Wigner Translations and the Observer Dependence of the Position of Massless Spinning Particles

Abstract

The Wigner little group for massless particles is isomorphic to the Euclidean group SE(2).Applied to momentum eigenstates, or to infinite plane waves, the Euclidean "Wigner translations" act as the identity. We show that when applied to finite wavepackets the translation generators move the packet trajectory parallel to itself through a distance proportional to the particle's helicity.We relate this effect to the spin Hall effect of light and to the Lorentz-frame dependence of the position of a massless spinning particle.

Chiral fermions, massless particles and Poincare covariance

“…Similar side jumps which depend only on the kinematics of the problem appear, for example, in impurity scattering caused by spin-orbit interaction [27]. This phenomenon seems also to have its counterpart in optics in the form of the relativistic Hall effect of light [28][29][30][31][32][33][34]15] (see also [35][36][37]). …”

“…Another nice argument in favor of "exotic" transformation has been given in Ref. [16] (see also [15]) where the zero impact parameter collision of two massless particles of nonvanishing helicities was considered. By applying the Lorentz boost along the direction of motion of one incoming particle it is shown there that such a boost must result in "side jump" in order to fulfill the angular momentum conservation law.…”

Position, spin, and orbital angular momentum of a relativistic electron

“…Importantly, relativistic transformations of the angular-momentum tensor also require the centroid of the electron carrying intrinsic AM to be shifted in the direction orthogonal to both v andz [54,55]. As a result, the expectation value of the electron coordinate in the moving frame becomes [54][55][56][57] …”

“…In fact, q = R, and its expectation value for a single-electron state corresponds to the center of the probability density (center of charge), while the center of energy is defined as r E = N / H = q . The difference is important, e.g., for the "relativistic Hall effect" (11), where the center of energy r E undergoes the transverse shift twice as large as the center of the probability density [54,55,57]. Second, the projected position operator R and the spin-Hall effect corresponding to it appeared in 1959 in the work of Adams and Blount [33] (up to some arithmetic inaccuracies therein).…”

Chiral wave-packet scattering in Weyl semimetals

“…[22][23][24][25][26][27][28][29][30][31][32] . The Weyl semimetal state has been confirmed by photoemission and magneto-transport experiments in the noncentrosymmetric TaAs family of compounds.…”