorthogonal, ideal cloning is possible as well if we consider probabilistic cloning machines (Duan and Guo 1998); that is, there is a nonvanishing probability that the machine fails and does not produce any clones at all (this means T is not unital). Optimal cloning (with deterministic operations) of two nonorthogonal qubit states j = j j ih j j, j = 1, 2, is considered for all N, M in (Bruß et al. (1998) and Chefles and Barnett (1999)) (using averaged global fidelity as the figure of merit). The crucial observation in this case is that the optimal clones are pure, that is, T (N j ) = jj ihj j and that the j lie in the subspace spanned by the (unattainable) ideal clones M . j Universal Mixed-State Cloning

X = S(H) means that absolutely nothing is known a priori about the input state . If the distance measure  is U(d) and permutation invariant (which is the case for all possible choices discussed in the section ‘‘The distance measure’’) the analysis from the section ‘‘Covariant cloning maps’’ shows that a universal and symmetric minimizer exists. An explicit solution, however, is not known, and even the physically most appropriate choice for  is unclear. In contrast to the pure-state case, this is a serious question, because the set of optimal cloners is, in this case, much more sensitive to changes in . In particular, correlations among the clones become crucial, and it is very likely that local and global figures of merit lead to very different solutions. To emphasize this difference, an operation which minimizes only local errors is sometimes called ‘‘broadcasting,’’ rather than cloning. A related problem with (at least) partial solutions (‘‘purification’’) will be discussed in the section ‘‘Purification.’’

631

Cloning of Gaussian States

If the Hilbert space is infinite dimensional, the restriction to a reasonable small set X of preferred states is crucial, because otherwise the search for minimizers becomes hopeless. A physically relevant class with nice mathematical properties are Gaussian states and in particular coherent states. Cloning of the latter has been studied in Cerf et al. (2005) for the case N = 1 (and M arbitrary). As in the section ‘‘Covariant cloning maps,’’ it can be shown that the search for optimal cloners can be restricted to those which are covariant with respect to phase space translations. This simplifies the problem significantly and leads to the result that the global error is minimized by Gaussian cloning maps, while in the local case the best cloner is non-Gaussian. Asymmetric Cloning

In all examples discussed up to now, we have considered symmetric cloners, that is, the quality of all clones is measured with equal weight. Alternatively, we can look for asymmetric cloners which produce clones with different quality and ask for the trade-off between them. This problem was first discussed in Cerf (2000) and later in Iblisdir et al. (2005). It can be regarded as a constraint optimization problem, where the error of the first M0 < M clones should be minimized under the constraint that the error of the rest is bounded by a fixed value. In Iblisdir et al. (2005), it is conjectured that for pure input states and local errors the optimal solution to this problem is given by   T ðÞ ¼ V    1ðMNÞ V ½17 where V is a linear combination of projections in the commutant of {UN j U 2 U(H)}. This conjecture is true (at least) for qubits in the case 1 ! n þ 1 and 1 ! 1 þ n.

Related Problems Instead of cloning, we can also try to approximate other impossible machines by channels which operate on multiple inputs. To this end, we only have to replace the figure of merit [6] by    ^ j T ðN Þ; ðÞ  1; ðTÞ ¼ sup 1  F tr ½18 2X;j

where  : S(H) ! S(H) is a (possibly nonlinear) functional which describes the task we want to approximate. The generalization all,  of all can be given similarly. If  has the appropriate continuity and symmetry properties, the discussion in the section ‘‘General properties’’ applies completely, that is, we can assume covariance and permutation

632 Optimal Transportation

invariance, and we can consider operations which use state estimation in an intermediate step. Purification

Consider N quantum systems, all originally prepared in the same pure state , and then subsequently exposed to the same (known) decoherence process, described by a depolarizing channel R. The task of purification is to produce M output systems which approximate the original pure input state as well as possible. Hence, the corresponding figure of merit arises with X = {R() j  pure} and () = R1 (). This problem is discussed for qubits in Cirac et al. (1999), Keyl and Werner (2001) and D’Ariano et al. (2005). The optimal purifier can be given explicitly for all N, M in terms of irreducible SU(2) representations. Surprisingly, it turns out that the output purity can be improved even if the number of outputs, M, is larger than the number of available input systems, N (although N should be large enough). If we measure purity in terms of local errors, it can be shown that, in the limit N ! 1, perfectly purified qubits can be produced at an infinite rate (i.e., the number of output systems per input system can become infinite). However, we have to pay for this result with extremely large correlations between the output systems. Therefore, the global error does not disappear asymptotically, if we insist on a nonvanishing rate. Universal Not

‘‘Universal not’’ (UNOT) is an operation which sends each pure state  to its orthocomplement. This is a positive but not a completely positive operation. Hence, it cannot be performed by any physical device. However, we can try to approximate it by a cloning map T operating on N input systems. The corresponding figure of merit [18] arises if X is the set of pure states and () = 1  . In Buzˇek et al. (1999), it is shown that the optimal solution to this problem (for all N and M) is to estimate and

reprepare as described in the section ‘‘Relationships with quantum state estimation.’’ Approximating UNOT is, therefore, significantly more difficult than (pure-state) cloning, where the optimal solution is always (for finite M) better than estimation. See also: Channels in Quantum Information Theory; Compact Groups and Their Representations; Positive Maps on C*-algebras.

Further Reading Bruß D, DiVincenzo DP, Ekert A et al. (1998) Optimal universal and state-dependent cloning. Physical Review A 57(4): 2368–2378. Buscemi F, D’Ariano GM, and Macchiavello C (2005) Economical phase-covariant cloning of qudits. Physical Review A 71: 042327. Buzˇek V, Hillery M, and Werner RF (1999) Optimal manipulations with qubits: universal-not gate. Physical Review A 60(4): R2626–R2629. Cerf NJ (2000) Asymmetric quantum cloning machines. Journal of Modern Optics 47: 187–209. Cerf NJ, Krueger O, Navez P, Werner RF, and Wolf MM (2005) Non-Gaussian cloning of quantum coherent states is optimal. Physical Review Letters 95: 070501. Chefles A and Barnett SM (1999) Strategies and networks for statedependent quantum cloning. Physical Review A 60: 000136. Cirac JI, Ekert AK, and Macchiavello C (1999) Optimal purification of single qubits. Physical Review Letters 82: 4344–4347. D’Ariano GM, Macchiavello C, and Perinotti P (2005) Superbroadcasting of mixed states. Physical Review Letters 95: 060503. Duan L-M and Guo G-C (1998) Probabilistic cloning and identification of linearly independent quantum states. Physical Review Letters 80: 4999–5002. Iblisdir S, Acin A, and Gisin N (2005) Generalised asymmetric quantum cloning, quant-ph/0505152. Keyl M and Werner RF (1999) Optimal cloning of pure states, testing single clones. Journal of Mathematical Physics 40: 3283–3299. Keyl M and Werner RF (2001) The rate of optimal purification procedures. Annales Henri Poincare´ 2: 1–26. Some open problems in quantum information theory, http:// www.imaph.tu-bs.de/qi/problems/problems–.html. Werner RF (1998) Optimal cloning of pure states. Physical Review A 58: 980–1003. Wootters WK and Zurek WH (1982) A single quantum cannot be cloned. Nature 299: 802–803.

