Massive fermions interacting via a harmonic oscillator in the presence of a minimal length uncertainty relation

TL;DR

This paper derives the relativistic energy spectrum for a Dirac oscillator under a minimal length uncertainty, extending the algebraic structure and confirming consistency with nonrelativistic results.

Contribution

It introduces a modified Dirac equation with a harmonic oscillator potential incorporating a minimal length, and constructs related algebraic operators.

Findings

Relativistic energy spectrum derived under minimal length conditions

Construction of creation and annihilation operators for the modified oscillator

Verification of the su(1,1) algebra structure in this context

Abstract

We derive the relativistic energy spectrum for the modified Dirac equation by adding a harmonic oscillator potential where the coordinates and momenta are assumed to obey the commutation relation $\left[\hat{x},\hat{p}\right]=i\hbar\left(1+\eta p^2\right)$. In the nonrelativistic limit, our results are in agreement with the ones obtained {previously}. Furthermore, the extension to the construction of creation and annihilation operators for the harmonic oscillators with minimal length uncertainty relation is presented. Finally, we show that the commutation relation of the $su(1, 1)\sim so(2,1)$ algebra is satisfied by the {operators $\hat{\mathcal{L}_{\pm}}$ and $\hat{\mathcal{L}_{z}}$}.