Spin connection in general relativity

Abstract

The spin connection in the Riemann space of general relativity defines equivalence of two spinors at infinitesimally neighboring events, and evidently carries information about the environment of charged test particles of the fermion type. In this paper, we consider the spin connection in the four-dimensional space of events as fundamental, and study its concomitants and the consequences of its existence. We find that, if the spin connection permits the existence of a field of Dirac operators γk and as associated Riemann geometry, it leaves the γk undetermined by a family of continuous transformations generated by γ 5 with a uniform mangitude. If the physics of fermions could be expressed solely in terms of the spin connection, the mean values of all observables would have to be invariant under this family of transformations, and the two-component fermion description proposed by Feynman, Gell-Mann, Sudarshan, and Marshak would follow. Another indeterminancy of the γk, for a fixed spin connection, consists of a family of scale transformations of uniform magnitude over the whole space of events. The transformations of this family are of no physical consequence, as they can be compensated by a uniform change in proper time scale. For a fixed spin connection, there are usually no other indeterminacies of the γk of the continuous kind. The existence of the spin connection implies a conservation law for a spin tensor density derived from the Dirac operators and the spin curvature tensor, whose trace is the Einstein tensor density.