The theory of spinors via involutions and its application to the representations of the Lorentz group

Abstract

A simple theory of the spinor representations of the complex orthogonal group O(d,C) in the d-dimensional Euclidean space V(d) is presented via a basic lemma on involutional transformations and Cartan’s theorem on O(d,C). The arbitrary gauge factors of the representations are reduced to ± signs by introducing appropriate phase conventions. The concept of an axial involution is introduced. The plane rotations in V(d) are introduced and used to construct the representations of the proper orthogonal group O+(d,C). The Lorentz group is treated as a subgroup of O(4,C). The general expression for the basic 2×2 irreducible representations A(L0) of the proper orthochronous Lorentz group G(L0) is obtained by direct reduction of the 4×4 spinor representation S(L0) by means of the basic lemma on the involutional transformations. It is completely parameterized by the angle and the axis of the spacial rotation and by the velocity of the pure Lorentz transformation. The finite dimensional irreducible representations of the Lorentz group G(L) are discussed. The transformations of electro-magnetic field under G(L) are discussed in the most general form.