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

Amedeo Altavilla, Swanhild Bernstein, Martha Lina Zimmermann

TL;DR

We develop a geometric $q$-deformation of the Fock space of holomorphic functions on $\mathbb{C}$ via $q$-analyticity, introducing $z_q^n$ and the Hilbert space $\mathcal{F}_q(\mathbb{C})$ with basis $\{z_q^n/\sqrt{[n]_q!]\}$. The reproducing kernel and a $q$-position/$q$-momentum pair with deformed commutation relations are established, and a unitary $q$-Bargmann transform realizes $\mathcal{F}_q(\mathbb{C})$ as a subspace of $L^2(\mathbb{C}; e^{-|z|^2}dxdy)$ via complex Hermite expansions. A complete bidimensional framework is developed through a tensor Bargmann transform $\mathcal{B}_q^{(2)}$, identifying $\{z_q^k\overline{z}_q^h\}$ with the corresponding basis in $L^2$ and yielding a coherent-state kernel $A^{(2)}$. Overall, the work provides a geometric-analytic approach to $q$-function theory that complements operator-theoretic models like ACKS, and it builds a unitary correspondence between $q$-Hermite data and $q$-analytic monomials.

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.

$q$-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

We develop a geometric -deformation of the Fock space of holomorphic functions on via -analyticity, introducing and the Hilbert space with basis . The reproducing kernel and a -position/-momentum pair with deformed commutation relations are established, and a unitary -Bargmann transform realizes as a subspace of via complex Hermite expansions. A complete bidimensional framework is developed through a tensor Bargmann transform , identifying with the corresponding basis in and yielding a coherent-state kernel . Overall, the work provides a geometric-analytic approach to -function theory that complements operator-theoretic models like ACKS, and it builds a unitary correspondence between -Hermite data and -analytic monomials.

Abstract

We introduce a -deformation of the Fock space of holomorphic functions on , based on a geometric definition of -analyticity. This definition is inspired by a standard construction in complex differential geometry. Within this framework, we define -analytic monomials and construct the associated -Fock space as a Hilbert space with orthonormal basis . The reproducing kernel of this space is computed explicitly, and -position and -momentum operators are introduced, satisfying -deformed commutation relations. We show that the -monomials can be expanded in terms of complex Hermite polynomials, thereby providing a realization of the -Fock space as a subspace of . Finally, we define a -Bargmann transform that maps suitable -Hermite functions into our -Fock space and acts as a unitary isomorphism. Our construction offers a geometric and analytic approach to -function theory, complementing recent operator-theoretic models.

Paper Structure

This paper contains 16 sections, 9 theorems, 124 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on $q$-Calculus
  3. $q$-Exponential Function
  4. Jackson Integral
  5. $q$-Analytic Functions
  6. $q$-Analytic Polynomials and Power Series
  7. A Family of $q$-Deformed Hermite Polynomials
  8. The $q$-Analog of the Gamma Function
  9. The $q$-Hermite Equation
  10. Recurrence Relation, Annihilation and Creation Operators
  11. Exponential Function and Orthogonality
  12. A $q$-Fock Space of $q$-analytic Functions
  13. Position and Momentum Operators
  14. Realization of $\mathcal{F}_q(\mathbb{C})$ in $L^{2}(\mathbb{C}; e^{-z\overline{z}} dxdy)$
  15. A $q$-Bargmann transform
  16. ...and 1 more sections

Key Result

Theorem 3.2

The $q$-Hermite polynomials satisfy the recurrence relation: Moreover, the $q$-derivative acts as an annihilation operator:

Figures (1)

  • Figure 1: The set generated by $S$ after $6$ iterations, where $q=0.6$.

Theorems & Definitions (45)

  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • Remark 2.11
  • ...and 35 more