Symmetries shared by the Poincar\'e Group and the Poincar\'e Sphere

TL;DR

This paper explores the mathematical connection between the Poincaré group and the Poincaré sphere, revealing shared symmetries through Lorentz group representations and linking particle mass invariance to polarization optics decoherence.

Contribution

It demonstrates that the Poincaré sphere and Poincaré group share underlying mathematical structures derived from Lorentz group representations, bridging optics and particle physics.

Findings

Shared symmetries derived from Lorentz group representations

Link between particle mass invariance and polarization decoherence

Detailed analysis of Wigner's little groups for internal symmetries

Abstract

Henri Poincar\'e formulated the mathematics of Lorentz transformations, known as the Poincar\'e group. He also formulated the Poincar\'e sphere for polarization optics. It is shown that these two mathematical instruments can be derived from the two-by-two representations of the Lorentz group. Wigner's little groups for internal space-time symmetries are studied in detail. While the particle mass is a Lorentz-invariant quantity, it is shown possible to address its variations in terms of the decoherence mechanism in polarization optics.