Solutions of the Maxwell equations and photon wave functions

TL;DR

This paper develops a six-component matrix form of Maxwell's equations, deriving photon wave functions, analyzing their properties under Lorentz transformations, and constructing Green functions for electromagnetic fields, with applications to radiation problems.

Contribution

It introduces a novel six-component formulation of Maxwell's equations, providing explicit photon wave functions and covariant Green functions, extending the analogy with Dirac equation solutions.

Findings

Photon wave functions are eigenfunctions of the Hamiltonian for electromagnetic fields.

The six-component Maxwell equation is Lorentz invariant, including sources.

A covariant Green function for the Maxwell equations is constructed and applied to radiation problems.

Abstract

Properties of six-component electromagnetic field solutions of a matrix form of the Maxwell equations, analogous to the four-component solutions of the Dirac equation, are described. It is shown that the six-component equation, including sources, is invariant under Lorentz transformations. Complete sets of eigenfunctions of the Hamiltonian for the electromagnetic fields, which may be interpreted as photon wave functions, are given both for plane waves and for angular-momentum eigenstates. Rotationally invariant projection operators are used to identify transverse or longitudinal electric and magnetic fields. For plane waves, the velocity transformed transverse wave functions are also transverse, and the velocity transformed longitudinal wave functions include both longitudinal and transverse components. A suitable sum over these eigenfunctions provides a Green function for the matrix Maxwell equation, which can be expressed in the same covariant form as the Green function for the Dirac equation. Radiation from a dipole source and from a Dirac atomic transition current are calculated to illustrate applications of the Maxwell Green function.