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

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

adapted complex structures and the geodesic flow in this paper we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact real analytic riemannian manifold motivated by the complexifier approach of t thiemann as well as certain formulas of v guillemin and m stenzel we obtain the polarization associated to the adapted complex structure by applying the imaginary time geodesic flow to the vertical polarization meanwhile at the level of functions we show that every holomorphic function is obtained from a function that is constant along the fibers by composition with the imaginary time geodesic flow we give several equivalent interpretations of this composition including a convergent power series in the vector field generating the geodesic flow

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

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

direct images fields of hilbert spaces and geometric quantization geometric quantization often produces not one hilbert space to represent the quantum states of a classical system but a whole family h s of hilbert spaces and the question arises if the spaces h s are canonically isomorphic axelrod et al j diff geo 33 787 902 1991 and hitchin commun math phys 131 347 380 1990 suggest viewing h s as fibers of a hilbert bundle h introduce a connection on h and use parallel transport to identify different fibers here we explore to what extent this can be done first we introduce the notion of smooth and analytic fields of hilbert spaces and prove that if an analytic field over a simply connected base is flat then it corresponds to a hermitian hilbert bundle with a flat connection and path independent parallel transport second we address a general direct image problem in complex geometry pushing forward a hermitian holomorphic vector bundle e to y along a non proper map y to s we give criteria for the direct image to be a smooth field of hilbert spaces third we consider quantizing an analytic riemannian manifold m by endowing tm with the family of adapted kahler structures from lempert and sz ke bull lond math soc 44 367 374 2012 this leads to a direct image problem when m is homogeneous we prove the direct image is an analytic field of hilbert spaces for certain such m but not all the direct image is even flat which means that in those cases quantization is unique

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

coherent state transforms and vector bundles on elliptic curves abstract we extend the coherent state transform cst of hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves we show that by applying the cst to appropriate distributions we obtain the space of level k rank n and genus one non abelian theta functions with the unitarity of the cst transform being preserved furthermore the shift in the level k k n appears in a natural way in this finite dimensional framework

probability of fuzzy events defined as denumerable additivity measure abstract in this paper the probability of fuzzy events is defined as a denumerable additivity measure this definition is based on a non conventional approach of separativity between fuzzy subsets the presented measure fulfils all properties analogous to the properties of classical probability in the crisp case further the notion of conditional probability of fuzzy events complete fuzzy repartition and independent fuzzy events are defined by means of the probability measure considered here connections between all the above notions are presented in this paper too finally the bayes formula is proved for the fuzzy case

quantization of abelian varieties distributional sections and the transition from kahler to real polarizations abstract we study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization real and mixed polarizations are interpreted as degenerate complex structures using a weak version of the equations of covariant constancy and the weil brezin expansion to describe distributional sections we give a unified analytical description of the quantization spaces for all non negative polarizations the blattner kostant sternberg bks pairing maps between half form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of sp 2 g r moreover these maps are shown to be unitary

coherent state transforms and abelian varieties we extend the coherent state transform cst of hall to the context of abelian varieties by considering them as quotients of the complexification of the abelian group k u 1 g we show that this transform applied to appropriate distributions on k gives all classical theta functions and that by defining on this space of theta functions an inner product related to the k averaged heat kernel the unitarity of the cst transform is still preserved

coherent states for n dimensional euclidean groups e n and their application the n dimensional euclidean group the group of rotations and translations of rn is the natural canonical group for quantizing a system whose configuration space is the sphere sn 1 n 2 since every unitary irreducible representation of e n is nonsquare integrable it follows that coherent states involving the whole group do not exist for such representations instead coherent states are defined based on unitary reducible representations of the group and it will be shown that with such states quantum dynamics on the sphere admits a path integral representation involving novel first order classical actions defined on e n as well as integrations over the entire group manifold at each time slice irreducibility of the quantization is achieved through a final limiting process