Optimal Transportation Y Brenier, Universite´ de Nice Sophia Antipolis, Nice, France ª 2006 Elsevier Ltd. All rights reserved.

The purpose of this article is to introduce some of the main ideas of optimal transportation theory. A lot more can be found in Villani’s book (Villani 2003), in a somewhat similar spirit. Supplementary information is also available in Ambrosio et al. (2005), Evans and Gangbo (1999), and Ru¨schendorf and Rachev (1990).

Transportation Maps Let us start by a rather abstract definition: Definition 1 Let X and Y be two topological spaces with Borel probability measures and , respectively. We say that a Borel map T : X ! Y is a transportation map between (X, ) and (Y, ) if, for each Borel subset A of Y, Z Z ðdxÞ ¼ ðdyÞ TðxÞ2A

y2A

Optimal Transportation

It is customary to say that T pushes forward  to , or to say that  is the image of  by T. An abstract measure-theoretic result asserts that there is always such a transportation map T, as soon as  has no atom (i.e., the  measure of any point x 2 X is zero).  0, Y =   1, A more concrete situation is when X =  where 0 and 1 are two smooth bounded open subsets of the d-dimensional Euclidean space Rd . In such a case, a classical result, due to Moser and improved by Dacorogna and Moser (1990), reads: Theorem 1 Let 0 and 1 be two smooth bounded open sets in Rd . Let 0 > 0 and 1 > 0 be two smooth functions on Rd such that Z Z 0 ðxÞdx ¼ 1 ðxÞdx ¼ 1 0

633

(The Monge–Ampe`re equation is a famous geometric PDE, related to the seeking of hypersurfaces with prescribed Gaussian curvature.) The main gain with respect to Moser’s construction is the property that the optimal map T has, at each x 2 0 , a Jacobian matrix DT(x) = D2 (x) which is a positivedefinite symmetric matrix. This property has been first exploited by McCann (1997) and later by many authors (see Villani (2003), for many references) to prove a large series of geometric and functional inequalities. A very fine example can be found in Barthe (1998). Let us just consider, as an elementary illustration, a short and sharp proof of the isoperimetric inequality using the optimal transportation map.

1

Then there is a smooth transportation map T  0 , 0 (x)dx) and (  1 , 0 (y)dy). Furtherbetween ( more, T is an orientation-preserving diffeomorphism and solves the Jacobian equation: 1 ðTðxÞÞ detðDTðxÞÞ ¼ 0 ðxÞ;

8x 2 0

½1

A Proof of the Isoperimetric Inequality Using Optimal Transportation Maps Let us recall the isoperimetric inequality: Theorem 3 Let  be a smooth bounded open subset in Rd . Then j@j  djB1 j1=d jj11=d

Transportation Maps with Convex Potentials An important property of Moser’s construction, which we did not state, is the possibility of prescribing the restriction of T along the boundary @0 . If one does not care about this latter property, one can improve Theorem 1 as follows (Caffarelli 1992):

holds true where B1 is the unit ball in Rd , jj and j@j, respectively, denote the d-dimensional volume of  and the (d  1)-dimensional Hausdorff measure of the boundary @. In addition, the inequality becomes an equality if and only if  is a ball. To prove this result, let us define densities: 1 ; x2 jj 1 1 ðyÞ ¼ ; y 2 B1 jB1 j

0 ðxÞ ¼

Theorem 2 Assume further that 1 is a uniformly strictly convex set. Then, there is a transportation map T with a smooth convex potential, namely TðxÞ ¼ DðxÞ;

8x 2 0

for some smooth convex function  defined on Rd and strictly convex on 0 . In addition, among  0 , 0 (x)dx) to all Borel maps T transporting (  1 , 1 (y)dy), D is the unique map that minimizes ( Z inf jTðxÞ  xj2 0 ðxÞdx ½2 T

and consider the associated optimal transportation  0 , 0 (x)dx) to (  1 , 0 (y)dy). From map D from ( the Monge–Ampe`re equation, 1 ðDðxÞÞ detðD2 ðxÞÞ ¼ 0 ðxÞ we get:

Rd

detðD2 ðxÞÞ ¼

where j  j denotes the Euclidean norm on Rd . Because of its characterization, T = D is often called the ‘‘optimal transportation map’’ with respect to the ‘‘transportation cost’’ [2]. Notice that, because of the Jacobian equation [1],  automatically is a classical solution to the Monge–Ampe`re equation: 1 ðDðxÞÞ detðD2 ðxÞÞ ¼ 0 ðxÞ;

8x 2 0

½3

jB1 j ; jj

x2

½4

Since the range of D on  is the unit ball B1 , we have Z Z I¼ DðxÞ  nðxÞdðxÞ  dðxÞ ¼ j@j @

@

where n(x) and d(x) respectively, denote the outward unit normal and the (d  1)-dimensional

634 Optimal Transportation

Hausdorff measure along @. Using the divergence theorem, we also have: Z I¼ ðxÞdx  2

where (x) = trace(D (x)) is the Laplacian of . From the geometric mean inequality, we know that, for any symmetric matrix A  0, ðdet AÞ1=d  1=d trace ðAÞ holds true, with equality if and only if A is equal to the identity matrix multiplied by a non-negative scalar factor. Thus, Z I  d ðdetðD2 ðxÞÞ1=d dx 

¼ djj11=d jB1 j1=d (because of [4]). So, we have obtained the isoperimetric inequality: j@j  djB1 j1=d jj11=d Let us now consider the case when this inequality becomes an equality. Then, necessarily, for each x 2 , A = D2 (x) satisfies det A = (trace(A)=d)d and, therefore, must be the identity matrix multiplied by a scalar factor  > 0, possibly depending on x. Because of [4], the determinant of D2 (x) is constant over . Thus,  > 0 must be constant. It follows that D(x) = (x  a), for some point a in Rd . Therefore,  must be the ball centered at a of radius 1=.

