## Abstract

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 ρ_{i} = Ψ̄Γ_{i}Ψ 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 = 2^{n} complex components, and the bispinor densities satisfy a system of (D- 1)^{2} homogeneous quadratic algebraic equations. The basis elements of the Clifford algebra {Γ_{i}} span a D^{2} = 2^{N}-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 dS^{2} = D dσ^{2} + σ^{2} dΩ^{2}, 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 language | English (US) |
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Pages (from-to) | 1991-1997 |

Number of pages | 7 |

Journal | Journal of Mathematical Physics |

Volume | 31 |

Issue number | 8 |

DOIs | |

State | Published - Jan 1 1990 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics