Algebraic solution for the vector potential in the Dirac equation

TL;DR

This paper reviews Radford's algebraic method for solving the vector potential in the Dirac equation, extending it to various spacetime dimensions and discussing constraints from the equation's rank.

Contribution

It introduces an algebraic approach to determine the vector potential in the Dirac equation, including constraints and extensions to general spacetimes.

Findings

Method applied to diverse dimensions

Addresses constraints from non-maximal rank

Extends to general spacetimes

Abstract

The Dirac equation for an electron in an external electromagnetic field can be regarded as a singular set of linear equations for the vector potential. Radford's method of algebraically solving for the vector potential is reviewed, with attention to the additional constraints arising from non-maximality of the rank. The extension of the method to general spacetimes is illustrated by examples in diverse dimensions with both $c$- and $a$-number wavefunctions.