General relativistic interpretation of some spinor wave equations

Abstract

The non-linear conform-invariant spinor wave equation proposed in an earlier paper is given a simple geometrical interpretation within the frame of general relativity. At each point of a conformal space-time manifold, an orthogonal frame of reference is defined, this frame being associated with a 4-spinorψ. It is shown that if the contracted torsion tensor ια relative to the frame vanishes (as in the case of a co-ordinate transformation) and the pseudo-vectorLα derived from the same torsion tensor remains constant and space-like, then the associated 4-spinorψ satisfies just the above mentioned wave equation which, in turn, determines the orthogonal frame of reference at every space-time point to within a conformal transformation of the co-ordinate system. In particular, if the pseudo-vectorLα also vanishes one obtains Dirac’s equation for a particle of rest mass zero. In a discussion of the physical meaning of the orthogonal frame of reference, the time-like basis vector and one of the space-like basis vectors are respectively related to the current density and the spin density 4-vectors associated with the wave equation.