Relating the solutions of the Dirac equation with a background electric potential to solutions with a background pseudoscalar potential

TL;DR

This paper demonstrates a method to relate solutions of the Dirac equation with electric and pseudoscalar potentials in (1+1) dimensions, showing how solving one provides solutions to the other and analyzing vacuum charge densities and anomalies.

Contribution

It establishes a direct relation between solutions of the Dirac equation with electric and pseudoscalar backgrounds, simplifying analysis of such systems.

Findings

Solutions of Dirac equation with electric potential can be related to those with pseudoscalar potential.

Vacuum charge density is computed for both cases.

Anomalies in quantum field theory influence the vacuum charge results.

Abstract

We compare two different solutions of the Dirac equation in (1+1) dimensions. One solution is for a fermion in the presence of an electric potential and the other is for a fermion in the presence of a pseudoscalar potential. It is shown that for properly defined potentials one can easily relate the solutions of one system to the solutions of the other. In effect, solving one problem gives the solution to both. In addition, the vacuum charge density is calculated in the both cases and it is shown how this result is impacted by the presence of anomalies in quantum field theory.