Harmonic and Dirac oscillators in a (2+1)-dimensional noncommutative space

TL;DR

This paper investigates harmonic and Dirac oscillators in a noncommutative (2+1)-dimensional space, revealing their equivalence to quantum models with an extra compact dimension through perturbative analysis.

Contribution

It introduces a novel approach to noncommutative space by using SL(2;R) generators, extending oscillator models and analyzing their spectra without constraints on noncommutativity parameters.

Findings

Models are equivalent to particles in a space with an extra compact dimension.

Spectrum analysis shows no constraints between coordinates and momenta noncommutativity.

Perturbation theory applied for small and large noncommutativity parameters.

Abstract

We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncom- mutative space where the noncommutativity is induced by a shift of the dynamical variables with generators of SL(2;R) in a unitary irreducible representation. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, finding no constraints between coordinates and momenta noncom- mutativity parameters. Since the representation space of the unitary irreducible representations SL(2;R) can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension. PACS: 03.65.-w; 11.30.Cp; 02.40.Gh