Generalized Coherent States for Polynomial Weyl-Heisenberg Algebras

TL;DR

This paper constructs and analyzes generalized coherent states for a polynomial Weyl-Heisenberg algebra that encompasses several classical algebras, providing new methods for finite and infinite-dimensional representations using complex and Grassmann variables.

Contribution

It introduces a unified framework for constructing Perelomov and Barut-Girardello coherent states for polynomial Weyl-Heisenberg algebras with arbitrary parameters.

Findings

Derived coherent states in finite and infinite dimensions.

Extended the Fock-Bargmann approach to generalized algebras.

Explored Grassmann variable methods for finite-dimensional states.

Abstract

It is the aim of this paper to show how to construct Perelomov and Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.