Group integrals and harmonic analysis (pedestrian style)

Transcription

1 Sonderforschungsbereich Transregio 12 Lecture Notes 1 Group integrals and harmonic analysis (pedestrian style) Thomas Guhr 1 and Stefanie Kramer 2 1 Fachbereich Physik, Universität Duisburg Essen 2 Fachbereich Mathematik, Ruhr Universität Bochum lecture given at the Langeoog workshop, February 2008

2 Sonderforschungsbereich Transregio 12 Lecture Notes 2 The topic of group integrals and harmonic analysis is in the middle of activities such as Random Matrix Theory, Supersymmetry, Group Theory and related. This is one motivation for this lecture. Another one is that there are still several open problems. Thus, there is considerable interest in these questions and plenty of ongoing research in many physics and mathematics groups not only in Random Matrix Theory and Disordered Systems but also in High Energy Physics, in statistical aspects of Quantum Chromodynamics, Signal Processing, Mathematical Statistics and, believe it or not, even in Finance and Economics. Coming back to physics and mathematics: There is also an intimate connection to the theory of exactly solvable systems. This is then the third reason why we want to discuss group integrals, because it closely relates to Heiner Kohler s lecture on exactly solvable systems at this workshop. Thus, this talk is also meant to be a preparation for Heiner Kohler s presentation. To a large extent, these lecture notes are a short version of the paper, T. Guhr and H. Kohler, Recursive Construction for a Class of Radial Functions I Ordinary Space, J. Math. Phys. 43 (2002) , math-ph/ , in which you also find historical remarks which are not repeated here. In the Appendix, some group theoretical and classification issues are sketched, following R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, John Wiley and Sons, New York, How that relates to classification issues in Random Matrix Theory (and Random Matrix Theory as such) is discussed in detail in F. Haake, Quantum Signatures of Chaos (2nd ed.), Springer Verlag, Berlin, A comment is in order: to keep the formulae transparent, all unimportant constants are set to one. Hence, if you need the results to be discussed here including those constants, check the literature listed above!

3 Sonderforschungsbereich Transregio 12 Lecture Notes 3 I Introduction A. Bessel functions Bessel functions are angular averages of plane waves. Consider a plane wave exp ( i k r ) (1) in a d dimensional space. Here, r = (x 1,..., x d ) is the position vector and k = (k 1,..., k d ) is the wave vector which points in the direction, in which the wave propagates. In this interpretation, the wave vector k is considered fixed. Later on we will use the fact that r and k can both be viewed as sets of d independent variables. We now want to average over all directions of the position vector, but not over its length r = r. Since the scalar product k r = kr cosϑ (2) depends, apart from the moduli k and r only on the angle ϑ between k and r, but not on the orientation of this pair of vectors in space, the average is χ (d) (kr) = dω r exp ( i k r ). (3) The integration is over the unit sphere Sr d 1 where the index r indicates that it is the sphere in the space of the r coordinates. The corresponding measure on the sphere dω r is referred to as solid angle in physics. By virtue of Eq. (2), the function χ (d) (kr) only depends on one variable, that is, on the product kr of the two moduli. Using ϑ as the polar angle, one can do all integrals except the ϑ integral itself and one arrives at χ (d) (kr) = π 0 exp (ikr cosϑ) sin d 2 ϑdϑ. (4) For d = 2 we find the well known Bessel function J 0 (kr), for d = 3, we find the function that is known as spherical Bessel function j 0 (kr) in physics. In mathematics, such functions are referred to as zonal spherical functions. We notice the property χ (d) (0) = 1. (5) The simple change of variables ξ = cosϑ (for d > 2) reveals a deep structural insight χ (d) (kr) = +1 1 exp (ikrξ) ( 1 ξ 2) (d 3)/2 dξ. (6) Hence, after the integration, odd and even dimensions are distinguished in the following way: For odd dimensions d = 3, 5, 7,... the exponent (d 3)/2 is an integer, implying

