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Geometric Aspects of Spinors

A Short Review
  • Andrzej Trautman
Conference paper
Part of the Fundamental Theories of Physics book series (FTPH, volume 55)

Abstract

Geometric properties of spinors are reviewed in connection with their role in complex and optical geometry. According to Cartan and Chevalley, a Weyl spinor φ ≠0,associated by the Dirac representation γ with a complex, 2m–dimensional vector space W, is called pure if the vector subspace N(φ) consisting of all elements ω of W such that γ(ω)φ = 0 is maximal, i.e. m-dimensional. If W is the complexification of a real space V with a scalar product of signature (2p+ε,2q+ε), where ε =0 or 1 and p+q+ε= m, then the real index of φ, γ = dim\(N(\varphi ) \cap \overline {N(\varphi )}\) in the generic case equals γ. Therefore, the direction of a generic pure spinor defines in V a complex (ε = 0) or an optical (ε = 1) structure. These observations are applied to a smooth, orientable 2m-dimensional spin manifold M with a bundle of directions of generic pure spinors. A section of this bundle - if it exists - defines an almost complex or an almost optical geometry, depending on whether γ = 0 or 1. With such a section one associates a bundle N of maximal, totally null subspaces of the complexified tangent spaces toM. Denoting by Z the module of sections of the bundleN, one considers the integrability conditions[Z,Z] ⊂Z In the pseudo-Euclidean case (γ = 0), the condition is equivalent to the vanishing of the Nijenhuis tensor of the almost complex structure; in the Lorentzian, 4-dimensional case, it is related to the geodetic, shear-free properties of the trajectories of the real line bundle Re\((N \cap \overline N )\)M.

Key words

Pure spinors optical and CR geometries algebraically special gravitational fields 

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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Andrzej Trautman
    • 1
  1. 1.Instytut Fizyki Teoretycznej UWWarszawaPoland

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