Poincar\'e Sphere and a Unified Picture of Wigner's Little Groups

TL;DR

This paper reveals that the Poincaré sphere's symmetries encompass Wigner's little groups, providing a unified geometric framework for understanding internal space-time symmetries of both massive and massless particles.

Contribution

It demonstrates that the Poincaré sphere can represent Wigner's little groups, unifying the description of particle symmetries for different mass states within a single geometric picture.

Findings

The Poincaré sphere contains Lorentz group symmetries.

It models the internal symmetries of particles with different masses.

A symmetry parameter allows transitioning from massive to massless particles.

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.