4 Sonderforschungsbereich Transregio 12 Lecture Notes 4 that (1 ξ 2 ) (d 3)/2 is a finite (!) polynomial, such that χ (d) (kr) consists of a finite number of terms combining sines, cosines and inverse powers of kr. However, for even dimensions d = 4, 6, 8,... (the case d = 2 is well known anyway), (d 3)/2 is half integer, (1 ξ 2 ) (d 3)/2 is an infinite series and χ (d) (kr) has infinitely many terms! We now turn to the differential equation. The plane wave is an eigenfunction of the Laplacean such that = 2 r 2 (7) exp ( i k r ) = k 2 exp ( i k r ). (8) We average both sides over the directions of k (yes, of k this is the point where we view, as anticipated above, r and k as two independent sets of variables). Hence, we now integrate over the sphere S d 1 k in the space of the vector k, dω k exp ( i k r ) = dω k k 2 exp ( i k r ) dω k exp ( i k r ) = k 2 dω k exp ( i k r ) χ (d) (kr) = k 2 χ (d) (kr) r χ (d) (kr) = k 2 χ (d) (kr). (9) In the second line, we may interchange the integration and the Laplacian. They commute, because the former acts in k space, the latter in r space. Moreover, on the right hand side we also may pull k 2 = k 2 out of the integral, because it only depends on the modulus k and is thus in invariant for the integration over the directions of k, that is, over the sphere Sk d 1. The logic behind this whole computation is that r and k appear on equal footing in the plane wave. Thus, the average over the directions of either k or r must yield the same result, namely χ d (kr). In the last line of Eq. (9), we use that χ d (kr) depends on the modulus of r only. Thus, we may replace the full Laplacean, which contains derivatives with respect to r and to all angles parametrizing the sphere Sr d 1, with its radial part Here we notice that r = 1 r d 1 r rd 1 r = 2 r + d 1 2 r r. (10) d d r = r d 1 drdω r (11) is the the transformation of the flat volume element d d r to spherical, that is, to modulus angles coordinates.

5 Sonderforschungsbereich Transregio 12 Lecture Notes 5 B. Random Matrix Theory Here, only a few very general things are discussed. We represent a physical system by an N N matrix H. Physics puts certain constraints on those matrices. Here, we are interested in the following three cases real symmetric β = 1 diagonalized by O(N) Hermitean β = 2 U(N) self dual (quaternion real) β = 4 USp(2N) A much deeper and comprehensive discussion of these symmetric spaces is given in the Holger Sebert s lecture at this workshop. We notice that there are further relevant cases, Alexander Altland and Martin Zirnbauer put forward classifications. We now setup a statistical model by choosing the elements of H at random from a probability distribution P(H). If A(H) describes a physical quantity, we are interested in averages of the form A = d[h]p(h)a(h). (12) This integral has to be understood as follows: We have the set of all these matrices H in one of the three classes defined above. In other words, we view the elements of these matrices H as variables and integrate over them. We must be a bit careful here. For example, if we want to integrate over all real symmetric matrices H we have to keep in mind that H nm = H mn such that only the elements on the diagonal and in the upper triangle, say, are independent. The d[h] is then just the volume element, that is, the flat measure simply the product of the differentials of all independent variables. In the vast majority of applications, it is reasonable to choose the probability distribution as an invariant function. This is also what we assume here. Hence, P(H) depends on H only via matrix invariants, that is, only on the eigenvalues of H. We diagonalize H = U 1 xu, x = diag (x 1... x N ), (13) where U is in the group specified in the last column of the table above. This means that we have P(H) = P(x). Often, A is an invariant function as well, A(H) = A(x). Thus, it is useful to transform the measure to these coordinates, d[h] = N (x) β d[x]dµ(u) (14)

