On some properties of the electromagnetic field and its interaction with a charged particle

TL;DR

This paper explores properties of null electromagnetic fields, introduces a natural vector potential, and proposes a modified Dirac equation coupling, providing solutions relevant to particles in electromagnetic waves.

Contribution

It introduces the natural vector potential for null electromagnetic fields and modifies the Dirac equation coupling to incorporate complex potentials.

Findings

Derived a general solution for null electromagnetic fields.

Identified a gauge-like condition similar to Dirac gauge.

Presented a solution analogous to Volkov's for a Dirac particle in a plane wave.

Abstract

A procedure for solving the Maxwell equations in vacuum, under the additional requirement that both scalar invariants are equal to zero, is presented. Such a field is usually called a null electromagnetic field. Based on the complex Euler potentials that appear as arbitrary functions in the general solution, a vector potential for the null electromagnetic field is defined. This potential is called natural vector potential of the null electromagnetic field. An attempt is made to make the most of knowing the general solution. The properties of the field and the potential are studied without fixing a specific family of solutions. A equality, which is similar to the Dirac gauge condition, is found to be true for both null field and Lienard-Wiechert field. It turns out that the natural potential is a substantially complex vector, which is equivalent to two real potentials. A modification of the coupling term in the Dirac equation is proposed, that makes the equation work with both real potentials. A solution, that corresponds to the Volkov's solution for a Dirac particle in a linearly polarized plane electromagnetic wave, is found. The solution found is directly compared to Volkov's solution under the same conditions.