Lorentz transformations, sideways shift and massless spinning particles

TL;DR

This paper explores how Lorentz transformations cause a helicity-dependent sideways shift in massless spinning particles, linking classical and quantum electromagnetic field properties and providing a straightforward derivation of the shift.

Contribution

It offers a general relation between transformation properties of electromagnetic fields at quantum and classical levels, deriving the helicity-dependent shift formula from classical Maxwell theory.

Findings

Lorentz transformations induce a helicity-dependent sideways shift.

Derived the shift formula using classical Maxwell theory.

Connected the shift to properties of Poincare algebra for massless particles.

Abstract

Recently (Phys. Rev. Lett. 114 (2015), 210402) the influence of the so called "Wigner translations" (more generally-Lorentz trans- formations) 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 trans- lations have an efect 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 proper- ties of electromagnetic field on quantum and classical levels. It allows for a straightforward derivation of the helicity-dependent transforma- tion 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.