Poincaré Sphere and a Unified Picture of Wigner’s Little Groups

Abstract

It is noted that the Poincar\'e sphere for polarization optics contains the symmetries of the Lorentz group. The sphere is thus capable of describing the internal space-time symmetries dictated by Wigner's little groups. For massive particles, the little group is like the three-dimensional rotation group, while it is like the two-dimensional Euclidean group for massless particles. It is shown that the Poincar\'e sphere, in addition, has a symmetry parameter corresponding to reducing the particle mass from a positive value to zero. The Poincar\'e sphere thus the gives one unified picture of Wigner's little groups for massive and massless particles.