The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration

TL;DR

This paper investigates solving the Dirac equation with external electromagnetic fields using separation of variables and noncommutative integration, discovering new solutions and symmetry operators beyond traditional methods.

Contribution

It introduces a novel approach to solving the Dirac equation, revealing new solutions and symmetry operators not accessible through standard separation of variables.

Findings

Found new solutions not obtainable by separation of variables.

Identified a nonlocal symmetry operator for specific electromagnetic fields.

Demonstrated the method on crossed electric and magnetic fields.

Abstract

Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operator