Lorentz transformations, sideways shift and massless spinning particles

Recently (Stone et al. (2015) [16]) the influence of the so called “Wigner translations” (more generally-Lorentz transformations) on circularly polarized Gaussian packets (providing the solution to Maxwell equations in paraxial approximation) has been studied. It appears that, within this approximation, the Wigner translations have an effect of shifting the wave packet trajectory parallel to itself by an amount proportional to the photon helicity. It has been suggested that this shift may result from specific properties of the algebra of Poincare generators for massless particles. In the present letter we describe the general relation between transformation properties of electromagnetic field on quantum and classical levels. It allows for a straightforward derivation of the helicity-dependent transformation rules. We present also an elementary derivation of the formula for sideways shift based on classical Maxwell theory. Some comments are made concerning the generalization to higher helicities and the relation to the coordinate operator defined long time ago by Pryce.