6 Sonderforschungsbereich Transregio 12 Lecture Notes 6 where the Jacobian is the Vandermonde determinant N (x) = (x n x m ) (15) n<m which annihilates the whole integrand whenever two eigenvalue coincide. What is dµ(u)? It is the invariant measure on a bit less than the group. Here it is sufficient that I briefly explain the concept of an invariant measure for groups. Consider a finite group G. For any function F of the group elements, we have = g GF(g) F(hg) with h G fixed! (16) g G since we sum over all elements and hg G implies that we just reorder terms. This is so because the map G G, g hg is bijective. For a continuous, infinite group, particularly a Lie group, the generalization is dµ(u)f(u) = dµ(u)f(u 0 U) with U 0 G fixed. (17) The invariant measure dµ(u) has to be determined such that this property holds. This is only to give the idea. In mathematics, one now has to distinguish left and right invariance of the measure, the condition under which one yields the other and so on. Thus in our present case, we have A = d[x] N (x) β dµ(u)p(x)a(x) = d[x] N (x) β P(x)A(x), (18) the group integration is trivial! But often one wants to study averages of the following form A H0 = d[h]p(h)a(h H 0 ) (19) with a fixed matrix H 0 which does not commute with H. The group integration is now highly non trivial. It can be cast into a standard form. The integral is a generalized convolution and we have A H0 = d[k] exp (itrh 0 K) p(k)a(k) (20) Since P(H) and A(H) are invariant, p(k) and a(k), their Fourier transforms, are invariant as well. Hence, only exp (itrh 0 K) contains the angles, that is, matrices parametrizing the group, as will become clear in the following.

7 Sonderforschungsbereich Transregio 12 Lecture Notes 7 II Matrix Bessel Functions A. Definition and properties Consider two N N matrices from the same space (β = 1, 2, 4), such that they have the same symmetries. Define the inner product or scalar product trhk = n,mh nm K mn (21) It generalizes the scalar product we used in the case of the Bessel functions, because it is independent of the overall orientation of H and K. Indeed if we rotate both of them with the same element W in the corresponding group, H W 1 HW and K W 1 KW, (22) the scalar product does not change. We then can generalize the plane wave exp(itrhk) is the plane wave in the matrix space. We introduce an eigenvalue angle decomposition H = U 1 xu, K = V 1 kv (23) k = diag(k 1,..., k N ) are the eigenvalues of K and average over the directions U, Φ (β) N (x, k) = dµ(u) exp (itrhk) = dµ(u) exp ( itru 1 xuk ) (24) where we used the invariance of the measure dµ(u) in the last step. Thus, U corresponds to cosϑ in Section A.. The functions Φ (β) N (x, k) are Matrix Bessel functions. They obviously have the symmetry Φ (β) N (x, k) = Φ (β) N (k, x). (25) Completely in analogy to the ordinary Bessel functions, we can derive the differential equation. Define a Laplacean = tr 2 H 2 (26) where / H is a matrix gradient. Formally, this Laplacian can be viewed as a proper product of the two matrix gradients / H which yields again a matrix, followed by the trace operation. The plane waves are, once more, the eigenfunctions exp(itrhk) = trk 2 exp(itrhk) (27)

8 Sonderforschungsbereich Transregio 12 Lecture Notes 8 We now integrate both sides over V which diagonalizes K and find dµ(v ) exp(itrhk) = dµ(v )trk 2 exp(itrhk) dµ(v ) exp(itrhk) = trk 2 dµ(v ) exp(itrhk) Φ (β) N (x, k) = trk2 Φ (β) N (x, k) x Φ (β) N (x, k) = trk 2 Φ (β) N (x, k). (28) Again, we employ in the second line that integration and Laplacian commute, because the act in different spaces the integration in K space and the Laplacian in H space. Moreover, trk 2 = trk 2 does not contain the angles, that is, the variables V parametrizing the group manifold and can thus be pulled out of the integral. In the last step we use that Φ (β) N (x, k) only depends on the eigenvalues x. The Laplacean has derivatives with respect to eigenvalues x and angles U. The angular part is in analogy to the ordinary Bessel functions given by x = = N n=1 N n=1 1 N (x) β 2 x 2 n + n<m N (x) β x n x n ( β x n x n x m Unfortunately, the letter is used for x as well as for N (x). x m ). (29) This differential equation establishes a direct link to the theory of interacting particles see Heiner Kohler s lecture. B. Recursive construction Physicists like and often need explicit results. Unfortunately, the calculation of Φ (β) N (x, k) for β = 1, 4 is very difficult. We now want to show that the matrix Bessel function satisfy the remarkable recursion relation for β = 1, 2, 4 where Φ (β) N (x, k) = x and the measure is given by dµ(x ) exp (i(trx trx )k N ) Φ (β) N 1 (x, k) (30) = diag ( x 1,...,x N 1 ) k = diag ( k, k N ), k = diag (k 1,...,k N 1 ) (31) dµ(x ) = N 1(x ) (x n x m (x) )(β 2)/2 d[x ]. (32) β 1 N n,m

