Symmetry of Dirac Equation and Corresponding Phenomenology

Abstract

It has been suggested that the high symmetries in the Schrödinger 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 $\frac{(1+\beta )}{2}V(r)$ where $V(r)$ is $\frac{-e^2}{r}$ 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.

15pages, 3 figures, accepted by International Journal of Modern Physics A

References in corpus (3)

• U(3) and Pseudo-U(3) Symmetry of the Relativistic Harmonic Oscillator
• Singularity Structures in Coulomb-Type Potentials in Two Body Dirac Equations of Constraint Dynamics
• Dynamical symmetry of Dirac hydrogen atom with spin symmetry and its connection with Ginocchio's oscillator

Cited by in corpus (5)

• Hidden pseudospin and spin symmetries and their origins in atomic nuclei
• Pseudospin symmetry in single particle resonances in spherical square wells
• Confinement of fermions in tachyon matter
• Phenomenological study on the significance of the scalar potential and Lamb shift
• Re-study on the contribution of scalar potential and spectra of $c\bar{c}$, $b\bar{b}$ and $b\bar{c}(\bar{b}c)$ families