the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

the inverse segal bargmann transform for compact lie groups in this paper i give a new inversion formula theorem 1 for the generalized segal bargmann transform introduced in b c hall j funct anal 122 1994 103 151 the inversion formula may be viewed as a formula for theinverseheat operator for a compact lie group i then use this formula to give a new direct proof of the unitary of thek invariant form of the segal bargmann transform theorem 2 the proof of the inversion formula relies on an identity theorem 5 which relates the laplacian for a compact lie groupkto the laplacian for the non compact dual symmetric space kc k

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

devil staircase behavior of dynamical invariants in chaotic scattering a crisis in chaotic scattering is characterized by the merging of two or more nonattracting chaotic saddles the fractal dimension of the resulting chaotic saddle increases through the crisis we present a rigorous analysis for the behavior of dynamical invariants associated with chaotic scattering by utilizing a representative model system that captures the essential dynamical features of crisis our analysis indicates that the fractal dimension and other dynamical invariants are a devil staircase type of function of the system parameter our results can also provide insight for similar devil staircase behaviors observed in the parametric evolution of chaotic saddles of general dissipative dynamical systems and in communicating with chaos 2000 elsevier science b v all rights reserved

on shannon entropies in μ deformed segal bargmann analysis we consider a deformation of the segal bargmann transform which is a unitary map from a deformed quantum configuration space onto a deformed quantum phase space the deformed segal bargmann space both of these hilbert spaces have canonical orthonormal bases we obtain explicit formulas for the shannon entropy of some of the elements of these bases we also consider two reverse log sobolev inequalities in the deformed segal bargmann space which have been proved in a previous work and show that a certain known coefficient in them is the best possible

short time behavior of hermite functions on compact lie groups abstract let p t x be the gaussian heat kernel on r n at time t the classical hermite polynomials at time t may be defined by a rodriguez formula given by h x t p t x p t x where is a constant coefficient differential operator on r n recent work of gross 1993 and hijab 1994 has led to the study of a new class of functions on a general compact lie group g in analogy with the r n case these hermite functions on g are obtained by the same formula wherein p t x is now the heat kernel on the group x is replaced by x 1 and is a right invariant differential operator let g be the lie algebra of g composing a hermite function on g with the exponential map produces a family of functions on g we prove that these functions scaled appropriately in t approach the classical hermite polynomials at time 1 as t tends to 0 both uniformly on compact subsets of g and in l p g where 1 p is a gaussian measure on g similar theorems are established when g is replaced by g k where k is some closed connected subgroup of g

uniqueness of ground states for schrodinger operators over loop groups abstract over the space l of based loops in a compact connected lie group g there is a natural gradient operator induced by the left action on l of the hilbert lie group k 0 which consists of finite energy elements of l denote by pinned brownian motion measure on l the schrodinger operator v acting in l 2 l is shown to have a unique ground state over each homotopy class in l the proof of uniqueness is reduced to the proof of ergodicity of the left action of k 0 on l for a simply connected group g of compact type ergodicity in turn is proved by first characterizing the ito expansion coefficients of k 0 invariant functions and then regularizing such functions via a smooth approximation of multiple stratonovich integrals as a byproduct of this method one obtains an isometric isomorphism of l 2 g heat kernel measure at time one with a natural completion of the universal enveloping algebra of g thus extending to the noncommutative case the well known isometry between l 2 r d gauss measure and the space of symmetric tensors over r d

