A gauge symmetric approach to Fierz identities

Abstract

Seventy-five years ago Cartan invented spinors by mapping C2 onto isotropic (null) vectors in C3. In recent work this map was extended and it was shown that bispinors are isomorphic to a class of Yang–Mills vector triplets Fk =Ek +iHk which satisfy the following SU(2)×U(1) gauge invariant constraint: Fj ⋅ Fk =s2 δjk, where s2 = (1)/(3) Fk ⋅ Fk (k summed from 1 to 3). Thus bispinors have inherent SU(2) ×U(1) gauge symmetry. In this paper it is shown, using the extended Cartan map and the gauge symmetry of the constrained Yang–Mills fields, that all the Fierz identities reduce to a single equation. Moreover, this equation includes not only the 75 identities recently derived by Takahashi [Y. Takahashi, J. Math. Phys. 24, 1783 (1983)] but an additional 75 which come from interchanging gauge and vector components. It is further shown that the Fierz identities for bispinors can be generalized to any multiplet, Ψ∈C2n, consisting of 2n−1 spinors (n=1 for spinors, n=2 for bispinors, n=3 for bispinor doublets, etc.). The generalized identities can also be used to show that the 2n−1 spinor multiplets are isomorphic to multiplets of constrained Yang–Mills vector fields.