Canonical quantization of the Dirac oscillator field in (1+1) and (3+1) dimensions

TL;DR

This paper develops a canonical quantization framework for the Dirac oscillator field in (1+1) and (3+1) dimensions, revealing it as a collection of decoupled quantum harmonic oscillators with relativistic energies.

Contribution

It introduces a novel canonical quantization procedure for the Dirac oscillator field, highlighting its structure as infinite decoupled harmonic oscillators in different dimensions.

Findings

Dirac oscillator field consists of infinite decoupled harmonic oscillators.

Energy quanta correspond to relativistic harmonic oscillator energies.

Quantization is feasible due to absence of Klein paradox and complete eigenfunctions.

Abstract

The main goal of this work is to study the Dirac oscillator as a quantum field using the canonical formalism of quantum field theory and to develop the canonical quantization procedure for this system in $(1+1)$ and $(3+1)$ dimensions. This is possible because the Dirac oscillator is characterized by the absence of the Klein paradox and by the completeness of its eigenfunctions. We show that the Dirac oscillator field can be seen as constituted by infinite degrees of freedom which are identified as decoupled quantum linear harmonic oscillators. We observe that while for the free Dirac field the energy quanta of the infinite harmonic oscillators are the relativistic energies of free particles, for the Dirac oscillator field the quanta are the energies of relativistic linear harmonic oscillators.