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.