Inverse approach to solutions of the Dirac equation for space-time dependent fields

TL;DR

This paper introduces an inverse method to generate solutions of the Dirac equation in space-time dependent electromagnetic fields, expanding the set of known solutions beyond traditional one-coordinate-dependent cases in lower dimensions.

Contribution

The authors develop an inverse approach that produces families of solutions for the Dirac equation in genuinely space-time dependent fields in 1+1 and 2+1 dimensions, surpassing previous limitations.

Findings

Generated new solution families for Dirac equation in space-time dependent fields.

Extended solution methods to lower-dimensional models with full space-time dependence.

Facilitated non-perturbative QED studies with broader solution sets.

Abstract

Exact solutions of the Dirac equation in external electromagnetic background fields are very helpful for understanding non-perturbative phenomena in quantum electrodynamics (QED). However, for the limited set of known solutions, the field often depends on one coordinate only, which could be the time $t$, a spatial coordinate such as $x$ or $r$, or a light-cone coordinate such as $ct-x$. By swapping the roles of known and unknown quantities in the Dirac equation, we are able to generate families of solutions of the Dirac equation in the presence of genuinely space-time dependent electromagnetic fields in $1+1$ and $2+1$ dimensions.