Crumeyrolle’s Bivectors and Spinors

  • Pertti Lounesto
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 321)

Abstract

Crumeyrolle often posed a rhetoric but baffling question: ‘What is a bivector?’ In this way Crumeyrolle tried to point out that bivectors do not exist in Clifford algebras, especially they do not exist in a canonical way in characteristic 2. However, there is a natural way to introduce bivectors in all other characteristics ≠ 2, because there is a one-to-one correspondence between quadratic forms and symmetric bilinear forms.

Crumeyrolle also emphasized geometric aspects of pure spinors because they are induced by maximal totally null subspaces of neutral quadratic spaces. The bilinear covariants of pure spinors are not directly related to the physical observables of the Dirac equation. In this paper a variant of Crumeyrolle’s spinoriality transformation is applied to extract the observables from Crumeyrolle’s spinors in such a way that they coincide with the bilinear covariants of standard column spinors, like those in Bjorken & Drell.

In short, this article solves a problem related to Crumeyrolle’s spinors and throws light on Crumeyrolle’s baffling question: ‘What is a bivector?’

Key words

Exterior algebra dimension grading contraction Clifford algebra parity grading spinor operators Weyl/pure spinors bilinear covariants spinoriality groups 
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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Pertti Lounesto
    • 1
  1. 1.Institute of MathematicsHelsinki University of TechnologyEspooFinland

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