Eulerian parametrization of Wigner’s little groups and gauge transformations in terms of rotations in two-component spinors

Abstract

A set of rotations and Lorentz boosts is presented for studying the three-parameter little groups of the Poincaré group. This set constitutes a Lorentz generalization of the Euler angles for the description of classical rigid bodies. The concept of Lorentz-generalized Euler rotations is then extended to the parametrization of the E(2)-like little group and the O(2,1)-like little group for massless and imaginary-mass particles, respectively. It is shown that the E(2)-like little group for massless particles is a limiting case of the O(3)-like or O(2,1)-like little group. A detailed analysis is carried out for the two-component SL(2,c) spinors. It is shown that the gauge degrees of freedom associated with the translationlike transformation of the E(2)-like little group can be traced to the SL(2,c) spins that fail to align themselves to their respective momenta in the limit of large momentum and/or vanishing mass.