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

hermite functions on compact lie groups ii abstract the usual formula for hermite polynomials on rd is extended to a compact lie group g yielding an isometry of l2 g p1 where p1 is the heat kernel measure at time one with a natural completion of the universal enveloping algebra of g the existence of such an isometry was first established by l gross here we present a simplified proof as well as the explicit form of the isometry

coherent state transforms for spaces of connections abstract the segal bargmann transform plays an important role in quantum theories of linear fields recently hall obtained a non linear analog of this transform for quantum mechanics on lie groups given a compact connected lie groupgwith its normalized haar measure h the hall transform is an isometric isomorphism froml2 g h to h g c l2 g c whereg c the complexification ofg h g c the space of holomorphic functions ong c and an appropriate heat kernel measure ong c we extend the hall transform to the infinite dimensional context of non abelian gauge theories by replacing the lie groupgby a certain extension of the space a g of connections modulo gauge transformations the resulting coherent state transform provides a holomorphic representation of the holonomyc algebra of real gauge fields this representation is expected to play a key role in a non perturbative canonical approach to quantum gravity in 4 dimensions

the complex time segal bargmann transform abstract we introduce a new form of the segal bargmann transform for a connected lie group k of compact type we show that the heat kernel t x t 0 x k has a space time analytic continuation to a holomorphic function c z re 0 z k c where k c is the complexification of k the new transform is defined by the integral b f z k c z k 1 f k d k z k c if s 0 and d s s the disk of radius s centered at s this integral defines a holomorphic function on k c for each f l 2 k s we construct a heat kernel density s on k c such that for all s as above b s b l 2 k s is an isometric isomorphism from l 2 k s onto the space of holomorphic functions in l 2 k c s when t s the transform b t t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type when t 0 2 s the transform b s t coincides with the one introduced by the first two authors

regular articlehermite functions on compact lie groups i the usual formula for hermite polynomials on rd is extended to a compact lie group g yielding an isometry of l2 g p1 where p1 is the heat kernel measure at time one with a natural completion of the universal enveloping algebra of g the existence of such an isometry was first established by l gross here we present a simplified proof as well as the explicit form of the isometry

the segal bargmann transform on a symmetric space of compact type abstract we study the segal bargmann transform on a symmetric space x of compact type mapping l 2 x into holomorphic functions on the complexification x c we invert this transform by integrating against a dual heat kernel measure in the fibers of a natural fibration of x c over x we prove that the segal bargmann transform is an isometry from l 2 x onto the space of holomorphic functions on x c which are square integrable with respect to a natural measure these results extend those of b hall in the compact group case

yang mills theory and the segal bargmann transform we use a variant of the segal bargmann transform to study canonically quantized yang mills theory on a space time cylinder with a compact structure group k the non existent lebesgue measure on the space of connections is approximated by a gaussian measure with large variance the segal bargmann transform is then a unitary map from the l 2 space over the space of connections to a holomorphicl 2 space over the space of complexified connections with a certain gaussian measure this transform is given roughly bye t1a 2 followed by analytic continuation here 1a is the laplacian on the space of connections and is the hamiltonian for the quantized theory on the gauge trivial subspace consisting of functions of the holonomy around the spatial circle the segal bargmann transform becomes e t1 k 2 followed by analytic continuation where 1k is the laplacian for the structure group k this result gives a rigorous meaning to the idea that 1a reduces to 1k on functions of the holonomy by letting the variance of the gaussian measure tend to infinity we recover the standard realization of the quantized yang mills theory on a space time cylinder namely 1 1k is the hamiltonian and l 2 k is the hilbert space as a byproduct of these considera tions we find a new one parameter family of unitary transforms from l 2 k to certain holomorphicl 2 spaces over the complexification of k this family of transformations interpolates between the two previously known unitary transformations our work is motivated by results of landsman and wren and uses probabilistic techniques similar to those of gross and malliavin

the segal bargmann transform for path groups letkbe a connected lie group of compact type and letw k denote the set of continuous paths ink starting at the identity and with time interval 0 1 thenw k forms an infinite dimensional group under the operation of pointwise multiplication let denote the wiener measure onw k we construct an analog of the segal bargmann transform forw k letkcbe the complexification ofk w kc the set of continuous paths inkcstarting at the identity and the wiener measure onw kc our transform is a unitary map ofl2 w k onto the holomorphic subspace ofl2 w kc by analogy with the classical transform our transform is given by convolution with the wiener measure followed by analytic continuation we prove that the transform forw k is nicely related by means of the ito map to the classical segal bargmann transform for the path space in the lie algebra ofk

