Symmetry operators and separation of variables in the $(2+1)$-dimensional Dirac equation with external electromagnetic field

TL;DR

This paper classifies symmetry operators for the Dirac equation with electromagnetic fields in 2+1 dimensions, enabling separation of variables and solving the equation in specific electromagnetic potentials.

Contribution

It provides a complete classification of symmetry operators and separation of variables for the Dirac equation in (2+1)-dimensional spacetime with external electromagnetic fields.

Findings

Classified all mutually commuting symmetry operators in flat (2+1) spacetime.

Identified electromagnetic potentials allowing variable separation.

Achieved explicit solutions for the Dirac equation in these potentials.

Abstract

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a $(2+1)$-dimensional Minkowski (flat) space. For each of the sets we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.