We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on the symplectic plane, apply it to the Schrödinger polarization and study the corresponding evolution of polarized sections. The flow is periodic in imaginary time and the evolution of polarized sections has interesting features. On the time intervals for which the polarization is real or Kähler, the half--form corrected time evolution of polarized sections is given by unitary operators which turn out to be equivalent to the classical Segal-Bargmann transforms (which are usually associated to the quadratic elliptic Hamiltonian $H=\frac12 p^2$ and to the heat operator). At the right endpoint of these intervals, the evolution of polarized sections is given by the Fourier transform from the Schrödinger to the momentum representation. In the complementary intervals of imaginary time, the polarizations are anti--Kähler and the Hilbert space of polarized sections collapses to ${\mathcal H}= \{0\}$.
Hyperbolic quadratic Hamiltonians thus give rise to a new factorization of the Segal-Bargmann transform, which is very different from the usual one, where one first applies a bounded contraction operator (the heat kernel operator), mapping $L^2$--states to real analytic functions with unique analytic continuation, and then one applies analytic continuation. In the factorization induced by an hyperbolic complexifier, both factors are unbounded operators but their composition is, in the Kähler or real sectors, unitary.
In another paper [KMNT], we explore the application of the above family of unitary transforms to the definition of new holomorphic fractional Fourier transforms.
We give identifications of the $q$-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed $q$-Gaussian variables. We prove that, when defined on the random matrix model of Śniady for the $q$-Gaussian variable, the classical Segal-Bargmann transform converges to the $q$-deformed Segal-Bargmann transform in the large $N$ limit. We also show that the $q$-deformed Segal-Bargmann transform can be recovered as a limit of a mixture of classical and free Segal-Bargmann transform.
The Segal-Bargmann transform is a Lie algebra and Hilbert space isomorphism between real and complex representations of the oscillator algebra. The Segal-Bargmann transform is useful in time-frequency analysis as it is closely related to the short-time Fourier transform. The Segal-Bargmann space provides a useful example of a reproducing kernel Hilbert space. Coupled supersymmetries (coupled SUSYs) are generalizations of the quantum harmonic oscillator that have a built-in supersymmetric nature and enjoy similar properties to the quantum harmonic oscillator. In this paper, we will develop Sega…
We present various relations among Versions A, B and C of the Segal-Bargmann transform. We get results for the Segal-Bargmann transform associated to a Coxeter group acting on a finite dimensional Euclidean space. Then analogous results are shown for the Segal-Bargmann transform of a connected, compact Lie group for all except one of the identities established in the Coxeter case. A counterexample is given to show that the remaining identity from the Coxeter case does not have an analogous identity for the Lie group case. A major result is that in both contexts the Segal-Bargmann transform for…
We present an explanation of how the $μ$-deformed Segal-Bargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echevarria and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of Segal-Bargmann analysis. This explanation relates the $μ$-deformed Segal-Bargmann transforms to the generalized Segal-Bargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the $μ$-deformed Segal-Bargmann transform can be understood as Hall type transforms. In particular, we define a $μ$-deformatio…
Some positive results about the composition of Toeplitz operators on the Segal-Bargmann space are presented. A Wick symbol where it is not possible to construct its associated Toeplitz operator is given.
This paper gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function in L^p(R^d) with respect to a Gaussian measure. The proof relies on a family of inversion formulas for the Segal-Bargmann transform, which can be "tuned" to give the best estimates for a given value of p.
We consider a $μ$-deformation of the Segal-Bargmann transform, which is a unitary map from a $μ$-deformed quantum configuration space onto a $μ$-deformed quantum phase space (the $μ$-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 $μ$-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.
We consider a deformation of Segal-Bargmann space and its transform. We study L^p properties of this transform and obtain entropy-entropy inequalities (Hirschman) and entropy-energy inequalities (log-Sobolev) that generalize the corresponding known results in the undeformed theory.
We develop isometry and inversion formulas for the Segal--Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.
