12 - Flags, poles and dipoles

Summary

The classification of spinors is commonly based on representation theory, irreducible representations of the Lorentz group SO₊ (1, 3). Thus, one customarily speaks about Dirac, Major ana and Weyl spinors. In this chapter spinors are classified in a new way by their bilinear covariants, multivectors of observables. The new classification is geometric, since it is based on multivectors, and physical, since it is based on observables. The classification reveals new spinors, called flag-dipole spinors, which reside between the Weyl, Majorana and Dirac spinors.

Dirac spinors describe the electron, and for them Ω ≠ 0. Weyl and Majorana spinors describe the neutrino. Weyl spinors are eigenspinors of the helicity projection operators ½(1 ± γ₀₁₂₃), and their bilinear covariants satisfy Ω = 0, S = 0, K ≠ 0. Majorana spinors are eigenspinors of the charge conjugation operator, with eigenvalues ±1, and their bilinear covariants satisfy Ω = 0, S ≠ 0, K = 0. [Weyl and Majorana spinors are usually introduced by properties of matrices, see Benn & Tucker 1987 and Crumeyrolle 1990.]

The flag-dipole spinors satisfy Ω = 0 [and cannot be Dirac spinors] and S ≠ 0, K ≠ 0 [and so they are neither Weyl nor Majorana spinors]. Unlike Weyl and Majorana spinors, the flag-dipole spinors do not form a real linear subspace, because they are characterized by a quadratic constraint. Therefore the superposition principle is violated, and the flag-dipole spinors cannot describe fermions.