Symmetries Shared by the Poincaré Group and the Poincaré Sphere

Henri Poincaré formulated the mathematics of Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré 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 to be possible to address its variations in terms of the decoherence mechanism in polarization optics.