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

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.