Deriving Electromagnetic Fields From the Spinor Solution of the Massless Dirac Equation

Abstract

In this paper, we present the explicit derivation of the electromagnetic (EM) field solution of Maxwell equations starting from the Dirac equation, used in describing the so-called spinor wave function of quantum particles. In particular, we show that if the four-component vector (spinor) solution of the Dirac equation for zero mass is identified with the four-potential of the EM field, then, under the Lorentz gauge, fields derived from that potential satisfy Maxwell equations. <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Vice versa</i> , the four-potential could be used to express a spinor solution, provided that the latter satisfies the Lorenz gauge. Some examples in the frequency domain clarify this connection. A crucial choice is needed: the EM potential has to be assumed as a linear combination of positive- and negative solutions of the spinor. This work may help to clarify the controversial relation between Maxwell and Dirac equations, while presenting an original way to derive the EM fields, leading, perhaps, to novel concepts in EM simulations.