Gauge Invariance and Rest Mass

Abstract

IT is well known that the main difference between Maxwell's equations and Proca's1 equations is that, whereas in the former the potentials Aµ admit the gauge transformation and describe photons of zero rest mass, in the latter there is no such transformation of the potentials which describe quanta of non-zero rest mass. This is often taken to mean that in a gauge-invariant electrodynamics the rest mass of the photon must vanish. In view of the present interest in the self-energy of the photon, it seems worth while to mention that there exist gauge-invariant wave equations which can describe quanta of non-zero rest mass. For example, the equations where f is any scalar function of the field-strengths and their derivatives of any order, are consistent with the transformation (1) of potentials and with the equation θνjν, = 0, expressing the conservation of charge. It is easy to arrange for f to be small for slowly varying or weak fields, so that, in these limiting cases, the equations reduce to Maxwell's. But the arbitrary function f must contain some parameters which have no counterpart in Maxwell's equations, and which in certain types of theory lead to the appearance of field quanta of non-zero rest mass.