We use a variant of the classical Segal-Bargmann transform to understand the canonical quantization of Yang-Mills theory on a space-time cylinder. This transform gives a rigorous way to make sense of the Hamiltonian on the gauge-invariant subspace. Our results are a rigorous version of the widely accepted notion that on the gauge-invariant subspace the Hamiltonian should reduce to the Laplacian on the compact structure group. We show that the infinite-dimensional classical Segal-Bargmann transform for the space of connections, when restricted to the gauge-invariant subspace, becomes the genera…
We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal-Bargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Saïd and Ørsted and independently by Soltani) and the Dunkl heat kernel, due to Rösler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal-Bargmann analysis on Euclidean space. Hall's results (trivi…
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.
We study certain densely defined unbounded operators on the Segal-Barg\-mann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding $D$-complex and study properties of the corresponding complex Laplacian $\tilde \Box_D = D D^* + D^* D,$ where $D$ is a differential operator of polynomial type.
We consider the generalized Segal-Bargmann transform C_t for a compact group K, introduced in B. C. Hall, J. Funct. Anal. 122 (1994), 103-151. Let K_C denote the complexification of K. We give a necessary-and-sufficient pointwise growth condition for a holomorphic function on K_C to be the image under C_t of a C-infinity function on K. We also characterize the image under C_t of Sobolev spaces on K. The proofs make use of a holomorphic version of the Sobolev embedding theorem.
This note explains how the two measures used to define the $μ$-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald function…
We apply a special case, the restriction principle (for which we give a definition simpler than the usual one), of a basic result in functional analysis (the polar decomposition of an operator) in order to define $C_{μ, t}$, the $C$-version of the Segal-Bargmann transform, associated to a finite Coxeter group acting in $\mathbb{R}^N$ and a given value $t>0$ of Planck's constant, where $μ$ is a multiplicity function on the roots defining the Coxeter group. Then we immediately prove that $C_{μ, t}$ is a unitary isomorphism. To accomplish this we identify the reproducing kernel function of the ap…
We consider the Segal-Bargmann transform for 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 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.
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_C. 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 quantizati…
For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as density. Here K is a maximal compact subgroup of G, and g_C=k_C+p_C is a complexified Cartan decomposition. In this realization the space of k-finite vectors consists of holomorphic polynomials on X. The reproducing kernel of the Fock space is calculated explicitly in terms of an I-Bessel function. We further find an explicit formula of a generalized Segal-Bargmann tran…
Let G/K be a Riemannian symmetric space of the complex type, meaning that G is complex semisimple and K is a compact real form. Now let Γ be a discrete subgroup of G that acts freely and cocompactly on G/K. We consider the Segal--Bargmann transform, defined in terms of the heat equation, on the compact quotient Γ\G/K. We obtain isometry and inversion formulas precisely parallel to the results we obtained previously for globally symmetric spaces of the complex type. Our results are as parallel as possible to the results one has in the dual compact case. Since there is no known Gutzmer formula i…
Both direct and reverse log-Sobolev inequalities, relating the Shannon entropy with a $μ$-deformed energy, are shown to hold in a family of $μ$-deformed Segal-Bargmann spaces. This shows that the $μ$-deformed energy of a state is finite if and only if its Shannon entropy is finite. The direct inequality is a new result, while the reverse inequality has already been shown by the authors but using different methods. Next the $μ$-deformed energy of a state is shown to be finite if and only if its Dirichlet form energy is finite. This leads to both direct and reverse log-Sobolev inequalities that …
This paper describes results of the author with B. K. Driver and T. Kemp concerning the large-N limit of the Segal--Bargmann transform for the unitary group U(N). We consider the transform on matrix-valued functions that are polynomials in a single variable in U(N). We show that in the large-N limit, the transform maps functions of this type to single-variable polynomial functions on the complex group GL(N;C). This result was conjectured by Ph. Biane and was also proved independently by G. Cébron. The first main ingredient in our proof of this result is an "asymptotic product rule" for the Lap…
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 multiplication operators and differential operators, conjugated by the generalized Segal-Bargmann transform. The main results show that the resulting operators on the genera…
This set of lecture notes gives an introduction to holomorphic function spaces as used in mathematical physics. The emphasis is on the Segal-Bargmann space and the canonical commutation relations. Later sections describe more advanced topics such as the Segal-Bargmann transform for compact Lie groups and the infinite-dimensional theory.
