Reflections, spinors, and projections on a Minkowski space underlie Dirac's equation

Abstract

Reflections and spinors on a Minkowski space are obtained by the methods of Cartan. The group of all such reflection transformations has as its subgroup the Poincaré group of Lorentz transformations. Two types of spinors are shown to exist for a Minkowski space. Spinor reflections and Lorentz transformations exist for both types of spinors. Associated with these spinors are two Hermitian, orthogonal projection operators which together spectrally resolve the identity operator of a two-dimensional complex vector space. These spinors and their associated projection operators are applied to find the structure of a 4 × 4 matrix G which is equivalent to the relativistic conservation law of the energy and momentum of a single moving particle. These spinor-calculus procedures demonstrate that G is singular of rank 2, and as a consequence the solutions of G θ = 0 consists of all bispinor elements of the null space of G. This equation and its solutions are equivalent to those of Dirac's quantum-mechanical equation of an electron in the Fourier domain of frequency and wavenumber. Finally some properties of 2-spinors and 4-spinors found herein are shown to extend naturally to n dimensions.