Essential Fierz identities for a fermionic field

TL;DR

This paper interprets Fierz identities for a fermionic field, linking them to the algebraic class of the associated spin 2-form and providing conditions on current densities, with implications for particle physics models.

Contribution

It offers a new interpretation of Fierz identities based on the algebraic class of the spin 2-form, connecting them to eigenvector equations and current constraints.

Findings

Fierz identities relate to the algebraic class of the spin 2-form S.

When S ≠ 0, identities derive from eigenvector equations in 3+1 decomposition.

When S = 0, identities impose orthogonality and equimodularity on currents.

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 $S \neq 0$, 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.