9 Sonderforschungsbereich Transregio 12 Lecture Notes 9 the integration over x has the limits x n x n x n+1. Before sketching the derivation, let us discuss this formula, which we derived some years ago. Consider β = 2, that is, the unitary case. Then the term coupling x and x becomes simply unity and the integral is elementary and yields a re-derivation of a well known result, Φ (2) N (x, k) = det [exp (ix nk m )] n,m=1,...,n N (x) N (k). (33) This is the celebrated Itzykson Zuber integral which coincides with the unitary Harish Chandra integral, see the references given in the article mentioned in the foreword. We now sketch the derivation. We use an element V = diag (Ṽ, 1) with Ṽ being in the N 1 dimensional subgroup we have by invariance of the measure Φ (β) N (x, k) = dµ(u) exp ( itru 1 xuk ) = dµ(ṽ ) dµ(u) exp ( itru 1 xuk ) = dµ(u) dµ(ṽ ) exp ( itru 1 xuv 1 kv ) (34) We then write U = [U 1 U 2 U N 1 U N ] = [B U N ] (35) where the N (N 1) rectangular matrix B collects the first N 1 columns of U. The last one, U N, parametrises a unit sphere in the N dimensional space. We have B B = 1 N 1 BB = Moreover, the scalar product becomes N 1 n=1 U n U n = 1 N U N U N (36) tru 1 xuv 1 kv = tr H K + H NN k N (37) with H NN = U NxU N and with the (N 1) (N 1) matrices H = B xb and K = Ṽ 1 kṽ. (38) The measure decomposes into the measure on the sphere and a conditional measure, because all U n are orthonormal to each other, dµ(u) = dµ(u N )dµ(b). (39) Collecting everything, we have Φ (β) N (x, k) = dµ(u N ) exp (ih NN k N ) dµ(ṽ ) dµ(b) exp ( itr H K ). (40)

10 Sonderforschungsbereich Transregio 12 Lecture Notes 10 We now want to do the U N integral. To this end, we change to variables x by defining the eigenequation ( ) ( 1N U N U n x 1N U N U ) N E n = x ne n, n = 1,...,(N 1) (41) with some eigenvectors E n. From that equation one finds, first, the relation and, second, that the x are the eigenvalues of H, trx trx = U N xu N = H NN (42) H = B xb = Ũ 1 x Ũ (43) with some Ũ which is, by construction, in the (N 1) group. Third, we find dµ(u N ) = dµ(x ). (44) The differentials of some unimportant phases are contained in dµ(u N ), but not in dµ(x ). Collecting everything, we arrive at Φ (β) N (x, k) = dµ(x ) exp (i(trx trx )) k N dµ(b) dµ(ṽ ) exp ( ) itrũ 1 x ŨṼ 1 kṽ = dµ(x ) exp (i(trx trx )k N ) dµ(b)φ (β) N 1(x, k) = dµ(x ) exp (i(trx trx )k N ) Φ (β) N 1 (x, k) (45) which is the assertion. Our coordinates x are related to, but different from the Gelfand Tzetlin coordinates. We call ours radial Gelfand Tzetlin coordinates, because they map an integration over a sphere onto an integration over radial coordinates. The details are given in the article mentioned in the foreword.

11 Sonderforschungsbereich Transregio 12 Lecture Notes 11 III Spherical Functions beyond Matrices and Lie Groups We have derived a differential equation for Φ (β) N (x, k) for β = 1, 2, 4. We can now view β as an arbitrary real number β > 0 and ask for solutions with the symmetry condition Φ (β) N (x, k) = Φ(β) N (k, x). We have shown that our recursion formula gives these functions even for arbitrary β > 0. Consider now all even β = 2p with p = 1, 2, 3,.... Then the measure dµ(x ) becomes a purely rational function. We conjectured that this implies that Φ (β) N (x, k) = Φ (2p) N (x, k) consist of a finite number of terms only, just like the Bessel functions for odd dimensions d, χ (d) (kr). We have calculated that explicitly for β = 4 up to N = 4.

