The wave-function description of the electromagnetic field

TL;DR

This paper introduces a complex prepotential function for electromagnetic fields that is Lorentz covariant, enabling a new integral and differential formulation of Maxwell's equations with potential applications in field analysis.

Contribution

It defines a novel Lorentz covariant prepotential for electromagnetic fields based on a complex scalar, extending the scalar potential framework of Whittaker and linking it to Dirac matrices.

Findings

Prepotential $S$ is a complex, Lorentz covariant function of spacetime.

The Faraday vector can be derived from $S$ via convolution with Dirac alpha matrices.

Maxwell equations are reformulated in terms of the prepotential $S$.

Abstract

For an arbitrary electromagnetic field, we define a prepotential $S$, which is a complex-valued function of spacetime. The prepotential is a modification of the two scalar potential functions introduced by E. T. Whittaker. The prepotential is Lorentz covariant under a spin half representation. For a moving charge and any observer, we obtain a complex dimensionless scalar. The prepotential is a function of this dimensionless scalar. The prepotential $S$ of an arbitrary electromagnetic field is described as an integral over the charges generating the field. The Faraday vector at each point may be derived from $S$ by a convolution of the differential operator with the alpha matrices of Dirac. Some explicit examples will be calculated. We also present the Maxwell equations for the prepotential.