The O(3,2) Symmetry derivable from the Poincar\'e Sphere

TL;DR

This paper reveals that the Poincaré sphere's symmetry extends to an O(3,2) group, encompassing mass variation in Lorentz transformations, which is not possible with the Lorentz group alone.

Contribution

It demonstrates that the Poincaré sphere contains an O(3,2) symmetry that accounts for mass variation, expanding the understanding of Lorentz symmetries in polarization optics.

Findings

The Poincaré sphere's symmetry includes the Lorentz group.

An O(3,2) symmetry related to mass variation is identified.

Illustrative calculation supports the theoretical claim.

Abstract

Henri Poincar\'e formulated the mathematics of the Lorentz transformations, known as the Poincar\'e group. He also formulated the Poincar\'e sphere for polarization optics. It is noted that his sphere contains the symmetry of the Lorentz group applicable to the momentum-energy four-vector of a particle in the Lorentz-covariant world. Since the particle mass is a Lorentz-invariant quantity, the Lorentz group does not allow its variations. However, the Poincar\'e sphere contains the symmetry corresponding to the mass variation, leading to the $O(3,2)$ symmetry. An illustrative calculation is given.