Rotation and Lorentz transformations in 2 × 2 and 4 × 4 complex matrices and in polarized-light physics Original Research Article

Hermitian 2 × 2 matrices exhibit basic 3D rotational and 4D Lorentz transformation properties. These matrices arise naturally in representations of the time-averaged pair products or intensities of any two-element wave, giving rise to the light Stokes-parameter transformation properties on the Poincaré sphere. Equivalent transformations are obtained for 4 × 4 anticommuting Hermitian Dirac matrices with two types of unitary matrices, corresponding to rotation and Lorentz transformations. Using exponential matrix representations, the 4 × 4 form can be related to the 2 × 2 form. The 4 × 4 representation has physical significance for the subset of intensity-distinguishable two-element standing-wave modes of a cavity, e.g. light standing waves. There is a basic resemblance between (1) the temporal differential equation for two-element standing waves in time, three observable “Stokes” parameters, and frequency and (2) the Dirac equation for spin-1/2 free-space particle states in time, three momenta, and particle rest mass. This resemblance is the basis for an optical analog with relativistic quantum mechanics which we describe.