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

TL;DR

This paper reveals that a simplified two-parameter form of the $ABCD$ matrix in optics encodes fundamental space-time symmetries of elementary particles, linking optical devices to particle physics concepts.

Contribution

It demonstrates that the equi-diagonal form of the $ABCD$ matrix captures essential space-time symmetry information of particles, bridging optics and particle physics.

Findings

The trace of the matrix indicates particle mass properties.

Optical devices like laser cavities exemplify the matrix's physical relevance.

The matrix's form encodes space-time symmetry classes of particles.

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.