Monge’s Optimal Transportation Problem Theorem 2 is one of the numerous avatars of the socalled optimal transportation theory that goes back to Monge’s mass transfer problem which addressed in 1781 the ‘me´moire sur la the´orie des de´blais et des remblais’ and was completely renewed by Kantorovich in the 1940s (see e.g., Ru¨schendorf and Rachev (1990) for instance). Let us quote a typical result, similar to Theorem 2, but without regularity assumptions on the data (see Brenier and Caffarelli (1992)): Theorem 4 Let 0 be a non-negative Lebesgue integrable function on Rd , such that Z 0 ðxÞdx ¼ 1 Rd

Then for any Borel probability measure 1 (dy) with compact support on Rd , there is a unique map T transporting 0 (x)dx to 1 (dy), which minimizes Z jTðxÞ  xj2 0 ðxÞdx R

d

where j  j denotes the Euclidean norm on Rd . In addition, there is a Lipschitz continuous convex function  defined on Rd such that T(x) = D(x) for 0 almost every x 2 Rd , which implies: Z Z f ðDðxÞÞ0 ðxÞdx ¼ f ðyÞ1 ðdyÞ Rd

Rd

for all continuous functions f on Rd . Theorem 2, which can be interpreted as a regularity result with respect to Theorem 4, is the main output of Caffarelli’s regularity theory for transportation maps with convex potentials (Caffarelli 1992). Caffarelli’s analysis starts by a proof that  actually is a weak solution of the Monge–Ampe`re equation [3] in the sense of Alexandrov and is strictly convex. Then, Caffarelli shows that D2  is Ho¨lder continuous, as soon as 0 and 1 are Ho¨lder continuous. Notice that the convexity assumption for 1 is crucial to insure the regularity of the convex potential. Caffarelli provided counter-examples when 1 is made of two separate balls attached together by a sufficiently thin pipe. Surprisingly enough, results such as Theorem 4 are related to concrete applications in, for example, astrophysics, image processing, etc. (Frisch et al. 2002, Haker and Tannenbaum 2003).

The Kantorovich Optimal Transportation Problem The Monge optimal transportation problem can be solved using the Kantorovich duality method, based on the key concept of ‘‘generalized transportation maps,’’ also called ‘‘transportation plans’’ or ‘‘doubly stochastic measures.’’ The abstract definition is: Definition 2 Let X and Y be two topological spaces with Borel probability measures  and , respectively. We say that a Borel probability measure  on X  Y is a generalized transportation map, or a transportation plan, if its marginals are, respectively,  and , namely Z Z ðdx; dyÞ ¼ ðdxÞ x2A;y2Y x2A Z Z ½5 ðdx; dyÞ ¼ ðdyÞ x2X;y2B

y2B

for all Borel subsets A and B of X and Y, respectively. The Monge–Kantorovich (MK) optimal transportation problem amounts, given a ‘‘transportation

Optimal Transportation

cost,’’ that is, a continuous function c : X  Y ! R, to find a minimizer for Z ½6 IMK ¼ inf cðx; yÞðdx; dyÞ 

where  is subject to be a transportation plan between (X, ) and (Y, ). Notice that this problem is convex (and can be seen as an infinite-dimensional linear program) and its dual problem can be easily computed (using, e.g., Rockafellar’s theorem in convex analysis and assuming, for simplicity, that both X and Y are compact). Theorem 5 IMK

We have Z  Z aðxÞðdxÞ þ bðyÞðdyÞ ¼ sup

635

or, more generally, c(x, y) = k(x  y), where k is a uniformly strictly convex function. A typical result is: Theorem 5 Let 0 be a non-negative Lebesgue integrable function on Rd , with unit integral, and 1 (dy) be a Borel probability measure with compact support on Rd . Let k be a uniformly strictly convex function on Rd . Then the MK problem Z IMK ¼ inf kðy  xÞðdx; dyÞ 

where  is subject to be a transportation plan between 0 (x)dx and 1 (dy) on Rd , has a unique solution of form ðdx; dyÞ ¼ ðy  TðxÞÞðdxÞ

½7

a;b

where (a, b) is any pair of continuous functions, defined on X and Y, respectively, and subject to:

where T is the unique minimizer of the Monge problem: Z IM ¼ inf kðTðxÞ  xÞ0 ðxÞdx T

aðxÞ þ bðyÞ  cðx; yÞ;

8x 2 X; 8y 2 Y

Of course, each transportation map T, in the sense of Definition 1, can be seen as a transportation plan  in the Kantorovich framework, just by setting ðdx; dyÞ ¼ ðy  TðxÞÞðdxÞ which means Z

ðdx; dyÞ ¼

x2A; y2B

Z ðdxÞ x2A;TðxÞ2B

for all Borel subsets A and B of X and Y, respectively. Then, we have Z Z cðx; yÞðdx; dyÞ ¼ cðx; TðxÞÞðdxÞ So, the MK problem can be seen as a ‘‘relaxed’’ version of the ‘‘classical’’ optimal transportation problem a` la Monge: Z IM ¼ inf cðx; TðxÞÞðdxÞ ½8 T

where T is subject to be a transportation map between (X, ) and (Y, ). Indeed, we have IMK  IM . It turns out that, in many important situations, there is no gap between these two values, which makes the MK problem a perfectly convenient convex substitute for the original, nonconvex, Monge transportation problem. This is, in particular, the case of the situation considered in Theorem 4, when the cost function is just cðx; yÞ ¼ jx  yj2

among all transportation maps T between 0 (x)dx and 1 (dy) on Rd . In addition IMK = IM . Proof for Theorem 5 (Sketch) For simplicity, we assume that 0 and 1 are both compactly supported in a ball B in Rd and we limit ourselves to the simplest cost function k(x) = jxj2 =2. We first denote by M the set of all Borel regular probability measures on B  B having 0 (x)dx and 1 (dy) as marginals, which means Z Z f ðxÞ ðdx; dyÞ ¼ f ðxÞ0 ðxÞdx ZBB ZB f ðyÞ ðdx; dyÞ ¼ f ðyÞ1 ðdyÞ BB

B

for all continuous functions f on Rd . From Theorem 7, we deduce: Z max x  y ðdx; dyÞ 2M BB Z ¼ inf ½ðxÞ0 ðxÞ þ ðxÞ1 ðxÞdx B

where the infimum is taken over all pairs (, ) of continuous functions on B satisfying ðxÞ þ ðyÞ  x  y;

8x 2 B; 8y 2 B

Then, it can be established that the infimum is attained by a pair (, ) such that  is the restriction of a Lipschitz continuous convex function defined on Rd , and for 0 (x)dx almost every point of Rd ,  coincides with the Legendre–Fenchel transform of , LFðÞðyÞ ¼ sup ðx  y  ðxÞÞ x2Rd

636 Optimal Transportation

Moreover, if = opt 2 M maximizes (dx, dy), then

R

BB

x  y

ðxÞ þ ðyÞ ¼ x  y holds for opt -almost every (x, y) 2 Rd  Rd . Using well-known properties of the Legendre–Fenchel transform in convex analysis, one deduces that opt is necessarily of the form opt ðdx; dyÞ ¼ ðy  DðxÞÞ0 ðxÞ dx which implies Z Rd Rd

f ðyÞ opt ðdx; dyÞ ¼

Z Rd

f ðDðxÞÞ0 ðxÞ dx

for all continuous functions f on Rd and achieves the proof since the second marginal of opt is 1 (dy).

