On the Quaternionic Short-Time Fourier and Segal–Bargmann Transforms

In this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal–Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb’s uncertainty principle. We provide also the reproducing kernel associated with the Gabor space considered in this setting.

Compositions of states and observables in Fock spaces

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [ 29 , 13 ]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework.

The -Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle

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 , the -version of the Segal-Bargmann transform, associated with a finite Coxeter group acting in and a given value of Planck's constant, where is a multiplicity function on the roots defining the Coxeter group. Then we immediately prove that is a unitary isomorphism. To accomplish this we identify the reproducing kernel function of the appropriate Hilbert space of holomorphic functions. As a consequence we prove that the Segal-Bargmann transforms for Versions , , and are also unitary isomorphisms though not by a direct application of the restriction principle. The point is that the -version is the only version where a restriction principle, in our definition of this method, applies directly. This reinforces the idea that the -version is the most fundamental, most natural version of the Segal-Bargmann transform.