Rotation and lorentz transformations in 2x2 and 4x4 complex matrices and in polarized-light physics

Hermitian 2 x 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 Poincare sphere. Equivalent transformations are obtained for 4 x 4 anticommuting Hermitian Dirac matrices with two types of unitary matrices, corresponding to rotation and Lorentz transformations. Using exponential matrix representations, the 4 x 4 form can be related to the 2 x 2 form. The 4 x 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. (C) Elsevier Science Inc., 1997.