On the time dependence of the photon Compton wavelength

Abstract

The Proca equation is a generalisation of the Maxwell equations that modifies the potential of a constant charge at rest from the Coulomb to the Yukawa form and describes massive photons. In this paper, we assume time-dependent photon mass and solve the Proca equation for a charge at rest. It is found that the vector potential is non-vanishing and that the scalar potential has the form $$\Phi \bigl (r,x^0\bigr )=f\bigl (r,x^0\bigr )/r$$ Φ ( r , x 0 ) = f ( r , x 0 ) / r with $$x^0=ct$$ x 0 = c t and, thus, is space-time dependent. The solution shows that an oscillating potential pulse leaves the charge at $$x^0=0$$ x 0 = 0 and propagates with the speed of the light to modify the initial Yukawa potential. Experimentally the change of sign of the potential field at the charge location can be interpreted as a change of the sign of the charge. The picture that emerges from the theory is, then, that of charges that at the initial time emit massive radiation that fills the Universe. The period of the field oscillation is, however, increasing and suggests that after a long time the potential may become stabilised. A noteworthy consequence of the theory is that the present form of the Maxwell electrodynamics is only temporarily true.