The Wasserstein Distance Optimal transportation theory is strongly related to the geometric analysis of probability measures. For simplicity, let us just consider the space Prob(B) of all Borel probability measures  supported by some fixed ball B in Rd . This space is compact for the weak topology of measures. An equivalent definition of this topology is provided by the distance d, naturally attached to the MK problem: dð0 ; 1 Þ ¼ inf

Z

2

1=2

jx  yj ðdx; dyÞ

½9

BB

where  is subject to be a transportation plan between 0 and 1 on B. (Of course, more general convex functions k can be used to define the cost function.) It has become popular to call this distance as Wasserstein distance (or its generalizations for various k). It turns out that Prob(B) equipped with this distance has a formal Riemannian structure (Otto 2001, Ambrosio et al. 2005). For instance, given two probability measures 0 (x)dx and 1 (x)dx, we can define a ‘‘shortest path’’ t ! (t,  ) 2 Prob(B) such that (0) = 0 , (1) = 1 , just by setting: ðt; dxÞ ¼

Z

ða þ ðDðaÞ  aÞt  xÞ0 ðaÞda; B

8t 2 ½0; 1

where D is the optimal transportation map between 0 and 1 on B. This idea, which is somewhat related to the geometric analysis of hydrodynamics and various concepts of generalized flows Arnol’d and Khesin 1998, Brenier, was successfully used by McCann (1997) and Otto (2001). In particular, the concept of convexity along these geodesic paths on Prob(B) has been pointed out by McCann (1997) to be a crucial tool for new proofs of geometric and functional inequalities. Otto, and other contributors (see Ambrosio et al. (2005) for a comprehensive discussion), observed that many important parabolic or dissipative evolution PDEs can be described as ‘‘gradient flows’’ (or ‘‘steepest descent’’) of such functionals, with respect to the Wasserstein metric.

Further Reading Ambrosio L, Gigli N, and Savare´ G (2005) Gradient Flows, ETH Lectures in Mathematics. Birkha¨user. Arnol’d VI and Khesin B (1998) Topological Methods in Hydrodynamics. Berlin: Springer. Barthe F (1998) Inventiones Mathematicae 134: 335–361. Brenier Y Polar factorization and monotone rearrangement of vector-valued functions. Brenier Y (1991) Minimal geodesics on groups of volumepreserving maps. Communications on Pure and Applied Mathematics 44: 375–417. Brenier Y (1999) Minimal geodesics on groups of volumepreserving maps. Communications on Pure and Applied Mathematics 52: 411–452. Caffarelli L (1992) Boundary regularity of maps with convex potentials. Communications on Pure and Applied Mathematics 45: 1141–1151. Cullen M, Norbury J, and Purser J (1991) Generalised Lagrangian solutions for atmospheric and oceanic flows. SIAM Journal of Applied Mathematics 51: 20–31. Dacorogna B and Moser J (1990) Ann. Inst. H. Poincare´ Anal. NL 7: 1–26. Evans C and Gangbo W (1999) Mem. Amer. Math. Soc. 137: 653. Frisch U, Matarrese S, Mohayaee R, and Sobolevski A (2002) Nature 417: 260–262. Haker S and Tannenbaum A (2003) New Approach to Monge– Kantorovich with Applications to Computer Vision and Image Processing, IMA Series on Applied Mathematics. Springer. McCann R (1997) Advances in Mathematics 128: 153–179. Otto F (2001) CPDEs 26: 101–174. Ru¨schendorf L and Rachev ST (1990) Journal of Multivariate Analysis 32: 48–54. Ru¨schendorf L and Rachev ST (1998) Mass Transportation Problems. Berlin: Springer. Villani C (2003) Topics in Optimal Transportation, AMS Graduate Studies in Mathematics.

Ordinary Special Functions

637

Ordinary Special Functions The residue of (z) at z = n is equal to (1)n =n!. Legendre’s duplication formula is

W Van Assche, Katholieke Universiteit Leuven, Leuven, Belgium ª 2006 Elsevier Ltd. All rights reserved.

22z1 ð2zÞ ¼ pffiffiffi ðzÞðz þ 1=2Þ 

Introduction The exponential function, the logarithm, the trigonometric functions, and various other functions are often used in mathematics and physics. They are transcendental functions in the sense that they cannot be obtained by a finite number of operations as a solution of an algebraic (polynomial) equation. Typically, they are obtained by a Taylor series expansion. Many other higher transcendental functions arise in mathematical physics, often as solutions of differential equations. A precise knowledge of the behavior of such functions, their relation with other functions, addition, multiplication and composition properties, representations as an infinite series, or as an integral, often shed a lot of light onto the problem in which they arise. If they are sufficiently useful to a large audience, then they usually get a name and they will be called special functions. In what follows, we describe a few of these special functions of one variable, but clearly this is just a tip of the iceberg. Many other special functions exist and we refer to the classical tables of Abramowitz and Stegun (1964) and the Bateman manuscript project (Erde´lyi et al. 1953–55) for more special functions. Nowadays, there have been numerous q-extensions of special functions (see q-Special Functions).

Gamma and Beta Function The gamma function is defined by Z 1 ðzÞ ¼ tz1 et dt; 0:

½1

0

It satisfies the functional equation (z þ 1) = z(z) and since (1) = 1 we have (n þ 1) = n! for n 2 N. The gamma function therefore extends the factorial function for integers to complex numbers. The functional equation  ðzÞð1  zÞ ¼ ½2 sin z allows to continue the gamma function analytically to 1, then all the zeros are real. All the zeros are simple (except possibly at the origin). Each of the functions J (z), Y (z), H(1) (z), or H(2) (z) satisfies the recurrence relation za1 ðzÞ þ zaþ1 ðzÞ ¼ 2a ðzÞ and the differential–recurrence relation a1 ðzÞ  a1 ðzÞ ¼ 2a0 ðzÞ

Ordinary Special Functions

Hypergeometric Series

Modified Bessel Functions

The modified Bessel equation is x2 y00 þ xy0  ðx2 þ  2 Þy ¼ 0

½9

Clearly J (ix) is a solution of this equation. The ‘‘modified Bessel function of the first kind’’ is defined as I ðxÞ ¼ e

 i=2

J ðxe

i=2

 < arg x  =2 ½10

Þ;

