The Theory of Spinors

Abstract

The concept of a “spinor” emerged from the work of E. Cartan on the representations of simple Lie algebras. However, it was not until Dirac employed a special case in the construction of his relativistically invariant equation for the electron with “spin” that the notion acquired its present name or its current stature in mathematical physics. In this chapter we present an elementary introduction to the algebraic theory of spinors in Minkowski spacetime and illustrate its utility in special relativity by recasting in spinor form much of what we have learned about the structure of the electromagnetic field in Chapter 2. We shall not stray into quantum mechanics and, in particular, will not discuss the Dirac equation (for this, see the encyclopedic monograph[PR] of Penrose and Rindler). Since it is our belief that an intuitive appreciation of the notion of a spinor is best acquired by approaching them by way of group representations, we have devoted this first section to an introduction to these ideas and how they arise in special relativity. Since this section is primarily motivational, we have not felt compelled to prove everything we say and have, at several points, contented ourselves with a reference to a proof in the literature.