Solution of Maxwell's Equations in Terms of a Spinor Notation: the Direct and Inverse Problem

Abstract

Maxwell's equations for fields with sources in media in which the dielectric constant and permeability are unity are written in terms of a spinor notation which resembles the one used for Dirac's equation for the electron. One can introduce Green's functions and expansions in terms of complete sets of orthogonal functions, analogous to those used in the quantum theory of the electron, to solve Maxwell's equations in more compact form than in terms of the conventional vector notation.In addition, the new notation enables us to solve in a simple way an "inverse radiation problem" which we describe as follows:Consider at time $t<0$ the electromagnetic field to be zero. At time $t=0$ sources are turned on and then later turned off. The electromagnetic field, which results after this process has been completed, will be a radiation field. We can solve the problem of finding the nature of the sources which will lead to a prescribed final radiation field. It is shown that, in general, the sources are not unique but additional conditions can be given which will make them so.