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The construction of spinor fields on manifolds with smooth degenerate metrics

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  • 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 express‐ ion 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.
  • Keywords: General Relativity Theory, Topology, Unified−Field Theories, Space−Time, Mathematical Manifolds, Dirac Equation, Metrics, Spinor Fields, Gravitational Fields
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  • Schray, J., Dray, T., Manogue, C. A., Tucker, R. W., & Wang, C. (1996, August). The construction of spinor fields on manifolds with smooth degenerate metrics. Journal of Mathematical Physics, 37(8), 3882-3896. doi:10.1063/1.531607
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  • 37
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  • 8
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  • This work was partially supported by NSF Grant No. PHY-9208494 (C.A.M., T.D., and J.S.), a Fulbright Grant (C.A.M.), and the Human Capital and Mobility Programme of the European Union (R.W.T.). C.W. is grateful to the Committee of Vice-Chancellors and Principals for an Overseas Research Studentship, Lancaster University for a Peel Studentship, and the School of Physics and Chemistry at Lancaster University for a School Studentship.
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