This paper explains some of the ideas behind a prior joint work of the author with Bruce Driver on the canonical quantization of Yang-Mills theory on a spacetime cylinder. The idea is that the generalized Segal-Bargmann transform for a compact group can be obtained from the ordinary Segal-Bargmann transform by imposing gauge symmetry.
We describe a construction of wavelets (coherent states) in Banach spaces generated by ``admissible'' group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example we consider operator valued Segal-Bargmann type spaces and the Weyl functional calculus. Keywords: Wavelets, coherent states, Banach spaces, group representations, covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann spaces, Weyl functional calculus (quantization), second quantization, bosonic field.
We deal with a family of generalized coherent states associated to the hyperbolic Landau levels of the Schrödinger operator with uniform magnetic field on the Poincaré disk. Their associated coherent state transforms constitute a class of generalized second Bargmann transforms.
We use methods from the Fock space and Segal-Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods, and in support vector machines (SVMs) classification algorithms. Complex analysis techniques allow us to consider several notions linked to the RBF kernels like the feature space and the feature map, using the so-called Segal-Bargmann transform. We show also how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis,…
The first two parts of this article surveys results related to the heat-kernel coherent states for a compact Lie group K. I begin by reviewing the definition of the coherent states, their resolution of the identity, and the associated Segal-Bargmann transform. I then describe related results including connections to geometric quantization and (1+1)-dimensional Yang--Mills theory, the associated coherent states on spheres, and applications to quantum gravity. The third part of this article summarizes recent work of mine with Driver and Kemp on the large-N limit of the Segal--Bargmann transform …
When considering a charged particle evolving in the Poincaré disk under influence of a uniform magnetic field with a strength proportional to +1, we construct for all hyperbolic Landau level ε^γ_$m$ m = 4m(-m), m 2 Z+ \[0, /2] a family of coherent states transforms labeled by (,m) and mapping isometrically square integrable functions on the unit circle with respect to the measure sin^γ-2m (θ/2) dθonto spaces of bound states of the particle. These transforms are called circular Bargmann transforms.
We introduce two extensions of the Segal-Bargmann coherent state transform from $L^2({\mathbb R},dx)$ to Hilbert spaces of slice monogenic and axial monogenic functions and study their properties. These two transforms are related by the dual Radon transform. Representation theoretic and quantum mechanical aspects of the new representations are studied.
This paper outlines a covariant theory of operators defined on groups and homogeneous spaces. A systematic use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is systematically illustrated by a representative collection of examples. Keywords: Lie groups and algebras, convolution operator, relative convolution, representation theory, pseudo-differential operators, PDO, singular integral operator, SIO, the Heisenberg group, SL(2,R), induced representation, Fock-Segal-Bargmann (FSB) representation, Shrodinger representation, covar…
We study a transform, inspired by coherent state transforms, from the Hilbert space of Clifford algebra valued square integrable functions $L^2({\mathbb R}^m,dx)\otimes {\mathbb C}_{m}$ to a Hilbert space of solutions of the Weyl equation on ${\mathbb R}^{m+1}= {\mathbb R} \times {\mathbb R}^m$, namely to the Hilbert space ${\mathcal M}L^2({\mathbb R}^{m+1},dμ)$ of ${\mathbb C}_m$-valued monogenic functions on ${\mathbb R}^{m+1}$ which are $L^2$ with respect to an appropriate measure $dμ$. We prove that this transform is a unitary isomorphism of Hilbert spaces and that it is therefore an analo…
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 $-τ$…
Free field theories on a four dimensional, globally hyperbolic spacetime, whose dynamics is ruled by a Green hyperbolic partial differential operator, can be quantized following the algebraic approach. It consists of a two-step procedure: In the first part one identifies the observables of the underlying physical system collecting them in a *-algebra which encodes their relational and structural properties. In the second step one must identify a quantum state, that is a positive, normalized linear functional on the *-algebra out of which one recovers the interpretation proper of quantum mechan…
