ELKO, flagpole and flag-dipole spinor fields, and the instanton Hopf fibration

Abstract

In a previous paper we explicitly constructed a mapping that leads Dirac spinor fields to the dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields). ELKO spinor fields are prime candidates for describing dark matter, and belong to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification, based on the relations and values taken by their associated bilinear covariants. Such a mapping between Dirac and ELKO spinor fields was obtained in an attempt to extend the Standard Model in order to encompass dark matter. Now we prove that such a mapping, analogous to the instanton Hopf fibration map $S^3...S^7\to S^4$, prevents ELKO to describe the instanton, giving a suitable physical interpretation to ELKO. We review ELKO spinor fields as type-(5) spinor fields under the Lounesto spinor field classification, explicitly computing the associated bilinear covariants. This paper is also devoted to investigate some formal aspects of the flag-dipole spinor fields, which correspond to the class-(4) under the Lounesto spinor field classification. In addition, we prove that type-(4) spinor fields (corresponding to flag-dipoles) and ELKO spinor fields (corresponding to flagpoles) can also be entirely described in terms of the Majorana and Weyl spinor fields. After all, by choosing a projection endomorphism of the spacetime algebra Cl(1,3) it is shown how to obtain ELKO, flagpole, Majorana and Weyl spinor fields, respectively corresponding to type-(5) and -(6) spinor fields, uniquely from limiting cases of a type-(4) (flag-dipole) spinor field, in a similar result obtained by Lounesto.

17 Pages, RevTeX, accepted for publication in Adv. Appl. Clifford Alg

Cited by in corpus (60)

• Dynamics of Teleparallel Dark Energy
• Self-interacting Elko dark matter with an axis of locality
• Classification of Singular Spinor Fields and Other Mass Dimension One Fermions
• The theory of local mass dimension one fermions of spin one half
• Unfolding Physics from the Algebraic Classification of Spinor Fields
• Dynamical dispersion relation for ELKO dark spinor fields
• Exotic Dark Spinor Fields
• Very special relativity as relativity of dark matter: the Elko connection
• A modified theory of gravity with torsion and its applications to cosmology and particle physics
• An Identification of the Dirac Operator with the Parity Operator
• The most general cosmological dynamics for ELKO Matter Fields
• Conformal Gravity with the most general ELKO Matter
• Localization of 5D Elko Spinors on Minkowski Branes
• Lorentz invariant dark-spinor and inflation
• On the bilinear covariants associated to mass dimension one spinors
• Spinor Fields Classification in Arbitrary Dimensions and New Classes of Spinor Fields on 7-Manifolds
• Questing for Algebraic Mass Dimension One Spinor Fields
• Constraints on mass dimension one fermionic dark matter from the Yukawa interaction
• A generally-relativistic gauge classification of the Dirac fields
• ELKO and Dirac Spinors seen from Torsion
• Super-inflation and generation of first order vector perturbations in ELKO
• Elko and mass dimension one field of spin one half: causality and Fermi statistics
• Effective lagrangian for a mass dimension one fermionic field in curved spacetime
• The most general ELKO Matter in torsional f(R)-theories
• Attractor behaviour in ELKO cosmology
• On the Spinor Representation
• Torsion Gravity for Dirac Fields
• General Dynamics of Spinors
• The connection between Dirac dynamic and parity symmetry
• Weyl and Majorana Spinors as Pure Goldstone Bosons
• Polar Form of Spinor Fields from Regular to Singular: the Flag-Dipoles
• Fermionic dark matter-photon quantum interaction: A mechanism for darkness
• Generalized non-associative structures on the 7-sphere
• Non-Associativity in the Clifford Bundle on the Parallelizable Torsion 7-Sphere
• Spin-half mass dimension one fermions and their higher-spin generalizations
• A hint towards mass dimension one Flag-dipole spinors
• Fermionic degeneracy and non-local contributions in flag-dipole spinors and mass dimension one fermions
• Spinor Fields, Singular Structures, Charge Conjugation, ELKO and Neutrino Masses
• Geometry, Zitterbewegung, Quantization
• ELKO in Polar Form
• Symmetries and unitary interactions of mass dimension one fermionic dark matter
• Torsion Axial Vector and Yvon-Takabayashi Angle: Zitterbewegung, Chirality and all that
• The role of singular spinor fields in a torsional gravity, Lorentz-violating, framework
• Least-Order Torsion-Gravity for Dirac Fields, and their Non-Linearity terms
• Singular spinors and their connection
• Non-Trivial Effects of Sourceless Forces for Spinors: Toward an Aharonov-Bohm gravitational effect?
• Carroll spinors
• Flag-dipole spinors: On the dual structure derivation and $\mathcal{C}$, $\mathcal{P}$ and $\mathcal{T}$ symmetries
• Generalized Elko Theory
• Type-4 spinors: transmuting from Elko to single-helicity spinors
• The Tensorial Connections
• Singular spinors as expansion coefficients of local spin-half fermionic and bosonic fields: On the two-fold Wigner degeneracy
• Re-normalizable Chern-Simons Extension of Propagating Torsion Theory
• The Restricted Inomata-McKinley spinor-plane, homotopic deformations and the Lounesto classification
• Additional fermionic fields onto parallelizable 7-spheres
• Necessity of Tensorial Connections for Spinorial Systems
• Geometry of spinors: doubly-chiral plane-wave expansion
• Wigner multiplets in QFT: from Wigner degeneracy to Elko fields
• The Most General Propagator in Quantum Field Theory
• Chern-Simons Extension of ESK Theory