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 ct, where c is the speed of light, vanishes as t→∞t→∞ as an inverse power of tt.