Extended theory of special relativity and photon mass

Abstract

Although it is well known that Einstein's special theory of relativity and Maxwell's theory of electromagnetic waves are in some sense incompatible with de Broglie's presumption (1) of the photon rest mass, a lot of endeavours are used to determine an upper l imit of the photon rest mass (2). The only way to introduce the photon rest mass in the relativistic theory has ever been believed is to replace the Maxwell equations by the Proca equation. In this case, however, not only the freedom of gauge invariance found in conventional electrodynamies is completely lost, but also the existence of a third degree of freedom, i.e. longitudinal polarization, should be allowed (2). According to detailed investigations, the effects of longitudinal photons have been turned out to be negligibly small and very difficult to observe (2on). Recently, by the extended Lorentz transformation, a possibility was shown to modify Einstein 's theory of special relativity in such a way that the photon had an effective mass (4). The purpose of the present note is devoted to improve the previous theory on the basis of our philosophical hypothesis that fluctuation associated with quantum theory cannot be completely removed from our space-time. When the extended Lorentz transformation for energy-momentum is writ ten in the form (4)