[Paper Review] Quantum deformed algebras : Coherent states and special functions

J. D. Bukweli-Kyemba, Mahouton Norbert Hounkonnou|arXiv (Cornell University)|Jan 1, 2013

Algebraic structures and combinatorial models63 references14 citations

TL;DR

This paper introduces a generalized ${\cal R}(p,q)$-deformed Heisenberg algebra that unifies and extends $q$- and $(p,q)$-deformations, enabling the construction of new coherent states, hypergeometric series, and special functions such as ${\cal R}(p,q)$-deformed Rogers-Szegۦ polynomials and continuous Hermite polynomials. The framework generalizes known results, provides a Hopf algebra structure, and extends differentiation and integration calculi in noncommutative settings.

ABSTRACT

The Heisenberg algebra is first deformed with the set of parameters ${q, l, λ}$ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called ${\cal R}(p,q)$-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual $q$- and $(p,q)$-differentiation and integration. A Hopf algebra structure compatible with the ${\cal R}(p,q)$-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called ${\cal R}(p,q)$-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous ${\cal R}(p,q)$-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.

Motivation & Objective

• To generalize the Heisenberg algebra using parameters $\{q, l, \lambda\}$ to construct a new family of generalized coherent states.
• To develop a unified method for deriving structure functions from commutation relations of deformed single-mode oscillator algebras.
• To generalize two-parameter deformed Heisenberg algebras related to meromorphic functions, leading to ${\cal R}(p,q)$-deformed special functions.
• To extend $q$- and $(p,q)$-calculus to a noncommutative framework via a new algebraic structure compatible with Hopf algebra properties.
• To define and fully characterize ${\cal R}(p,q)$-deformed Rogers-Szeg€¶ and Hermite polynomials, including recursion relations and generating functions.

Proposed method

• Deform the standard Heisenberg algebra using parameters $\{q, l, \lambda\}$ to generate generalized coherent states satisfying Klauder’s criteria.
• Derive matrix elements of relevant operators and prove sub-Poissonian statistics for the main deformed states, enabling the definition of a generalized metric.
• Construct a unified formalism to determine structure functions directly from commutation relations of deformed oscillator algebras.
• Introduce the ${\cal R}(p,q)$-deformed algebra as a generalization of two-parameter deformed Heisenberg algebras linked to meromorphic functions.
• Define new hypergeometric series and a ${\cal R}(p,q)$-binomial theorem as generalizations of known $(p,q)$-series.
• Establish a noncommutative differential and integral calculus by defining $\partial_{p,q,h}^{\mu,\nu}$ and related operators, and construct a compatible Hopf algebra structure.

Experimental results

Research questions

• RQ1 How can the Heisenberg algebra be deformed using three parameters $\{q, l, \lambda\}$ to generate generalized coherent states with sub-Poissonian statistics?
• RQ2 What is the general method to derive structure functions from commutation relations of deformed single-mode oscillator algebras?
• RQ3 How can the ${\cal R}(p,q)$-deformed algebra be constructed to unify and extend known $q$- and $(p,q)$-deformations?
• RQ4 What are the properties and recursion relations of the newly defined ${\cal R}(p,q)$-deformed Rogers-Szeg€¶ and Hermite polynomials?
• RQ5 How can a noncommutative calculus, including differentiation and integration, be generalized within the ${\cal R}(p,q)$-algebra framework, and what Hopf algebra structure is compatible with it?

Key findings

• The generalized coherent states constructed from the $\{q,l,\lambda\}$-deformed Heisenberg algebra exhibit sub-Poissonian statistics, confirming their nonclassical nature and enabling the definition of a generalized metric on the system's geometry.
• A unified method is established to derive structure functions directly from commutation relations of deformed oscillator algebras, generalizing known results and enabling systematic construction of coherent states.
• The ${\cal R}(p,q)$-deformed hypergeometric series are introduced and shown to generalize known $(p,q)$-hypergeometric series, with a new ${\cal R}(p,q)$-binomial theorem derived as a special case.
• The ${\cal R}(p,q)$-deformed Rogers-Szeg€¶ polynomials are fully characterized, including their three-term recursion relation and difference equation, generalizing standard and $q$-deformed versions.
• Continuous ${\cal R}(p,q)$-deformed Hermite polynomials are defined via $\mathbb{H}_n(\cos\theta; p,q,\mu,\nu,h)$, satisfying a three-term recursion relation involving $p$- and $q$-deformed parameters.
• A Hopf algebra structure is explicitly constructed to be compatible with the ${\cal R}(p,q)$-algebra, extending the framework to noncommutative geometry and quantum groups.