Spinors beyond Dirac, Weyl and Majorana: the Flag-Dipoles

TL;DR

The paper revisits the Lounesto classification of 1/2-spin spinor fields and concentrates on flag-dipole spinors, whose physical realization has been elusive. It develops a polar-geometric framework, introducing a velocity vector $U^a$, a companion vector $X^a$, and a chiral angle α to recast the Dirac equation into reduced polar field equations for flag-dipoles, with all four spinor classes connected through consistent bilinears. By deriving a reduced flag-dipole system and providing two explicit exact solutions in spherical and cylindrical symmetry, the authors demonstrate that flag-dipole spinors can satisfy the Dirac equation in ways that are not Dirac, Weyl, or Majorana, while revealing rich spin and angular-momentum structures including soliton-like features. These results broaden the landscape of spinor solutions, motivate potential physical realizations, and invite further exploration of tensorial connections and broader backgrounds to generalize the constructions.

Abstract

We recall the Lounesto classification of 1/2-spin spinor fields, based on the vanishing of spinorial bilinear quantities: the classes are the regular spinor fields (i.e. the Dirac field), as well as singular spinor fields, also known as flag-dipole spinor fields, admitting two limiting sub-classes, given by the dipole spinors (i.e. the Weyl spinor) and the flagpole spinors (i.e. the Majorana spinor). We discuss each class in terms of its representatives, with particular emphasis upon the flag-dipole spinor fields.