Symmetry generators and quantum numbers for fermionic circularly symmetric systems

Abstract

The description of spin-1/2 quantum relativistic particles under the effect of external potentials which only act on a particular plane of motion is important for several physical systems. In this paper we derive, by a simple method, the generators for the continuous symmetries of a modified 3+1 Dirac equation suitable for this scenario, 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 problem, which arises when the vector and scalar potentials have the same magnitude and the 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.