Bispinor geometry for even-dimensional space-time

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 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–Walker space. In particular, the line element may be written as dS2=D dσ 2+σ 2 dΩ2, where σ=Ψ̄Ψ is 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)).

Type-4 spinors: transmuting from Elko to single-helicity spinors

“…Classical spinors are objects of the space that carries the usual τ = (1/2, 0) ⊕ (0, 1/2) representation of the Lorentz group, that can be thought as being sections of the vector bundle P Spin e 1,3 (M) × τ C 4 . Given a spinor field ψ ∈ sec P Spin e 1,3 (M) × τ C 4 , the bilinear covariants are sections of the bundle (T M) [1,2]. Indeed, the well-known Lounesto spinor classification is based upon bilinear covariants and the underlying multivector structure.…”

“…The above identities are fundamental not only for classification, but also to further assert the inversion theorem [2]. Within the Lounesto classification scheme, a non vanishing J is crucial, since it enables to define the so called boomerang [1] which has an ample geometrical meaning to assert that there are precisely six different classes of spinors.…”

“…The spinors types-(1), (2) and (3), are called Dirac spinor fields (regular spinors). The spinor field (4) is called flagdipole [7], while the spinor field (5) is named flag-pole [13,16,17].…”

“…Since it was proposed by Lounesto [1], no physical or experimental evidence was found, therefore, we do not know what kind of particle is described by the mentioned class of spinors. The classical path that several physicists and mathematicians normally use to construct such spinors is based on the so-called inversion theorem [2,3], however, the last mentioned protocol do not explicitly provide the spinor's components. In other words, it only shows how components might be connected, therefore, we do not have enough information about how it emerges from the space-time symmetries.…”

Subliminal aspects concerning the Lounesto’s classification

“…up to uor knowledge, (11) is only fulfilled if α = β, otherwise the Dirac dynamic is not reached. The last result combined with Table 1 in [7] lead to the observation that only spinors belonging to class 2 within Lounesto's classification, under the requirement α = β, satisfy the Dirac equation.…”