Maxwell equations in matrix form, squaring procedure, separating the variables, and structure of electromagnetic solutions

TL;DR

This paper explores the matrix formalism of Maxwell's equations using the Riemann-Silberstein-Majorana-Oppenheimer approach, constructing solutions via squaring procedures and analyzing wave separation in various geometries.

Contribution

It introduces a matrix formalism for Maxwell equations, constructs solutions from Klein-Gordon solutions, and investigates wave separation in this framework.

Findings

Constructed four formal solutions of Maxwell equations from Klein-Gordon solutions.

Analyzed the separation of physical electromagnetic waves in matrix formalism.

Detailed examination of plane and cylindrical wave solutions.

Abstract

The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell electrodynamics in vacuum is investigated within the matrix formalism. The matrix form of electrodynamics includes three real 4 \times 4 matrices. Within the squaring procedure we construct four formal solutions of the Maxwell equations on the base of scalar Klein -- Fock -- Gordon solutions. The problem of separating physical electromagnetic waves in the linear space \lambda_{0}\Psi^{0}+\lambda_{1}\Psi^{1}+\lambda_{2}\Psi^{2}+ lambda_{3}\Psi^{3} is investigated, several particular cases, plane waves and cylindrical waves, are considered in detail.