Maxwell's Equations in Complex Variables

TL;DR

This paper explores Maxwell's equations using complex variables, differential forms, and the Hodge star operator, revealing new insights into solutions, harmonic conditions, and gauge interpretations in complex and Minkowski spaces.

Contribution

It introduces a complex variable framework for Maxwell's equations, connecting holomorphic functions, harmonic solutions, and gauge conditions in a novel way.

Findings

Holomorphic functions lead to nontrivial solutions.

A simple necessary and sufficient condition for harmonic solutions.

Interpretation of Lorenz gauge in terms of the codifferential operator.

Abstract

This paper provides a view of Maxwell's equations from the perspective of complex variables. The study is made through complex differential forms and the Hodge star operator in $\mathbb{C}^2$ with respect to the Euclidean and the Minkowski metrics. It shows that holomorphic functions give rise to nontrivial solutions, and the inner product between the electric and the magnetic fields is considered in this case. Further, it obtains a simple necessary and sufficient condition regarding harmonic solutions to the equations. In the end, the paper gives an interpretation of the Lorenz gauge condition in terms of the codifferential operator.