Dirac oscillator in dynamical noncommutative space

Ilyas Haouam

doi:10.14311/ap.2021.61.0689

Abstract

In this paper, we address the energy eigenvalues of two-dimensional Dirac oscillator perturbed by a dynamical noncommutative space. We derived the relativistic Hamiltonian of Dirac oscillator in the dynamical noncommutative space, in which the space-space Heisenberg-like commutation relations and noncommutative parameter are position-dependent. Then, we used this Hamiltonian to calculate the first-order correction to the eigenvalues and eigenvectors, based on the language of creation and annihilation operators and using the perturbation theory. It is shown that the energy shift depends on the dynamical noncommutative parameter τ . Knowing that, with a set of two-dimensional Bopp-shift transformation, we mapped the noncommutative problem to the standard commutative one.

Highlights

In the last few decades, physicists and mathematicians have developed a mathematical theory called noncommutative geometry, which has quickly become a topic of great interest and has been finding applications in many areas of modern physics, such as high energy [1], cosmology [2, 3], gravity [4], quantum physics [5,6,7] and field theory [8, 9]
While n are non-negative integers, we explicitly observe that our eigenvalues are non-degenerated, this case can be explained by the fact that the particle is restricted to moving in two dimensions, and the third dimension does not contribute in the form of energy
We have investigated the effects of a dynamical noncommutative (DNC) space on the 2D Dirac oscillator (DO) in the presence of an external magnetic field in terms of creation and annihilation operator languages and through properly chosen canonical pairs of coordinates and its corresponding momenta in a complex NC space

Summary

Introduction

In the last few decades, physicists and mathematicians have developed a mathematical theory called noncommutative geometry, which has quickly become a topic of great interest and has been finding applications in many areas of modern physics, such as high energy [1], cosmology [2, 3], gravity [4], quantum physics [5,6,7] and field theory [8, 9]. The NC space-time structures were first mentioned by Snyder in 1947 [10, 11], in which he introduced noncommutativity in the hope of regularizing the divergencies that plagued quantum field theory. Motivated by the attempts to understand the string theory, the quantum gravitation and black holes through NC spaces and by seeking to highlight more phenomenological implications, we consider the Dirac oscillator (DO) within a two-dimensional dynamical noncommutative (DNC) space ( known as a position-dependent NC space). In sub-section 3.3, based on the perturbation theory and Fock basis, the energy spectrum including the dynamical noncommutativity effect is obtained, we summarize the results and discussions.

Review of dynamical noncommutativity

Two-dimensional Dirac oscillator in dynamical noncommutative space

Conclusion