Complex analytic realizations for quantum algebras

TL;DR

This paper introduces a method to obtain complex analytic realizations of certain deformed quantum algebras using deformation mappings and coherent states, providing explicit series-based realizations that reduce to known forms as deformation parameters approach unity.

Contribution

The paper presents a novel approach to derive explicit complex analytic realizations for deformed algebras like $q$-oscillators and $su_q(2)$, extending the analytic framework in quantum algebra representations.

Findings

Explicit series realizations for $q$-oscillators and $su_q(2)$, $su_q(1,1)$ algebras.

Realizations reduce to classical Bargmann representations as $q o 1$.

Method connects deformation mappings with coherent state techniques.

Abstract

A method for obtaining complex analytic realizations for a class of deformed algebras based on their respective deformation mappings and their ordinary coherent states is introduced. Explicit results of such realizations are provided for the cases of the $q$-oscillators ($q$-Weyl-Heisenberg algebra) and for the $su_{q}(2)$ and $su_{q}(1,1)$ algebras and their co-products. They are given in terms of a series in powers of ordinary derivative operators which act on the Bargmann-Hilbert space of functions endowed with the usual integration measure. In the $q\rightarrow 1$ limit these realizations reduce to the usual analytic Bargmann realizations for the three algebras.