A dual and conjugate system for $q$-Gaussians for all $q$

TL;DR

This paper develops a dual and conjugate system for $q$-Gaussians on $q$-deformed Fock space, proving the existence of free Gibbs potential and finiteness of free Fisher information for all $q$ in (-1,1).

Contribution

It introduces explicit formulas for dual and conjugate systems of $q$-Gaussians and extends results on free Gibbs potential and Fisher information to all $q$ in (-1,1).

Findings

Explicit formulas for dual and conjugate systems of $q$-Gaussians.

Proof of free Gibbs potential existence for all $q$ in (-1,1).

Finiteness of non-microstates free Fisher information for all $q$ in (-1,1).

Abstract

We provide a concrete formula for a dual system as well as for a conjugate system of $q$-Gaussians represented on the $q$-deformed Fock space. Moreover, using this formula, we prove the existence of a free Gibbs potential and that the non-microstates free Fisher information is finite for any $q$ with $-1<q<1$, which is an improvement on a previous result of Y. Dabrowski. We also indicate how our results can be extended to the more general setting of mixed $q_{ij}$-relations.