639

P1 n A power series n = 0 cn z is said to be hypergeometric when the ratio cnþ1 =cn is a rational function of the index n. Most series that one finds in calculus textbooks are hypergeometric series and some of them define important special functions. When ðn þ a1 Þðn þ a2 Þ    ðn þ ap Þ cnþ1 ¼ ðn þ b1 Þðn þ b2 Þ    ðn þ bq Þðn þ 1Þ cn then we write the corresponding series as

so that

p Fq

If  is not an integer, then I (x) and I (x) are two linearly independent solutions of [9], and when  = n is an integer one has In (x) = In (x). The ‘‘modified Bessel function of the second kind’’ is defined by K ðxÞ ¼

 a1 ; a2 ; . . . ; ap  z b1 ; b2 ; . . . ; bq  1 X ða1 Þn ða2 Þn    ðap Þn zn ¼ ðb1 Þn ðb2 Þn    ðbq Þn n! n¼0 

1 X

ðx=2Þ2k I ðxÞ ¼ ðx=2Þ k!ð þ k þ 1Þ k¼0 

 ½I ðxÞ  I ðxÞ 2 sin 

Some special cases of modified Bessel functions are rffiffiffiffiffiffi 2 sinh x I1=2 ðxÞ ¼ x rffiffiffiffiffiffi 2 cosh x I1=2 ðxÞ ¼ x

where (a)n = a(a þ 1)(a þ 2)    (a þ n  1), with (a)0 = 1, is the rising factorial or Pochhammer symbol. When p and q are small, one also uses the notation p Fq (a1 , . . . , ap ; b1 , . . . , bq ; z) where a semicolon (;) is used to separate the parameters in the numerator from the parameters in the denominator and also to separate the parameters from the variable z. Some special cases are:

 the exponential series 0 F0 ð; ; zÞ

¼

and

1 n X z n¼0

rffiffiffiffiffiffi  x e K1=2 ðxÞ ¼ 2x

1 F0 ð1; ; zÞ

¼

1 X n¼0

¼ expðzÞ

zn ¼

1 1z

 the binomial series

0

and

1 F0 ð; ; zÞ ¼

Z

n!

 the geometric series

One has the integral representation Z 1 K ðzÞ ¼ ez cosh x cosh x dx

ðz=2Þ I ðzÞ ¼ pffiffiffi  ð þ 1=2Þ

½11

1  X  n z ¼ ð1 þ zÞ n n¼0

1

2 1=2 zx

ð1  x Þ

e

dx

1

whenever 1=2. The ‘‘Airy functions’’ are given by pffiffiffiffiffiffiffiffi pffiffiffi  z=3 z K1=3 ð Þ AiðzÞ ¼ I1=3 ð Þ  I1=3 ð Þ ¼  3 pffiffiffiffiffiffiffiffi   BiðzÞ ¼ z=3 I1=3 ð Þ þ I1=3 ð Þ where = 2z2=3=3. They are both a solution of Airy’s differential equation y00 ðzÞ  zyðzÞ ¼ 0

 the logarithmic function 2 F1 ð1; 1; 2; zÞ

¼

1 X zn 1 ¼  logð1  zÞ z n þ 1 n¼0

 the Bessel function ðz=2Þ 0 F1 ð;  þ 1; z2 =4Þ ¼ ð þ 1ÞJ ðzÞ For generic values of the parameters, we see that the hypergeometric series converges everywhere in the complex plane when q  p, it converges for jzj < 1 when p = q þ 1, and for p > q þ 1 it is only defined at

640 Ordinary Special Functions

z = 0. When one of the numerator parameters is a negative integer, say a1 = m, then the series is terminating and defines a polynomial of degree m. None of the denominator parameters is allowed to be a negative integer m, unless there is a numerator parameter which is a negative integer k with k < m. For q  p, the hypergeometric series therefore defines an entire function which is the corresponding hypergeometric function. For p = q þ 1, the hypergeometric series only converges in the open unit disk, but sometimes it can be continued analytically to a larger domain in the complex plane. The analytic continuation of the hypergeometric series is then called the hypergeometric function. Take for example the geometric series, then it is clear that the hypergeometric series converges in the open unit disk, but the corresponding hypergeometric function is defined in the whole complex plane with a simple pole at z = 1. The logarithmic function log (1  z) has a hypergeometric series in the open unit disk, but it can be continued analytically to the complex plane with a cut along [1, 1) and a branch point at z = 1. Gauss Hypergeometric Function

¼

1 X ðaÞn ðbÞn n z ðcÞn n! n¼0

½12

which is often denoted by F(a, b; c; z). It is a solution of the hypergeometric equation

¼

ðcÞ ðbÞðc  bÞ

Z

1

0

xb1 ð1  xÞcb1 dx ð1  zxÞa

½14

for 0 and 0. This allows to find the analytic continuation from the open unit disk to the complex plane. A useful result is the Gauss summation formula 2 F1 ða; b; c; 1Þ

¼

ðcÞðc  a  bÞ ðc  aÞðc  bÞ

0 The special case for a terminating series is known as the Chu–Vandermonde sum 2 F1 ðn; a; c; 1Þ

¼

ðc  aÞn ðcÞn

Pfaff’s transformation is  a F ða; b; c; zÞ ¼ ð1  zÞ F 2 1 2 1 a; c  b; c;



z z1

2 F1 ða; b; c; zÞ

¼ ð1  zÞcab 2 F1 ðc  a; c  b; c; zÞ

Confluent Hypergeometric Function

The hypergeometric series 1 F1 (a; c; z) defines an entire function in the complex plane and satisfies the differential equation zy00 ðzÞ þ ðc  zÞy0 ðzÞ  ayðzÞ ¼ 0

zð1  zÞy00 ðzÞ þ ½c  ða þ b þ 1Þzy0 ðzÞ  abyðzÞ ¼ 0

2 F1 ða; b; c; zÞ

and Euler’s transformation is

The most famous hypergeometric function is the Gauss hypergeometric function defined for jzj < 1 by the hypergeometric series 2 F1 ða; b; c; zÞ

Euler gave the integral representation

½13

and this solution is regular at z = 0. Obviously, 2 F1 (a, b; c; z) = 2 F1 (b, a; c; z). The six functions 2 F1 (a  1, b; c; z), 2 F1 (a, b  1; c; z), and 2 F1 (a, b; c  1; z) are called contiguous to 2 F1 (a, b; c; z) and there are 15 linear relations (with coefficients which are linear functions of z) between 2 F1 (a, b; c; z) and any two contiguous functions. Two of these relations are ð2a  c  az þ bzÞFða; b; c; zÞ þ ðc  aÞFða  1; b; c; zÞ þ aðz  1ÞFða þ 1; v; c; zÞ ¼ 0 and cða  ðc  bÞzÞFða; b; c; zÞ  acð1  zÞFða þ 1; b; c; zÞ þ ðc  aÞðc  bÞzFða; b; c þ 1; zÞ ¼ 0

