Symmetry of Dirac Equation and Corresponding Phenomenology

TL;DR

This paper investigates the symmetry properties of the Dirac equation with Coulomb and harmonic oscillator potentials, analyzing how quantum electrodynamics (QED) and quantum chromodynamics (QCD) influence symmetry restoration and breaking.

Contribution

It demonstrates that symmetry restoration in the Dirac equation due to QED effects is minimal and discusses the complex symmetry behavior in QCD.

Findings

Symmetry restoration in QED is very small.

Higher order effects may introduce scalar potential components.

QCD symmetry behavior is more complex and warrants further study.

Abstract

It has been suggested that the high symmetries in the Schr\"odinger equation with the Coulomb or harmonic oscillator potentials may remain in the corresponding relativistic Dirac equation. If the principle is correct, in the Dirac equation the potential should have a form as ${(1+\beta)\over 2}V(r)$ where $V(r)$ is ${-e^2\over r}$ for hydrogen atom and $\kappa r^2$ for harmonic oscillator. However, in the case of hydrogen atom, by this combination the spin-orbit coupling term would not exist and it is inconsistent with the observational spectra of hydrogen atom, so that the symmetry of SO(4) must reduce into SU(2). The governing mechanisms QED and QCD which induce potential are vector-like theories, so at the leading order only vector potential exists. However, the higher order effects may cause a scalar fraction. In this work, we show that for QED, the symmetry restoration is very small and some discussions on the symmetry breaking are made. At the end, we briefly discuss the QCD case and indicate that the situation for QCD is much more complicated and interesting.