Dirac general covariance and tetrads. I. Clifford and Lie bundles and torsion

Abstract

The isomorphic map of Clifford and Lie bundles to arbitrary coordinate atlases using a global orthonormal tetrad field on a parallelizable space-time is used to construct a fully covariant Dirac spinor theory. The Klein–Gordon equation exhibits a natural spin-torsion coupling of the Einstein–Cartan form, the torsion coming from the tetrad field. The tetrad connection coefficients are explicitly derived in addition to their relationship to the usual Levi–Cività coefficients. Various topological conditions for vanishing torsion are given. The Dirac and adjoint Dirac equations are obtained from a simple Lagrangian and the structure of the adjoint equation is discussed.