½15

This hypergeometric series (and the differential equation) are formally obtained from 2 F1 (a, b; c; z=b) by letting b ! 1, which gives a confluence of two of the singularities at z = 1. This is the reason why the differential equation [15] is known as the confluent hypergeometric equation. The solution ða; c; zÞ ¼ 1 F1 ða; c; zÞ

½16

is called a confluent hypergeometric function, and a second linearly independent solution of [15] is z1c (c  a þ 1, 2  c; z). The function ða; c; zÞ ¼

ð1  cÞ ða; c; zÞ ða  c þ 1Þ ðc  1Þ 1c þ z ða  c þ 1; 2  c; zÞ ðaÞ

½17

Ordinary Special Functions

is therefore also a solution of eqn [15]. The following integral representations hold: ða; c; zÞ ¼

ðcÞ ðaÞðc  aÞ

Z

1

ezx xa1 ð1  xÞca1 dx

0

1 ðaÞ

Z

1

ezx xa1 ð1 þ xÞca1 dx

0

whenever 0. The ‘‘Whittaker functions’’ are defined as M; ðzÞ ¼ ez=2 zc=2 ða; c; zÞ W; ðzÞ ¼ ez=2 zc=2 ða; c; zÞ with  = a þ c=2 and = (c  1)=2. They are a solution of the Whittaker equation  1  1  4 2 y00 ðzÞ þ  þ þ yðzÞ ¼ 0 4 z 4z2 The ‘‘parabolic cylinder functions’’ are also confluent hypergeometric functions. They are given by D ðzÞ ¼ 2=2 ez

2

=4

ð=2; 1=2; z2 =2Þ

¼ 2ð1Þ=2 ez

2

=4

with Cn An1 > 0 for every n  1. For the monic polynomials Pn (x) = pn (x)=kn , with kn = 1=(A0 A1 A2    An1 ) this relation becomes Pnþ1 ðxÞ ¼ ðx  bn ÞPn ðxÞ  a2n Pn1 ðxÞ with bn = Bn and a2n = An1 Cn . This recurrence relation gives rise to a tridiagonal matrix 1 0 b0 a1 0 0 0 0 0 C B a1 b1 a2 0 0 C B 0 C B 0 a2 b2 a3 0 C B J ¼ B 0 0 a3 b 3 a4 0 C C B . . C B @ 0 0 0 a4 . . . . A .. 0 0 0 0 .

whenever 0, and ða; c; zÞ ¼

641

zðð1  Þ=2; 3=2; z2 =2Þ

which is formally symmetric and which is called the ‘‘Jacobi matrix.’’ The spectral measure of this operator, acting on ‘2 (N), is equal to the orthogonality measure whenever this symmetric operator can be extended to a self-adjoint operator. If this is not possible in a unique way – a situation which can occur for unbounded operators only – then every self-adjoint extension of J gives rise to a spectral measure which can be used for the orthogonality conditions [18]. In this case, there are infinitely many positive measures which can be used in the orthogonality relations and all these measures have the same moments Z mn ¼ xn d ðxÞ R

When  is a non-negative integer, one finds Hermite polynomials Hn ðzÞ ¼ 2n=2 ez

2

=2

pffiffiffi Dn ð 2zÞ

Some families of orthogonal polynomials have additional properties which are quite useful in many practical and physical applications, such as the following:

 The derivatives p0n are again a family of orthogonal polynomials (Hahn property).

 The polynomials pn satisfy a second-order linear differential equation of the form

Classical Orthogonal Polynomials A family of polynomials {pn (x), n 2 N}, where pn has degree n, is orthogonal on the real line if there is a positive measure on the real line for which Z

pn ðxÞpm ðxÞd ðxÞ ¼ hn m;n

½18

R

Usually the measure is absolutely continuous, in which case d (x) = w(x) dx with w a non-negative density function on the real line, or is discrete and supported on a finite or at most countable set. Any family of orthogonal polynomials satisfies a ‘‘threeterm recurrence relation’’ xpn ðxÞ ¼ An pnþ1 ðxÞ þ Bn pn ðxÞ þ Cn pn1 ðxÞ

½19

ðxÞy00 ðxÞ þ y0 ðxÞ ¼ n yðxÞ where  is a polynomial of degree at most 2,  is a polynomial of degree 1, both independent of n, and n is a real number (Bochner property).  The polynomials can be obtained by a Rodrigues formula wðxÞpn ðxÞ ¼ Cn

dn ðwðxÞn ðxÞÞ dxn

where w is a non-negative function and  a polynomial of degree at most 2 (Hildebrand property). There are three families of orthogonal polynomials on the real line which have these three properties, and

642 Ordinary Special Functions

each of these three properties characterizes these families. These are the Hermite polynomials, the Laguerre polynomials, and the Jacobi polynomials. In a more general situation when the orthogonality relation is described by a linear functional and the functional is not required to be positive, one has an additional family of Bessel polynomials. The densities w(x) for these families all satisfy a first-order differential equation [(x)w(x)]0 = (x)w(x), where  is a polynomial of degree at most 2 and  a polynomial of degree 1. This equation is known as the ‘‘Pearson equation.’’

Hermite polynomials are relevant for the analysis of the quantum harmonic oscillator, and the lowering and raising operators there correspond to creation and annihilation. Laguerre Polynomials

Laguerre polynomials Ln (x) are for  > 1 orthogonal with respect to the gamma density w(x) = x ex on [0, 1): Z

1

0

Hermite Polynomials

Hermite polynomials Hn (x) are orthogonal with 2 respect to the normal density w(x) = ex : Z 1 2 Hn ðxÞHm ðxÞex dx ¼ 2n n! n;m

ðn þ 1ÞLnþ1 ðxÞ ¼ ð2n þ  þ 1  xÞLn ðxÞ  ðn þ ÞLn1 ðxÞ and the differential equation is xy00 ðxÞ þ ð þ 1  xÞy0 ðxÞ þ nyðxÞ ¼ 0

Hnþ1 ðxÞ ¼ 2xHn ðxÞ  2nHn1 ðxÞ and the polynomials satisfy the second-order differential equation