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

geometric quantization parallel transport and the fourier transform in quantum mechanics the momentum space and position space wave functions are related by the fourier transform we investigate how the fourier transform arises in the context of geometric quantization we consider a hilbert space bundle mathcal h over the space mathcal j of compatible complex structures on a symplectic vector space this bundle is equipped with a projectively flat connection we show that parallel transport along a geodesic in the bundle mathcal h to mathcal j is a rescaled orthogonal projection or bogoliubov transformation we then construct the kernel for the integral parallel transport operator finally by extending geodesics to the boundary for which the metaplectic correction is essential we obtain the segal bargmann and fourier transforms as parallel transport in suitable limits

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

boolean representations of fuzzy quantum spaces abstract let f be a fuzzy quantum space by a boolean representation of f we mean a boolean algebra f with a mapping h f onto f such that for each state s and for each observable y on f there is a state probability measure s and an observable y on f such that the following diagram commutes r is the real line and b r is the borel algebra we prove that 1 a boolean representation always exists 2 there is a minimal and a maximal boolean representation 3 a minimal boolean representation possesses the uniqueness property for observables further all states resp observables on f correspond to states resp observables on f this makes it possible to transfer many results from boolean algebras to fuzzy quantum spaces we argue that the noncompatibility encountered in fuzzy quantum spaces originates rather from the ambiguous representation of events by fuzzy sets than from the systems they describe

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

partial coherent state transforms g t invariant kahler structures and geometric quantization of cotangent bundles of compact lie groups abstract in this paper we study the analytic continuation to complex time of the hamiltonian flow of certain g t invariant functions on the cotangent bundle of a compact connected lie group g with maximal torus t namely we will take the hamiltonian flows of one g g invariant function h and one g t invariant function f acting with these complex time hamiltonian flows on g g invariant kahler structures gives new g t invariant but not g g invariant kahler structures on t g we study the hilbert spaces h corresponding to the quantization of t g with respect to these non invariant kahler structures on the other hand by taking the vertical schrodinger polarization as a starting point the above g t invariant hamiltonian flows also generate families of mixed polarizations p 0 c im 0 each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a kahler structure on the leaves of a foliation of t g the geometric quantization of t g with respect to these mixed polarizations gives rise to unitary partial coherent state transforms corresponding to ksh maps as defined in 11 12

coherent state transforms and the mackey stone von neumann theorem mackey showed that for a compact lie group k the pair k c 0 k has a unique non trivial irreducible covariant pair of representations we study the relevance of this result to the unitary equivalence of quantizations for an infinite dimensional family of k k invariant polarizations on t k the kahler polarizations in the family are generated by complex time hamiltonian flows applied to the schrodinger vertical real polarization the unitary equivalence of the corresponding quantizations of t k is then studied by considering covariant pairs of representations of k defined by geometric prequantization and of representations of c 0 k defined via heisenberg time evolution followed by time geometric quantization induced evolution we show that in the semiclassical and large imaginary time limits the unitary transform whose existence is guaranteed by mackey s theorem can be approximated by composition of the time geometric quantization induced evolution with the time evolution

the bargmann representation for the quantum mechanics on a sphere the bargmann representation is constructed corresponding to the coherent states for a particle on a sphere introduced in kowalski and rembielinski j phys a 33 6035 2000 the connection is discussed between the introduced formalism and the standard approach based on the hilbert space of square integrable functions on a sphere s2

complex time evolution in geometric quantization and generalized coherent state transforms abstract for the cotangent bundle t k of a compact lie group k we study the complex time evolution of the vertical tangent bundle and the associated geometric quantization hilbert space l 2 k under an infinite dimensional family of hamiltonian flows for each such flow we construct a generalized coherent state transform cst which is a unitary isomorphism between l 2 k and a certain weighted l 2 space of holomorphic functions for a particular set of choices we show that this isomorphism is naturally decomposed as a product of a heisenberg type evolution for complex time within l 2 k followed by a polarization changing geometric quantization evolution for complex time in this case our construction yields the usual generalized segal bargmann transform of hall we show that the infinite dimensional family of hamiltonian flows can also be understood in terms of thiemann s complexifier method which generalizes the construction of adapted complex structures

