Parameters of Lorentz Matrices and Transitivity in Polarization Optics

TL;DR

This paper explores the properties of Lorentz matrices within polarization optics, presenting a method to parameterize them and deriving transitivity relations that aid in analyzing Mueller matrices in optical experiments.

Contribution

It introduces a novel parameterization method for Lorentz matrices and derives transitivity equations relevant for polarization optics analysis.

Findings

Parameterization of Lorentz matrices using Dirac basis.

Explicit method to construct Lorentz parameters from matrices.

Transitivity relations applicable to polarization optics.

Abstract

In the context of applying the Lorentz group theory to polarization optics in the frames of Stokes-Mueller formalism, some properties of the Lorentz group are investigated. We start with the factorized form of arbitrary Lorentz matrix as a product of two commuting and conjugate $4\times 4$-matrices, $L(q,q)= A(q_{a}) A^*(q_{a}); a= 0,1,2,3$. Mueller matrices of the Lorentzian type M=L are pointed out as a special sub-class i n the total set of $4\times 4$ matrices of the linear group GL(4,R). Any arbitrary Lorentz matrix is presented as a linear combination of 16 elements of the Dirac basis. On this ground, a method to construct parameters q_a by an explicitly given Lorentz matrix L is elaborated. It is shown that the factorized form of L=M matrices provides us with a number of simple transitivity equations relating couples of initial and final 4-vectors, which are defined in terms of parameters q_a of the Lorentz group. Some of these transitivity relations can be interpreted within polarization optics and can be applied to the group-theoretic analysis of the problem of measuring Mueller matrices in optical experiments.