The functions ‘n (x) = x=2 ex=2 Ln (x) satisfy   þ 1 x 2   ðx‘0n Þ0 þ n þ ‘n ¼ 0 2 4 4x

y00 ðxÞ  2xy0 ðxÞ þ 2nyðxÞ ¼ 0 The functions hn (x) = ex ential equation

2

=2

Hn (x) satisfy the differ-

Differentiation has the effect that   0 Ln ðxÞ ¼ Lþ1 n1 ðxÞ

h00n ðxÞ þ ð2n þ 1  x2 Þhn ðxÞ ¼ 0 The derivatives satisfy Hn0 (x) = 2nHn1 (x) (lowering 2 2 operation) and one also has [ex Hn (x)]0 = ex Hnþ1 (x) (raising operation). The Rodrigues formula is 2

ex Hn ðxÞ ¼ ð1Þn

dn x2 e dxn

and 

0 x ex Ln ðxÞ ¼ ðn þ 1Þx1 ex L1 nþ1 ðxÞ

The Rodrigues formula is

The polynomials can be written as a hypergeometric series Hn ðxÞ ¼ ð2xÞn 2 F0 ðn=2; ðn  1Þ=2; ; 1=x2 Þ

Hn ðxÞ ¼ n!

bn=2c X k¼0

ð1Þk ð2xÞn2k k!ðn  2kÞ!

Their generating function is

n¼0

Hn ðxÞ

x ex Ln ðxÞ ¼

1 dn nþ x ½x e  n! dxn

The hypergeometric expression is n!Ln ðxÞ ¼ ð þ 1Þn 1 F1 ðn;  þ 1; xÞ

or alternatively as

1 X

ðn þ Þ

m;n n!

The Pearson equation is [xw]0 = ( þ 1  x)w so that (x) = x and (x) =  þ 1  x. The recurrence relation is

1

Observe that the density satisfies w0 = 2xw so that  = 1 and (x) = 2x. The recurrence relation is

Ln ðxÞLm ðxÞx ex dx ¼

tn ¼ expð2xt  t2 Þ n!

and the generating function is 1 X n¼0

Ln ðxÞtn ¼ ð1  tÞ1 exp

 xt  t1

Laguerre polynomials occur as eigenfunctions of the hydrogen atom.

Ordinary Special Functions Jacobi Polynomials

Pn(, ) (x)

Jacobi polynomials are orthogonal for the beta density w(x) = (1  x) (1 þ x) on [1, 1] whenever  > 1 and  > 1: Z

1 1

Pð;Þ ðxÞPð;Þ ðxÞð1  xÞ ð1 þ xÞ dx n m þþ1

¼

2 ðn þ  þ 1Þðn þ  þ 1Þ

n;m ðn þ  þ  þ 1Þ 2n þ  þ  þ 1

The Pearson equation is [(1  x2 )w]0 = [     ( þ  þ 2)x]w and the differential equation is ð1  x2 Þy00 ðxÞ þ ½    ð þ  þ 2Þxy0 ðxÞ þ nðn þ  þ  þ 1ÞyðxÞ ¼ 0

643

These functions are more easily written by using the change of variable x = cos and then Tn ( cos ) = cos n and Un ( cos ) = sin (n þ 1) = sin .  The ‘‘Gegenbauer polynomials’’ or ultraspherical polynomials are Jacobi polynomials with equal parameters: Cn ðxÞ ¼ ð2Þn =ð þ 1=2Þn Pð1=2;1=2Þ ðxÞ n Gegenbauer polynomials are involved in the angular or spatial part of the wave function of physical systems in a central potential in both position and momentum space, and in the spatial part of the wave function of hydrogenic systems in momentum space, as well as in the eigenfunctions of several quantum-mechanical potentials, such as the relativistic harmonic oscillator.

Differentiation has the effect h

i0 ðþ1;þ1Þ Pð;Þ ðxÞ ¼ ðn þ  þ  þ 1Þ=2Pn1 ðxÞ n

and h

ð1  xÞ ð1 þ xÞ Pð;Þ ðxÞ n

i0 ð1;1Þ

¼ 2ðn þ 1Þð1  xÞ1 ð1 þ xÞ1 Pnþ1

ðxÞ

The Rodrigues formula is ð1  xÞ ð1 þ xÞ Pnð;Þ ðxÞ i ð1Þn dn h ð1  xÞnþ ð1 þ xÞnþ ¼ n n 2 n! dx In terms of hypergeometric series, one has Pð;Þ ðxÞ ¼ n

ð þ 1Þn n!   n; n þ  þ  þ 1  1  x 2 F1  2 þ1

Observe that one has Pn(, ) ( x) = (1)n Pn(, ) (x). Special cases of the Jacobi polynomials are as follows: 0)  The ‘‘Legendre polynomials’’ Pn (x) = P(0, (x). n

They appear when the Laplacian is separated in spherical coordinates as functions of the polar angle , for which x = cos .  The ‘‘Chebyshev polynomials’’ of the first kind Tn ðxÞ ¼ Pð1=2;1=2Þ ðxÞ=Pð1=2;1=2Þ ð1Þ n n and of the second kind Un ðxÞ ¼ ðn þ 1ÞPð1=2;1=2Þ ðxÞ=Pð1=2;1=2Þ ð1Þ n n

Other Classical Orthogonal Polynomials

Instead of restricting attention to the differential operator D = d=dx, one can also use the (forward) difference operator  for which f (x) = f (x þ 1)  f (x), the divided difference operator  for which  f (x) = f ((x))=(x) with a quadratic function , or certain q-difference operators and look for orthogonal polynomials that satisfy difference equations in the variable x. Together with the three-term recurrence relation (in the degree n), one then has families of polynomials satisfying a bispectral problem. For the difference operator and the divided difference operator, this gives several important families of orthogonal polynomials which all have a hypergeometric representation. These hypergeometric polynomials are usually listed in a table, and each level indicates the number of parameters and/or the order of the hypergeometric function. This table is known as Askey’s table and is given in Figure 1. The extension with q-difference operators involves basic hypergeometric series and q-extensions of classical orthogonal polynomials. ‘‘Charlier polynomials’’ Cn (x; a) are orthogonal with respect to the Poisson distribution 1 X

Cn ðk; aÞCm ðk; aÞ

k¼0

ak ¼ ea =an n;m k!

The recurrence relation is aCnþ1 ðx; aÞ þ ðx  n  aÞCn ðx; aÞ þ nCn1 ðx; aÞ ¼ 0 and the second-order difference equation is ayðx þ 1Þ þ ðn  x  aÞyðxÞ þ xyðx  1Þ ¼ 0

644 Ordinary Special Functions

Wilson

f (x þ i=2)  f (x  i=2) and one has Pn (x; ) = þ1=2 2 sin Pn1 (x; ). They are given by

Racah

Continuous dual Hahn

Continuous Hahn

Hahn

Dual Hahn

MeixnerPollaczek

Jacobi

Meixner

Krawtchouk

Laguerre

Charlier

Pn ðx; Þ

  ð2Þn in n;  þ ix  2i e 2 F1 ¼ 1  e 2 n!