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

on the reflection type decomposition of the adjoint reduced phase space of a compact semisimple lie group we consider a system with symmetries whose configuration space is a compact lie group acted upon by inner automorphisms the classical reduced phase space of this system decomposes into connected components of orbit type subsets to investigate hypothetical quantum effects of this decomposition one has to construct the associated costratification of the hilbert space of the quantum system in the sense of huebschmann in the present paper instead of the decomposition by orbit types we consider the related decomposition by reflection types conjugacy classes of reflection subgroups these two decompositions turn out to coincide e g for the classical groups su n and sp n we derive defining relations for reflection type subsets in terms of irreducible characters and discuss how to obtain from that the corresponding costratification of the hilbert space of the system to illustrate the method we give explicit results for some low rank classical groups

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

unitarity in quantization commutes with reduction let m be a compact kahler manifold equipped with a hamiltonian action of a compact lie group g in this paper we study the geometric quantization of the symplectic quotient m g guillemin and sternberg invent math 67 515 538 1982 have shown under suitable regularity assumptions that there is a natural invertible map between the quantum hilbert space over m g and the g invariant subspace of the quantum hilbert space over m

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

projective flatness in the quantisation of bosons and fermions we compare the quantisation of linear systems of bosons and fermions we recall the appearance of projectively flat connection and results on parallel transport in the quantisation of bosons we then discuss pre quantisation and quantisation of fermions using the calculus of fermionic variables we define a natural connection on the bundle of hilbert spaces and show that it is projectively flat this identifies up to a phase equivalent spinor representations constructed by various polarisations we introduce the concept of metaplectic correction for fermions and show that the bundle of corrected hilbert spaces is naturally flat we then show that the parallel transport in the bundle of hilbert spaces along a geodesic is a rescaled projection provided that the geodesic lies within the complement of a cut locus finally we study the bundle of hilbert spaces when there is a symmetry

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

holomorphic sobolev spaces and the generalized segal bargmann transform we consider the generalized segal bargmann transform ct for a compact group k introduced in b c hall j funct anal 122 1994 103 151 let kc denote the complexification of k we give a necessary and sufficient pointwise growth condition for a holomorphic function on kc to be in the image under ct of c 1 k we also characterize the image under ct of sobolev spaces on k the proofs make use of a holomorphic version of the sobolev embedding theorem

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

the hilbert space costratification for the orbit type strata of su 2 lattice gauge theory we construct the hilbert space costratification of g su 2 quantum gauge theory on a finite spatial lattice in the hamiltonian approach we build on previous work f furstenberg g rudolph and m schmidt j geom phys 119 66 81 2017 where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture in this picture each element of the classical stratification corresponds to the zero locus of a finite subset pi of the algebra r of g invariant representative functions on gcn viewing the invariants as multiplication operators p i on the hilbert space h the union of their images defines a subspace of h whose orthogonal complement h is the element of the costratification corresponding to to construct h one has to determine the images of the p i explicitly to accomplish this goal we construct an orthonormal basis in h and determine the multiplication law for the basis elements that is we determine the structure constants of r in this basis this part of our analysis applies to any compact lie group g for g su 2 the above procedure boils down to a problem in combinatorics of angular momentum theory using this theory we obtain the union of the images of the operators p i as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of wigner s 3nj symbols the latter are further expressed in terms of 9j symbols using these techniques we are also able to reduce the eigenvalue problem for the hamiltonian of this theory to a problem in linear algebra we construct the hilbert space costratification of g su 2 quantum gauge theory on a finite spatial lattice in the hamiltonian approach we build on previous work f furstenberg g rudolph and m schmidt j geom phys 119 66 81 2017 where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture in this picture each element of the classical stratification corresponds to the zero locus of a finite subset pi of the algebra r of g invariant representative functions on gcn viewing the invariants as multiplication operators p i on the hilbert space h the union of their images defines a subspace of h whose orthogonal complement h is the element of the costratification corresponding to to construct h one has to determine the images of the p i explicitly to accomplish this goal we construct an orthonormal basis in h and determine the multiplication law for the basis elements that is we determine the structure constants of r in thi

