Relativistic U(3) Symmetry and Pseudo-U(3) Symmetry of the Dirac Hamiltonian

TL;DR

This paper analytically explores the relativistic U(3) and pseudo-U(3) symmetries of the Dirac Hamiltonian with harmonic oscillator potentials, revealing higher symmetries in specific limits and their implications for nuclear spectra.

Contribution

It derives the relativistic generators for U(3) and pseudo-U(3) symmetries and discusses their eigenfunctions, eigenvalues, and potential approximate symmetries in nuclear physics.

Findings

Identifies spin and pseudospin limits with U(3) and pseudo-U(3) symmetry.

Derives generators for these symmetries in the relativistic context.

Suggests approximate symmetry in nuclear spectra if anti-nucleons are bound.

Abstract

The Dirac Hamiltonian with relativistic scalar and vector harmonic oscillator potentials has been solved analytically in two limits. One is the spin limit for which spin is an invariant symmetry of the the Dirac Hamiltonian and the other is the pseudo-spin limit for which pseudo-spin is an invariant symmetry of the the Dirac Hamiltonian. The spin limit occurs when the scalar potential is equal to the vector potential plus a constant, and the pseudospin limit occurs when the scalar potential is equal in magnitude but opposite in sign to the vector potential plus a constant. Like the non-relativistic harmonic oscillator, each of these limits has a higher symmetry. For example, for the spherically symmetric oscillator, these limits have a U(3) and pseudo-U(3) symmetry respectively. We shall discuss the eigenfunctions and eigenvalues of these two limits and derive the relativistic generators for the U(3) and pseudo-U(3) symmetry. We also argue, that, if an anti-nucleon can be bound in a nucleus, the spectrum will have approximate spin and U(3) symmetry.