The Construction of Spinor Fields on Manifolds with Smooth Degenerate Metrics

TL;DR

This paper develops a framework for defining and analyzing spinor fields on manifolds with smooth degenerate metrics, especially near hypersurfaces where the metric becomes singular, extending the Dirac theory in this context.

Contribution

It introduces the concept of complex spinor fibration and derives a local spinor connection without orthonormal frames, enabling Dirac equations on degenerate manifolds.

Findings

Constructed a Dirac equation near metric degeneracy.

Identified additional freedom in the spinor theory compared to standard spacetime Dirac theory.

Discussed implications for continuity of conserved currents.

Abstract

We examine some of the subtleties inherent in formulating a theory of spinors on a manifold with a smooth degenerate metric. We concentrate on the case where the metric is singular on a hypersurface that partitions the manifold into Lorentzian and Euclidean domains. We introduce the notion of a complex spinor fibration to make precise the meaning of continuity of a spinor field and give an expression for the components of a local spinor connection that is valid in the absence of a frame of local orthonormal vectors. These considerations enable one to construct a Dirac equation for the discussion of the behavior of spinors in the vicinity of the metric degeneracy. We conclude that the theory contains more freedom than the spacetime Dirac theory and we discuss some of the implications of this for the continuity of conserved currents.