The electronic and electromagnetic Dirac equations

Abstract

Maxwell’s equations and the Dirac equation are the first-order differential relativistic wave equation for electromagnetic waves and electronic waves respectively. Hence, there is a notable similarity between these two wave equations, which has been widely researched since the Dirac equation was proposed. In this paper, we show that the Maxwell equations can be written in an exact form of the Dirac equation by representing the four Dirac operators with $8\times8$ matrices. Unlike the ordinary $4\times4$ Dirac equation, both spin–1/2 and spin–1 operators can be derived from the $8\times8$ Dirac equation, manifesting that the $8\times8$ Dirac equation is able to describe both electrons and photons. As a result of the restrictions that the electromagnetic wave is a transverse wave, the photon is a spin–1 particle. The four–current in the Maxwell equations and the mass in the electronic Dirac equation also force the electromagnetic field to transform differently to the electronic field. We use this $8\times8$ representation to find that the Zitterbewegung of the photon is actually the oscillatory part of the Poynting vector, often neglected upon time averaging.

Keywords

• Dirac equation
• Maxwell’s equations
• Zitterbewegung
• Spin
• Lorentz transformation