On unique parametrization of the linear group GL(4.C) and its subgroups by using the Dirac matrix algebra basis

Abstract

A unifying overview of the ways to parameterize the linear group GL(4.C) and its subgroups is given. As parameters for this group there are taken 16 coefficients G = G(A,B,A_{k}, B_{k}, F_{kl}) in resolving matrix G in terms of 16 basic elements of the Dirac matrix algebra. Alternatively to the use of 16 tensor quantities, the possibility to parameterize the group GL(4.C) with the help of four 4-dimensional complex vectors (k, m, n, l) is investigated. The multiplication rules G'G are formulated in the form of a bilinear function of two sets of 16 variables. The detailed investigation is restricted to 6-parameter case G(A, B, F_{kl}), which provides us with spinor covering for the complex orthogonal group SO(3.1.C). The complex Euler's angles parametrization for the last group is also given. Many different parametrizations of the group based on the curvilinear coordinates for complex extension of the 3-space of constant curvature are discussed. The use of the Newmann-Penrose formalism and applying quaternion techniques in the theory of complex Lorentz group are considered. Connections between Einstein-Mayer study on semi-vectors and Fedorov's treatment of the Lorentz group theory are stated in detail. Classification of fermions in intrinsic parities is given on the base of the theory of representations for spinor covering of the complex Lorentz group.