A new matrix formulation of the Maxwell and Dirac equations

Abstract

Presented in this paper is a new matrix formulation of both the classical electromagnetic Maxwell equations and the relativistic quantum mechanical Dirac equation. These new matrix representations will be referred to as the Maxwell spacetime matrix equation and the Dirac spacetime matrix equation. Both are Lorentz invariant. Key to these new matrix formulations is an 8-by-8 matrix operator referred to here as the spacetime matrix operator. As it turns out, the Dirac spacetime matrix equation is equivalent to four new vector equations, which are similar in form to the four Maxwell vector equations. These new equations will be referred to as the Dirac spacetime vector equations. This allows these new vector equations to be as readily solved as solving a set of Maxwell vector equations. Based on these two new matrix approaches, two computer programs, encoded using Matlab software, have been developed and tested for determining the reflection and transmission characteristics of multilayer optical thin-film structures and multilayer quantum well-and-barrier structures. A listing of these software programs may be found in supplemental material associated with this article. Numerical results obtained based on the use of these computer programs are presented in the results section of this article.