‘‘Hahn and dual Hahn polynomials’’ are orthogonal on a finite set of points. Hahn polynomials are given by 

Hermite Figure 1 Askey’s table.

The forward difference operator has the effect Cn (x; a) = n=aCn1 (x; a) and the backward difference operator rf (x) = f (x)  f (x  1) has the effect r[ax =x!Cn (x;a)] = ax =x!Cnþ1 (x; a). The hypergeometric representation is Cn (x; a) = 2 F0 (n  x; ; 1=a). Observe that the variable x appears as a parameter of the hypergeometric series. ‘‘Krawtchouk polynomials’’ Kn (x; p, N) are orthogonal with respect to the binomial distribution:  N k Kn ðk; p; NÞKm ðk; p; NÞ p ð1  pÞNk k k¼0

N X

¼

ð1Þn n! n p ð1  pÞn n;m ðNÞn

where N is a positive integer and 0 < p < 1. They are given by Kn (x; p, N) = 2 F1 (n, x; N; 1=p) and correspond to Meixner polynomials for which the parameter  is a negative integer. ‘‘Meixner polynomials’’ mn (x; , c) are orthogonal with respect to the negative binomial distribution (Pascal distribution) 1 X

mn ðk; ; cÞmj ðk; ; cÞ

k¼0

ðÞk ck n! ¼

n;j k! cn ðÞn ð1  cÞ

where  > 0 and 0 < c < 1. They are given by mn (x; , c) = 2 F1 (n, x; ; 1  1=c). ‘‘Meixner–Pollaczek polynomials’’ Pn (x; ) are orthogonal on (1, 1): Z 1 Pm ðx; ÞPn ðx; Þeð2Þx jð þ ixÞj2 dx 1

¼

2ðn þ 2Þ ð2 sin Þ2 n!

m;n

where  > 0 and 0 <  < . The appropriate difference operator has an imaginary shift f (x) =

Qn ðx; ; ; NÞ ¼ 3 F2

 n; n þ  þ  þ 1; x  1  þ 1; N

and their orthogonality is with respect to a hypergeometric distribution on {0, 1, . . . , N}. The appropriate difference operator is the (forward) difference operator . They are related to the 3  j symbols or Wigner coefficients that arise when considering angular momenta in two quantum systems. Dual Hahn polynomials are given by  Rn ððxÞ; ; ; NÞ ¼ 3 F2

 n; x; x þ  þ þ 1  1  þ 1; N

where (x) = x(x þ  þ þ 1). They are obtained from the Hahn polynomials by interchanging the roles of n and x. They are orthogonal on the set {(0), (1), . . . , (N)}. The appropriate difference operator is the divided difference operator which acts on f as f ((x))=(x). ‘‘Continuous Hahn and dual Hahn polynomials’’ are orthogonal on the real line. The continuous Hahn polynomials are pn ðx; a; b; c; dÞ ða þ cÞn ða þ dÞn   n! n; n þ a þ b þ c þ d  1; a þ ix  3 F2 1 a þ c; a þ d

¼ in

and the appropriate difference operator is the difference operator with imaginary shift. The continuous dual Hahn polynomials are Sn ðx2 ; a; b; cÞ ¼ ða þ bÞn ða þ cÞn   n; a þ ix; a  ix  1 3 F2 a þ b; a þ c  and the appropriate difference operator is the divided difference operator which acts on f as f (x2 )= x2 . ‘‘Wilson polynomials’’ are the most general system of hypergeometric polynomials satisfying a bispectral problem. All the other classical orthogonal polynomials can be obtained from them by taking

Ordinary Special Functions

appropriate parameters or as limiting cases. They are given by Wn ðx2 ; a; b; c; dÞ ða þ bÞn ða þ cÞn ða þ dÞn   n; n þ a þ b þ c þ d  1; a þ ix; a  ix  ¼ 4 F3 1 a þ b; a þ c; a þ d and for R(a, b, c, d) > 0 (with nonreal parts appearing in conjugate pairs) they are orthogonal on the positive real line with respect to the weight function   ða þ ixÞðb þ ixÞðc þ ixÞðd þ ixÞ  wðxÞ ¼   ð2ixÞ ‘‘Racah polynomials’’ can be obtained from Wilson polynomials when the parameters are such that one of a þ b, a þ c, or a þ d is a negative integer N. They are given by Rn ððxÞ; ; ; ; Þ   n; n þ  þ  þ 1; x; x þ  þ þ 1  ¼ 4 F1 1  þ 1;  þ þ 1;  þ 1 where  þ 1 = N or  þ þ 1 = N or  þ 1 = N, and N is a non-negative integer. They are orthogonal on the finite set {(0), (1), . . . , (N)}, where (x) = x(x þ  þ þ 1). They arise as 6  j symbols in the coupling of three angular momenta. See also: Combinatorics: Overview; Compact Groups and their Representations; Integrable Systems:

645

Overview; Painleve´ Equations; q-Special Functions; Random Matrix Theory in Physics; Separation of Variables for Differential Equations.

Further Reading Abramowitz M and Stegun IA (1964) Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55 (reprinted 1984). New York: Dover. Andrews GE, Askey R, and Roy R (1999) Special Functions, Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge: Cambridge University Press. Bailey WN (1935) Generalized Hypergeometric Series, Cambridge Mathematical Tract, vol. 32. Cambridge: Cambridge University Press. Erde´lyi A, Magnus W, Oberhettinger F, and Tricomi FG (1953–1955) Higher Transcendental Functions, Bateman Manuscript Project, vols. 1–3. New York: McGraw-Hill. Gradshteyn IS and Ryzhik IM (1965) Table of Integrals, Series, and Products, chs 8–9, pp. 904–1080. New York: Academic Press. Koekoek R and Swarttouw R (1998) The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue. Reports of the faculty of Technical Mathematics and Informatics, no. 98–17, Delft University of Technology. Lozier D, Olver F, Clark C, and Boisvert R Digital Library of Mathematical Functions, http://dlmf.nist.gov. Nikiforov AF and Uvarov VB (1988) Special Functions of Mathematics Physics. Basel: Birkha¨user. Nikiforov AF, Suslov SK, and Uvarov VB (1991) Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics. Berlin: Springer. Szego¨ G (1939) Orthogonal Polynomials, American Mathematical Society Colloquium Publications XXIII, 4th edn., 1975. Providence, RI: American Mathematical Society. Watson GN (1922) A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press.