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Fortschritte der Physik

Two types of q‐deformed Wigner algebra

Won Sang Chung

Published May 7, 2014• 1 citation
Publisher

Abstract

Abstract In this paper we introduce two types of q‐deformed Wigner algebras. One is q‐deformation of Wigner algebra with q‐reflection symmetry and another is q‐deformation of Wigner algebra with reflection symmetry. For two types of q‐deformed Wigner algebras, we investigate the representation and the eigenvalue of the position operator. Like q‐calculus, we introduce the (q; ν)‐ numbers, (q; ν)‐derivatives and (q; ν)‐Hermite polynomials for two algebras. For the deformation parameter q = 1 ‐ ϵ with small ϵ, we discuss the thermodynamics of the particle obeying the q‐deformed Wigner algebra.

DOI
10.1002/prop.201400001

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Abstract In this paper we introduce two types of q‐deformed Wigner algebras. One is q‐deformation of Wigner algebra with q‐reflection symmetry and another is q‐deformation of Wigner algebra with reflection symmetry. For two types of q‐deformed Wigner algebras, we investigate the representation and the eigenvalue of the position operator. Like q‐calculus, we introduce the (q; ν)‐ numbers, (q; ν)‐derivatives and (q; ν)‐Hermite polynomials for two algebras. For the deformation parameter q = 1 ‐ ϵ with small ϵ, we discuss the thermodynamics of the particle obeying the q‐deformed Wigner algebra.
Deformation of the Heisenberg-Weyl algebra W s of creation-annihilation operators is studied and the problem of eigenvalues of the Hamiltonian for s deformed oscillators with interaction is solved within this algebra. At first, types of deformation are found for which solutions could be presented analytically and a simple q−deformation is considered by the means of the perturbation theory. Cases of reducing Hamiltonians which do not preserve the total particle number to that studied here are indicated. Investigations of new types of symmetries in different areas of mathematical physics by using the Inverse Scattering Method (ISM) and nonlinear differential equations, led to the appearance of notions ”Quantum Group”, ”Quantum Algebra” [1]. Nowadays, the deformed quantum statistics, parastatistics and corresponding algebras of creation-annihilation operators are widely studied. The common Heisenberg-Weyl algebra W s consists of generators ai, a + , ni, i =1 ,...,s ,
We study the thermostatistics of q-deformed bosons and fermions obeying the symmetric (q<-->q(-1)) algebra and show that it can be built on the formalism of q calculus. The entire structure of thermodynamics is preserved if ordinary derivatives are replaced by an appropriate Jackson derivative. In this framework, we derive the most important thermodynamic functions describing the q-boson and q-fermion ideal gases in the thermodynamic limit. We also investigate the semiclassical limit and the low-temperature regime and demonstrate that the nature of the q deformation gives rise to pure quantum statistical effects stronger than undeformed boson and fermion particles.
This paper addresses a construction of new q-Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q-deformed boson system is studied.