Essential Fierz identities for a fermionic field

Abstract

For a single fermionic field, an interpretation of the Fierz identities (which establish relations between the bilinear field observables) is given. They appear closely related to the algebraic class (regular or singular) of the spin 2-form $S$ associated to the spinor field. If $S\ne 0$, the Fierz identities follow from the 3+1 decomposition of the eigenvector equations for $S$ with respect to an inertial laboratory, which makes this interpretation suitable for fermionic particle physics models. When $S=0$, the Fierz identities reduce to three constraints on the current densities associated with the spinor field, saying that they are orthogonal, equimodular, the vector current being timelike and the axial one being spacelike.

17 pages

References in corpus (9)

• From Dirac spinor fields to ELKO
• Classification of Singular Spinor Fields and Other Mass Dimension One Fermions
• On the bilinear covariants associated to mass dimension one spinors
• Questing for Algebraic Mass Dimension One Spinor Fields
• Spin-half bosons with mass dimension three half: Evading the spin-statistics theorem
• Weyl and Majorana Spinors as Pure Goldstone Bosons
• Fierz bilinear formulation of the Maxwell-Dirac equations and symmetry reductions
• Relativistic dynamical inversion in manifestly covariant form
• Spin-1/2 "bosons'' with mass dimension 3/2 and fermions with mass dimension 1 cannot represent physical particle states

Cited by in corpus (2)

• A note on hidden classes in spinor classification
• Euler and Pontryagin currents of the Dirac operator