Relativistic harmonic oscillator

TL;DR

This paper explores the classical and quantum aspects of the relativistic harmonic oscillator, analyzing phase space evolution, distribution distortions, and potential quantum solution strategies using Hamiltonian mechanics and Lie algebraic techniques.

Contribution

It introduces a Hamiltonian formalism and Lie algebraic methods to analyze the relativistic harmonic oscillator's classical phase space dynamics and discusses quantum solution approaches.

Findings

Phase space evolution of relativistic particles under harmonic potential

Non-linear relativistic effects distort spatial and momentum distributions

Potential strategies for solving the relativistic quantum Salpeter equation

Abstract

We consider the relativistic generalization of the harmonic oscillator problem by addressing different questions regarding its classical aspects. We treat the problem using the formalism of Hamiltonian mechanics. A Lie algebraic technique is used to solve the associated Liouville equations, yielding the phase space evolution of an ensemble of relativistic particles, subject to a "harmonic" potential. The non-harmonic distortion of the spatial and momentum distributions due to the intrinsic non-linear nature of the relativistic contributions are discussed. We analyze the relativistic dynamics induced by two types of Hamiltonian, which can ascribed to those of harmonic oscillators type. Finally, we briefly discuss the quantum aspects of the problem by considering possible strategies for the solution of the associated Salpeter equation.