12 Sonderforschungsbereich Transregio 12 Lecture Notes 12 Appendix Symmetric Spaces and Classification Issues We try to give a quick presentation of the issue, first, in a mathematics and, second, in a physics language. This appendix is thus meant to provide some kind of dictionary. We find the Lie group by exponentiating the Lie algebra. Hence we have for the Lie algebra of the unitary group A = su(n). (A.1) These are the Hermitean N N matrices. Any operator T that maps a Lie algebra onto itself is called an automorphism, obeying (X, Y ) = (T X, T Y ) (A.2) where (, ) is the inner product, that is apart from factors, the trace in the defining representation. Any automorphism with T 2 = 1 is called involutive automorphism. It has eigenvalues ±1, for (A.3) (T 1)(T + 1) = T 2 1 = 0. (A.4) Now we decompose a compact and simple Lie algebra g into its subspaces such that g = k p with T k = 1 T p = 1 (A.5) These subspaces are orthogonal, for (a, b) = (T a, T b) = (+a, b) = (a, b) = 0 a k, b p. (A.6) One then shows that k is a subalgebra of g. Weyl s unitary trick is: define g = k ip (A.7) as a new algebra. It is closed under commutation with real (!) structure constants. The decomposition is known as Cartan s decomposition. In the regular or adjoint representation we have: R(k) is real and skew symmetric, R(p) is real and skew symmetric, and R(ip) is real and symmetric.

13 Sonderforschungsbereich Transregio 12 Lecture Notes 13 One shows that there are only three types of involutive automorphisms I. complex conjugation T = K, where K is the operation of complex conjugation II. T = 0 +1 N/2 1 N/ N M 0 III. T = 0 1 M They obey T gt 1 = g and T 2 = 1 N. (symplectic) (chiral) (A.8) We now apply I, that is, complex conjugation to SU(N) and find k = so(n) ip generates SO(N) real symmetric matrices. Then we apply II, that is, the symplectic conjugation to SU(N) (here, N must be even) and find k = usp(n) ip generates USp(N) Hermitean self dual. In physics, one views that in the framework of time reversal. For spinless particles, the time reversal operator is just the complex conjugation T = K T 2 = +1 (A.9) which then leads as above to the real symmetric matrices. For 1/2 spin particles, one wants that the total (!) angular momentum behaves under time reversal as j = l + s T jt 1 = j. This means that T must also act in spin space. A useful choice is T = iσ y K = exp (iπσ y /2)K with σ y = [ 0 i i 0 ] (A.10) (A.11) (A.12)

14 Sonderforschungsbereich Transregio 12 Lecture Notes 14 such that T 2 = 1. (A.13) Hence T is always antiunitary! We notice that the sign convention differs from the choice (A.3) in the mathematics literature. For M particles we have such that T = iσ 1y iσ My K = exp (iπs y /2 h) K (A.14) T 2 = ( 1) M. (A.15) In the case of an odd particle number, we have a half integer total angular momentum J and T 2 = 1. This leads to Kramers degeneracy, because if ψ is an eigenfunction of H, then T ψ is as well with same eigenvalue! Write the basis as { n, T n }, then a submatrix of the Hamiltonian is [ ] m H n m H T n h mn = (A.16) T m H n T m H T n and it can be shown that with τ = i σ = h mn = h (0) mn ([ 0 i i 0 ], 3 a=1 h (α) mn τ α [ ], [ i 0 0 +i ]) (A.17), (A.18) where all h (α) mn, α = 1, 2, 3, 4 are real and hmn = h + nm (A.19) such that h (0) mn = h(0) nm and h(α) mn = h(α) nm, α = 1, 2, 3. In summary, a time reversal invariant system is represented by Hermitean self dual (quaternion real) matrices, if J is 1/2 odd integer and not (!) invariant under rotations real symmetric matrices if J is integer (rotation invariance is no issue then) or if J is 1/2 odd integer and invariant under rotations (Kramers degeneracies are lifted).