Relativistic Ladder Operators for the Three-Dimensional Harmonic Oscillator

TL;DR

This paper introduces Lorentz covariant ladder operators for the relativistic 3D harmonic oscillator, providing a concise formulation of quantum constraints and connecting to the non-relativistic Schrödinger equation.

Contribution

It develops a Lorentz covariant formalism for ladder operators in the relativistic 3D harmonic oscillator, extending traditional non-relativistic methods.

Findings

Covariant ladder operators are expressed as conjugate 4-vectors.

Operators generate three independent eigenstate transitions.

Non-relativistic limit recovers Schrödinger equation.

Abstract

The quantum constraint equations for a relativistic three-dimensional harmonic oscillator are shown to find concise expression in terms of Lorentz covariant ladder operators. These ladder operators consist of two conjugate 4-vectors that are each constrained to generate three linearly independent combinations of ladder operator components for raising and lowering the eigenstates of the oscillator. Correspondence to the Schr\"{o}dinger equation for the harmonic oscillator in the non-relativistic limit is demonstrated.