Maximal velocity of photons in non-relativistic QED

We consider the problem of propagation of photons in the quantum theory of non-relativistic matter coupled to electromagnetic radiation, which is, presently, the only consistent quantum theory of matter and radiation. Assuming that the matter system is in a localized state (i.e. for energies below the ionization threshold), we show that the probability to find photons at time tt at the distance greater than View the MathML sourcect, where View the MathML sourcec is the speed of light, vanishes as t→∞t→∞ as an inverse power of tt