Symmetry generators and quantum numbers for fermionic circularly symmetric motion

TL;DR

This paper derives symmetry generators for the planar Dirac equation with circular symmetry, enabling classification of quantum states and exploring spin and pseudospin symmetries in relativistic fermionic systems.

Contribution

It introduces a simple method to derive symmetry generators for planar Dirac particles with circular symmetry, including spin and pseudospin symmetries, and analyzes their energy degeneracies.

Findings

Derived symmetry generators for planar Dirac equation with circular symmetry.

Established quantum numbers and labeling scheme for eigenspinors.

Compared degeneracies with spherically symmetric Dirac systems.

Abstract

The planar dynamics of spin-1/2 quantum relativistic particles is important for several physical systems. In this paper we derive, by a simple method, the generators for the continuous symmetries of the 3+1 Dirac equation for planar motion, when there is circular symmetry, i.e., the interactions depend only on the radial coordinate. We consider a general set of potentials with different Lorentz structures. These generators allow for several minimal complete sets of commuting observables and their corresponding quantum numbers. We show how they can be used to label the general eigenspinors for this problem. We also derive the generators of the spin and pseudospin symmetries for this planar Dirac problem, which arise when the vector and scalar potentials have the same magnitude and tensor potential and the space components of the four-vector potential are absent. We investigate the associated energy degeneracies and compare them to the known degeneracies in the spherically symmetric 3+1 Dirac equation.