constraints reduction and quantization theorems are proved that establish the unitary equivalence of the extended and reduced phase space quantizations of a constrained classical system with symmetry several examples are presented

berezin toeplitz quantization on lie groups let k be a connected compact semisimple lie group and kc its complexification the generalized segal bargmann space for kc is a space of square integrable holomorphic functions on kc with respect to a k invariant heat kernel measure this space is connected to the schrodinger hilbert space l 2 k by a unitary map the generalized segal bargmann transform this paper considers certain natural operators on l 2 k namely mul tiplication operators and differential operators conjugated by the general ized segal bargmann transform the main results show that the resulting operators on the generalized segal bargmann space can be represented as toeplitz operators the symbols of these toeplitz operators are expressed in terms of a certain subelliptic heat kernel on kc i also examine some of the results from an infinite dimensional point of view based on the work of l gross and p malliavin

the segal bargmann transform for noncompact symmetric spaces of the complex type we consider the generalized segal bargmann transform defined in terms of the heat operator for a noncompact symmetric space of the complex type for radial functions we show that the segal bargmann transform is a unitary map onto a certain l 2 space of meromorphic functions for general functions we give an inversion formula for the segal bargmann transform involving integration against an unwrapped version of the heat kernel for the dual compact symmetric space both results involve delicate cancellations of singularities

on the bks pairing for kahler quantizations of the cotangent bundle of a lie group a natural one parameter family of kahler quantizations of the cotangent bundle t k of a compact lie group k taking into account the half form correction was studied in fmmn in the present pa per it is shown that the associated blattner kostant sternberg bks pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work from the point of view of adpw the bks pairing map is a composition of uni tary coherent state transforms of k introduced in ha1 continuity of the hermitian structure on the quantum bundle in the limit when one of the kahler polarizations degenerates to the vertical real polar ization leads to the unitarity of the corresponding bks pairing map this is in agreement with the unitarity up to scaling with respect to a rescaled inner product of this pairing map established by hall

on the algebra of quantum observables for a certain gauge model we prove that the algebra of observables of a certain gauge model is generated by unbounded elements in the sense of woronowicz the generators are constructed from the classical generators of invariant polynomials by means of geometric quantization

on joint distribution of observables for f quantum spaces abstract in an f quantum space model for measurements of quantum mechanical observables based on fuzzy sets ideas we solve a problem of existence of a joint distribution for a given system of observables in a given f state of an f quantum space

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

geometric quantization complex structures and the coherent state transform abstract it is shown that the heat operator in the hall coherent state transform for a compact lie group k j funct anal 122 1994 103 151 is related with a hermitian connection associated to a natural one parameter family of complex structures on t k the unitary parallel transport of this connection establishes the equivalence of geometric quantizations of t k for different choices of complex structures within the given family in particular these results establish a link between coherent state transforms for lie groups and results of hitchin comm math phys 131 1990 347 380 and axelrod et al j differential geom 33 1991 787 902

coherent states over symplectic homogeneous spaces a generalization of the perelomov procedure for the construction of coherent states is proposed the new procedure is used to construct systems of coherent states in the carrier spaces of unitary irreducible representations of groups g s v where v is a vector space and s gl v the coherent states are shown to be labeled by the points in cotangent bundles t o of orbits o of s in v the dual of v it is proven that t o is a symplectic homogeneous space of g the generalized procedure for the construction of coherent states presented in this paper is shown to encompass as special cases the constructions known in the literature for the coherent states of the weyl heisenberg the ax b and the galilei and poincare groups moreover completely new sets of coherent states are constructed for the euclidean group e n where the perelomov construction fails