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Journal of Physics A Mathematical and General

SUq(1,1) and the relativistic oscillator

R. M. Mir‐Kasimov

Published Sep 21, 1991• 45 citations
Publisher

Abstract

It is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q-oscillator. The corresponding group of dynamical symmetry is the quantum group SUq(1,1). The deformation parameter being q=e- omega omega 0/ where omega 0=4mc2 and omega is a frequency of the oscillator. The deformed creation and annihilation operators are finite difference ones. The corresponding deformation of the Heisenberg-Weyl group and new coherent states are also considered.

DOI
10.1088/0305-4470/24/18/016

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It is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q-oscillator. The corresponding group of dynamical symmetry is the quantum group SUq(1,1). The deformation parameter being q=e- omega omega 0/ where omega 0=4mc2 and omega is a frequency of the oscillator. The deformed creation and annihilation operators are finite difference ones. The corresponding deformation of the Heisenberg-Weyl group and new coherent states are also considered.
The Quantum Group as a Symmetry - The Schrödinger equation of the NNN-dimensional qqq-deformed Harmonic Oscillator -
With the aim to construct a dynamical model with quantum group symmetry, the qqq-deformed Schrödinger equation of the harmonic oscillator on the NNN-dimensional quantum Euclidian space is investigated. After reviewing the differential calculus on the qqq-Euclidian space, the qqq-analog of the creation-annihilation operator is constructed. It is shown that it produces systematically all eigenfunctions of the Schrödinger equation and eigenvalues. We also present an alternative way to solve the Schrödinger equation which is based on the qqq-analysis. We represent the Schrödinger equation by the qqq-difference equation and solve it by using qqq-polynomials and qqq-exponential functions. The problem of the involution corresponding to the reality condition is discussed.
With the aim to construct a dynamical model with quantum group symmetry, the q-deformed Schrödinger equation of the harmonic oscillator on the N-dimensional quantum Euclidian space is investigated. After reviewing the differential calculus on the q-Euclidian space, the q-analog of the creation-annihilation operator is constructed. It is shown that it produces systematically all eigenfunctions of the Schrödinger equation and eigenvalues. We also present an alternative way to solve the Schrödinger equation which is based on the q-analysis. We represent the Schrödinger equation by the q-difference equation and solve it by using q-polynomials and q-exponential functions. The problem of the involution corresponding to the reality condition is discussed.
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 ,
Nonlinearity of electromagnetic field vibrations described by q-oscillators is shown to produce essential dependence of second correlation functions on intensity and deformation of Planck distribution. Experimental tests of such nonlinearity are suggested. 1 on leave from Lebedev Physical Institute, Moscow 1 In the framework of investigations on quantum groups [1] the special case of the Heisenberg-Weyl algebra was first considered by Biedenharn [2] and Macfarlane [3]. They introduced ”q-oscillator operators ” starting from a deformed commutation relation containing a dimensionless parameter q. The physical sense of the quantun q-oscillator up to now was not clarified. In [4] a generalization to several modes has been analyzed. An attempt to connect quantum q-oscillator with relativistic oscillator model has been discussed in [5]. The q-oscillators and quantum group SUq(2) have also been considered
q-deformed harmonic oscillator is established for any arbitrary complex q-deformation parameter (with real and imaginary q as its special cases) and the formalism so developed is used to construct the quantum group SUq(2). The eigenvalues of the Casimir of SUq(2) are now real, even for complex q-deformation (q=ea+ib). This happens for (a,b)≪1. Thus, the use of real part of energies in the earlier works of Gupta and collaborators7–14 is mathematically rigorous, since only small values of (a, b) are involved in all the nuclear and molecular physics problems studied7–14,18–20 so far.