The Language of Two-by-two Matrices spoken by Optical Devices

Abstract

With its three independent parameters, the $ABCD$ matrix serves as the beam transfer matrix in optics. If it is transformed to an equi-diagonal form, the matrix has only two independent parameters determined by optical devices. It is shown that this two-parameter mathematical device contains enough information to describe the basic space-time symmetry of elementary particles. If its trace is smaller than two, this matrix can represent the internal space-time symmetry of massive particles. If equal to two, the matrix is of massless particles. If the trace is greater than two, this matrix describes imaginary-mass particles. This matrix speaks Einstein's language for space-time structure of elementary particles. As for the optical devices, the laser cavity and the multilayer system are discussed as illustrative physical examples.