q-Fock Space of q-Analytic Functions and its realization in L²(ℂ; e^{-z z} dx dy)

2025/11/12 by Altavilla, Amedeo, Bernstein, Swanhild, Zimmermann, Martha Lina
#47B32 #81S05 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 46E22 #Secondary: 33D45

paper · doi:10.48550/arxiv.2511.09336

Abstract

We introduce a q-deformation of the Fock space of holomorphic functions on 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_qⁿ and construct the associated q-Fock space as a Hilbert space with orthonormal basis \z_qⁿ/√([n]_q!])\_{n≥ 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_qⁿ can be expanded in terms of complex Hermite polynomials, thereby providing a realization of the q-Fock space as a subspace of L²(ℂ; e^-|z|2 dx dy). 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.

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