Bispinor geometry for even-dimensional space-time

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_iΨ 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 {Γ_i} span a D² = 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 dS² = D dσ² + σ² dΩ², 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)).