Spinor Fields Classification in Arbitrary Dimensions and New Classes of Spinor Fields on 7-Manifolds

Abstract

A classification of spinor fields according to the associated bilinear covariants is constructed in arbitrary dimensions and metric signatures, generalizing Lounesto's 4D spinor field classification. In such a generalized classification a basic role is played by the geometric Fierz identities. In 4D Minkowski spacetime the standard bilinear covariants can be either null or non-null -- with the exception of the current density which is invariably different from zero for physical reasons -- and sweep all types of spinor fields, including Dirac, Weyl, Majorana and more generally flagpoles, flag-dipoles and dipole spinor fields. To obtain an analogous classification in higher dimensions we use the Fierz identities, which constrain the covariant bilinears in the spinor fields and force some of them to vanish. A generalized graded Fierz aggregate is moreover obtained in such a context simply from the completeness relation. We analyze the particular and important case of Riemannian 7-manifolds, where the Majorana spinor fields turn out to have a quite special place. In particular, at variance with spinor fields in 4D Minkowski spacetime that are classified in six disjoint classes, spinors in Riemannian 7-manifolds are shown to be classified, according to the bilinear covariants: (a) in just one class, in the real case of Majorana spinors; (b) in four classes, in the most general case. Much like new classes of spinor fields in 4D Minkowski spacetime have been evincing new possibilities in physics, we think these new classes of spinor fields in seven dimensions are, in particular, potential candidates for new solutions in the compactification of supergravity on a seven-dimensional manifold and its exotic versions.

15 pages

References in corpus (7)

• Classification of Singular Spinor Fields and Other Mass Dimension One Fermions
• Dynamical dispersion relation for ELKO dark spinor fields
• Exotic Dark Spinor Fields
• Hawking Radiation from Elko Particles Tunnelling across Black Strings Horizon
• The quadratic spinor Lagrangian, axial torsion current, and generalizations
• ELKO Spinor Fields: Lagrangians for Gravity derived from Supergravity
• Quantum tomography for Dirac spinors

Cited by in corpus (5)

• Revealing how different spinors can be: the Lounesto spinor classification
• Questing for Algebraic Mass Dimension One Spinor Fields
• Elko and mass dimension one field of spin one half: causality and Fermi statistics
• New fermions in the bulk
• The Heisenberg spinor field classification and its interplay with the Lounesto's classification