On the Restricted Lorentz Group and Groups Homomorphically Related to It

Abstract

A study is made of the real restricted Lorentz group, L, and of its relationship (a) to the group, SL(2C), of complex unimodular two‐dimensional matrices, and (b) to the group, O₃, of orthogonal transformations in a complex space of three dimensions. The discussion of case (a) is an improved version of the treatment by Wightman. Its notable features are, firstly, that it gives important formulas in new concise forms and their proofs in an elegant and economical manner, and, secondly, that it deals with the nontrivial matter of proving the internal consistency of the formalism. To illustrate the practical utility of the theory, the product of two nonparallel pure Lorentz transformations is studied. In the discussion of case (b), explicit formulas realizing the isomorphism of O₃ and L are obtained. These formulas are new and have been applied, for illustrative purposes, to the derivation of the transformation properties under L of the electromagnetic field vectors, regarded as a complex three‐vector (E + iH). A result analagous to the factorization of the general element of L into a spatial rotation and a pure Lorentz transformation, and to the polar decomposition of the general element of SL(2C), is derived for O₃. Insight into the relationship of O₃ to L is provided by considering the unimodular matrix description of the complex Lorentz group, and the contrasting specializations of it that lead to the unimodular matrix descriptions of its subgroups, O₃ and L.