On the relation between the Maxwell system and the Dirac equation

The relation between the two most important in mathematical physics first order systems of partial differential equations is among those topics which attract attention because of their general, even philosophical significance but at the same time do not offer much for the solution of particular problems concerning physical models. The Maxwell equations can be represented in a Dirac like form in different ways (e.g., [3], [5], [9]). Solutions of Maxwell’s system can be related to solutions of the Dirac equation through some nonlinear equations (e.g., [11]). Nevertheless, in spite of these significant efforts there remain some important conceptual questions. For example, what is the meaning of this close relation between the Maxwell system and the Dirac equation and how this relation is connected with the wave-particle dualism. In the present article we propose a simple equality involving the Dirac operator and the Maxwell operators (in the sense which is explained below). This equality establishes a direct connection between solutions of the two systems and moreover, we show that it is valid when a quite natural relation between the frequency of the electromagnetic wave and the energy of the Dirac particle is fulfilled. Our analysis is based on the quaternionic form of the Dirac equation obtained in [7] and on the quaternionic form of the Maxwell equations proposed in [6] (see also [8]). In both cases the quaternionic reformulations are completely equivalent to the traditional form of the Dirac and Maxwell systems.