The Zeeman Effect for the Relativistic Bound State

TL;DR

This paper investigates the Zeeman effect in relativistic bound states within a covariant quantum framework, revealing how magnetic splitting manifests when the defining vector becomes dynamic.

Contribution

It extends previous relativistic bound state models by incorporating a dynamical spacelike vector to analyze the Zeeman effect covariantly.

Findings

Recovered the standard Zeeman splitting in a covariant manner

Demonstrated the necessity of a dynamical $n_$ for magnetic effects

Connected relativistic bound state spectra with nonrelativistic results

Abstract

In the context of a relativistic quantum mechanics with invariant evolution parameter, solutions for the relativistic bound state problem have been found, which yield a spectrum for the total mass coinciding with the nonrelativistic Schr\"odinger energy spectrum. These spectra were obtained by choosing an arbitrary spacelike unit vector $n_\mu$ and restricting the support of the eigenfunctions in spacetime to the subspace of the Minkowski measure space, for which $(x_\perp )^2 = [x-(x \cdot n) n ]^2 \geq 0$. In this paper, we examine the Zeeman effect for these bound states, which requires $n_\mu$ to be a dynamical quantity. We recover the usual Zeeman splitting in a manifestly covariant form.