on the kakutani ito segal gross and segal bargmann hall isomorphisms recently gross has shown that the kakutani ito segal isomorphism theorem has an extension from the setting of gaussian measure on a vector space to heat kernel measure pt on a simply connected lie group g of compact type the isomorphism relates l2 pt to a certain completion of the universal enveloping algebra of g lie g gross proves this result using the kakutani ito segal theorem and an infinite dimensional calculus associated to g valued brownian motion hijab has greatly simplified and clarified gross proof hiiab s proof avoids most but not all of the infinite dimensional analysis in the original proof in this paper we will build on hijab s proof to give a completely finite dimensional non probabilistic proof of gross isomorphism theorem the proof given here relies heavily on hall s beautiful extension of the segal bargmann transformation to the setting of compact lie groups this theorem relating l2 pt to a certain l2 space of holomorphic functions l2 gc h on the complexified lie group gc will also be generalized to lie groups of compact type in the process it is shown how to characterize in terms of summability conditions on all the derivatives at the identity in gc those holomorphic functions which are in l2 gc

isometry theorem for the segal bargmann transform on a noncompact symmetric space of the complex type we consider the segal bargmann transform on a noncompact symmetric space of the complex type we establish isometry and surjectivity theorems for the transform in a form as parallel as possible to the results in the dual compact case the isometry theorem involves integration over a tube of radius r in the complexification followed by analytic continuation with respect to r a cancellation of singularities allows the relevant integral to have a nonsingular extension to large r even though the function being integrated has singularities

heat kernel analysis and cameron martin subgroup for infinite dimensional groups abstract the heat kernel measure t is constructed on gl h the group of invertible operators on a complex hilbert space h this measure is determined by an infinite dimensional lie algebra g and a hermitian inner product on it the cameron martin subgroup g cm is defined and its properties are discussed in particular there is an isometry from the l 2 t closure of holomorphic polynomials into a space h t g cm of functions holomorphic on g cm this means that any element from this l 2 t closure of holomorphic polynomials has a version holomorphic on g cm in addition there is an isometry from h t g cm into a hilbert space associated with the tensor algebra over g the latter isometry is an infinite dimensional analog of the taylor expansion as examples we discuss a complex orthogonal group and a complex symplectic group

segal bargmann transform functional calculus on matrix spaces and the theory of semi circular and circular systems letmd c be the space ofd dcomplex matrices and xdbe the subspace of hermitian matrices we study the segal bargmann transform of functions on xd with values inmd c which are given by functional calculus we show that whend the transform of such a map becomes close to the space of holomorphic functional calculus onmd c and that this yields in the limit an isometry between thel2space of wigner s semi circle distribution and the hardy space of the disk we relate this isometry to voiculescu s theory of circular and semi circular systems and we study its analogue when the segal bargmann transform is replaced by the hall transform on unitary groups of large dimensions

asymptotic behaviour of reproducing kernels of weighted bergman spaces let q be a domain in c f a nonnegative and g a positive function on q such that 1 g is locally bounded a2 the space of all holomorphic functions on q square integrable with respect to the measure f0gda where da is the 2n dimensional lebesgue measure and k x y the reproducing kernel for a2 it has been known for a long time that in some special situations such as on bounded symmetric domains q with g 1 and f the bergman kernel function the formula lim k x x 1 0 1 f x ae 00 holds true this fact even plays a crucial role in berezin s theory of quantization on curved phase spaces in this paper we discuss the validity of this formula in the general case the answer turns out to depend on loosely speaking how well the function log f can be approximated by certain pluriharmonic functions lying below it for instance holds if log f is convex and hence can be approximated from below by linear functions for any function g counterexamples are also given to show that in general may fail drastically or even be true for some x and fail for the remaining ones finally we also consider the question of convergence of k0 x y 1 0 for x y which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of f for instance if f is not real analytic at some point then kc x y must have zeroes for all at sufficiently large

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis