Bispinor geometry for even-dimensional space-time

J. P. Crawford

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


The geometric properties of Dirac spinor fields defined over even-dimensional space-time are explored with the aim of formulating the associated nonlinear sigma models. A spinor field Ψ may be uniquely reconstructed from the real bispinor densities ρi = Ψ̄ΓiΨ apart from an overall phase, so that the ρi constitute an alternate representation of the physical information contained in Ψ. For space-time of dimension N = 2n, the corresponding Dirac spinor has D = 2n complex components, and the bispinor densities satisfy a system of (D- 1)2 homogeneous quadratic algebraic equations. The basis elements of the Clifford algebra {Γi} span a D2 = 2N-dimensional space whose Cartan metric is flat pseudo-Riemannian; the bispinor densities reside in the (2D - 1)-dimensional curved subspace induced as an embedding by the algebraic constraints. The explicit geometric structure of the bispinor spaces are examined and found to be generalizations of Robertson-Walkerspace. In particular, the line element may be written as dS2 = D dσ2 + σ22, where σ = Ψ̄Ψis the scalar density and dΩ is the line element for the homogeneous space: SU(D/2,D/2)/S(U(1) ⊗ U(D/2 - 1,D/2)).

Original languageEnglish (US)
Pages (from-to)1991-1997
Number of pages7
JournalJournal of Mathematical Physics
Issue number8
StatePublished - 1990

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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