Two-Component Spinors in General Relativity

In this paper we develop a general-relativistic theory of two-component spinors somewhat along the lines pioneered more than twenty years ago by Weyl and by Infeld and van der Waerden. This formalism is in some respects more natural than the theory of four-component spinors. We begin by introducing into a four-dimensional manifold as our basic geometric structure a set of four 2×2 Hermitian matrices, ${\sigma }^{\mu }$, and we show that these matrices by themselves define uniquely a Riemannian metric with the usual signature of a space-time manifold. It turns out that one can describe the gravitational field and its laws very conveniently in this matrix formalism; at the same time the ${\sigma }^{\mu }$ enable one to construct invariant two-component and four-component spinor wave equations. We use these formal possibilities to define local Lorentz transformations and, in particular, the transformations corresponding to time reversal and to space inversion.