q-Deformed Segal–Bargmann Transform

• q-Deformed Segal–Bargmann transform is a generalization of the classical transform, replacing canonical functions with q-analogs and establishing new Hilbert spaces.
• It constructs q-analytic functions using recursive definitions, q-factorials, and q-Hermite polynomials to develop reproducible kernel Hilbert space structures.
• The framework bridges probabilistic, operator-algebraic, and function-theoretic methods, revealing deep connections to quantum groups and noncommutative probability.

The $q$-deformed Segal–Bargmann transform generalizes the classical Segal–Bargmann (or Bargmann) transform, a central integral transform linking real $L^2$-spaces with spaces of analytic functions, by introducing deformation parameters via $q$-calculi. This extension gives rise to a spectrum of new Hilbert spaces of $q$-analytic or $q$-polyanalytic functions, $q$-analogs of Hermite and related polynomials, novel $q$-commutation relations, and kernel structures that recover the classical theory as $q\rightarrow1$. This $q$-deformation framework provides a bridge between probabilistic, operator-algebraic, and function-theoretic approaches, with deep connections to noncommutative probability, quantum groups, and time–frequency analysis (Altavilla et al., 12 Nov 2025, Cébron et al., 2017, Arjika et al., 2017, Moize et al., 2020).

1. Construction of the $q$-Deformed Segal–Bargmann Transform

Given $L^2$0, several approaches construct $L^2$1-deformations of the classical Segal–Bargmann transform, all of which replace canonical objects—such as polynomials, measures, and kernels—with their $L^2$2-analogs.

$L^2$3-Fock Space and $L^2$4-Analyticity: A geometric approach defines recursively the $L^2$5-analytic monomials $L^2$6 via

• $L^2$7,
• $L^2$8 for $L^2$9 where $q$0, and corresponding $q$1-factorials $q$2 as $q$3 with $q$4. The $q$5-Fock space is

$q$6

with inner product $q$7 (Altavilla et al., 12 Nov 2025). The orthonormal basis is $q$8.

Source Hilbert Space: For the real variable, $q$9 uses a $q$0-exponential weight and a Jackson $q$1-integral,

$q$2

where $q$3.

The $q$4-Bargmann Transform: The transform $q$5 is

$q$6

where $q$7 and the integral kernel is $q$8 with $q$9-Hermite functions $q$0 (Altavilla et al., 12 Nov 2025, Moize et al., 2020).

2. Reproducing Kernel Hilbert Space Structure and $q$1-Polynomials

Reproducing Kernel: The $q$2-Fock space is an RKHS with the kernel

$q$3

$q$4 as $q$5 (Altavilla et al., 12 Nov 2025, Moize et al., 2020).

$q$6-Hermite Polynomials: In all frameworks, $q$7-Hermite polynomials $q$8 or their variants provide an ON basis for the real model; they are defined by

$q$9

with orthogonality determined by the associated $q$0-measure (Altavilla et al., 12 Nov 2025, Cébron et al., 2017).

Polyanalytic Extensions: The $q$1-deformed polyanalytic Segal–Bargmann transform $q$2 utilizes $q$3-analogs of higher Landau-level spaces, continuous $q$4-Hermite and Wall (little $q$5-Laguerre) polynomials, and constructs the image as spaces of $q$6-polyanalytic functions supported on quantum phase-space domains $q$7 (Arjika et al., 2017).

3. Operator-Theoretic and Algebraic Properties

$q$8-Position and $q$9-Momentum Operators: The $q$0-Fock model admits $q$1-position $q$2 and $q$3-momentum $q$4 operators defined by

$q$5

and oscillator-like operators $q$6 (Altavilla et al., 12 Nov 2025). Commutation relations satisfy

$q$7

which deforms the canonical commutator and recovers it as $q$8.

Unitarity: The transform $q$9 is a unitary isomorphism carrying the $q\rightarrow1$0-Hermite ON basis to the $q\rightarrow1$1-Fock ON basis (Altavilla et al., 12 Nov 2025, Moize et al., 2020). Isometry is maintained in all polyanalytic variants (Arjika et al., 2017).

$q\rightarrow1$2-Commutators: More generally, $q\rightarrow1$3 acts as the $q\rightarrow1$4-deformed commutator throughout these constructions.

4. Probabilistic and Random Matrix Approximations

$q\rightarrow1$5-Gaussian Variables: In noncommutative probability, $q\rightarrow1$6-Gaussian variables implement the $q\rightarrow1$7-commutation relation

$q\rightarrow1$8

The field operators generate Fock spaces whose $q\rightarrow1$9-structures support the $q$0-Segal–Bargmann transform (Cébron et al., 2017). Mixed $q$1-Gaussian constructions, using correlation matrices $q$2, interpolate between classical and free cases.

Random Matrix Limits: The $q$3-deformed SB transform can be obtained as a large $q$4 limit of classical matrix-valued transforms (Sniady model): $q$5 establishing probabilistic convergence from Gaussian ensembles and revealing connections to random matrix theory and the central limit theorems for mixed independence (Cébron et al., 2017).

5. Kernel Formulas, Domains, and Limiting Cases

A summary of central structural elements is provided in the table:

Element	$q$6-Deformed Formula	Classical Limit ($q$7)
$q$8-factorial	$q$9	$q$0
$q$1-analytic basis	$q$2 (as above)	$q$3
Reprod. kernel	$q$4	$q$5
Real measure	$q$6	$q$7
Transform kernel	$q$8	$q$9

The $L^2$00-deformed transforms generalize to two-parameter $L^2$01 and multidimensional forms, as well as to $L^2$02-deformed polyanalytic transforms and their phase-space regions $L^2$03 (Arjika et al., 2017).

6. Coherent State and Operator Realizations

Arik–Coon Oscillator: In the analytic case, the underlying algebra is the Arik–Coon oscillator: $L^2$04 for $L^2$05, with $L^2$06-coherent states $L^2$07 and overlap kernel $L^2$08 (Moize et al., 2020).

Resolution of the Identity: The $L^2$09-deformed coherent states and transforms satisfy

$L^2$10

establishing invertibility and completeness in the $L^2$11-polyanalytic setting (Arjika et al., 2017).

7. Limiting Behavior and Applications

In the limit $L^2$12:

• All deformed objects recover their classical counterparts: $L^2$13, $L^2$14, $L^2$15, and the $L^2$16-Segal–Bargmann transforms reduce to the classical isometries between $L^2$17 and analytic (or polyanalytic) Fock spaces (Altavilla et al., 12 Nov 2025, Moize et al., 2020).
• For $L^2$18, the theory interpolates towards free (Voiculescu-type) probability and kernels associated with noncommutative, free independence (Cébron et al., 2017).

Applications include the explicit construction of $L^2$19-deformed coherent states, $L^2$20-Landau levels, quantum phase-space models, and $L^2$21-deformed random point processes; they are foundational in advanced time–frequency analysis and emerging noncommutative function theory (Arjika et al., 2017, Moize et al., 2020).

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References:

• "$L^2$22-Fock Space of $L^2$23-Analytic Functions and its realization in $L^2$24" (Altavilla et al., 12 Nov 2025)
• "Segal-Bargmann transform: the $L^2$25-deformation" (Cébron et al., 2017)
• "Une $L^2$26-déformation de la transformation de Bargmann polyanalytique" (Arjika et al., 2017)
• "A q deformation of true-polyanalytic Bargmann transforms when q^{-1}> 1" (Moize et al., 2020)