$q$-Fock Space of $q$-Analytic Functions and its realization in $L^{2}(\mathbb{C}; e^{-z\bar z} \,\mathrm{d}x\,\mathrm{d}y)$

TL;DR

This paper introduces a geometric $q$-deformation of the Fock space of holomorphic functions, constructing a $q$-analytic framework with explicit kernels, operators, and a unitary $q$-Bargmann transform, linking to $q$-Hermite functions.

Contribution

It provides a new geometric and analytic construction of the $q$-Fock space, including explicit kernels, operators, and a $q$-Bargmann transform, expanding the understanding of $q$-analytic function theory.

Findings

Explicit $q$-reproducing kernel computed

$q$-position and $q$-momentum operators satisfy $q$-deformed relations

Realization of $q$-Fock space as a subspace of $L^2( ext{complex plane})$

Abstract

We introduce a $q$-deformation of the Fock space of holomorphic functions on $\mathbb{C}$, based on a geometric definition of $q$-analyticity. This definition is inspired by a standard construction in complex differential geometry. Within this framework, we define $q$-analytic monomials $z_q^n$ and construct the associated $q$-Fock space as a Hilbert space with orthonormal basis $\{z_q^n/\sqrt{[n]_q!]}\}_{n\ge 0}$. The reproducing kernel of this space is computed explicitly, and $q$-position and $q$-momentum operators are introduced, satisfying $q$-deformed commutation relations. We show that the $q$-monomials $z_q^n$ can be expanded in terms of complex Hermite polynomials, thereby providing a realization of the $q$-Fock space as a subspace of $L^2(\mathbb{C}; e^{-|z|^2}\,\mathrm{d}x\,\mathrm{d}y)$. Finally, we define a $q$-Bargmann transform that maps suitable $q$-Hermite functions into our $q$-Fock space and acts as a unitary isomorphism. Our construction offers a geometric and analytic approach to $q$-function theory, complementing recent operator-theoretic models.