Generalized Fierz Identities and the Superselection Rule for Geometric Multispinors

TL;DR

This paper explores the reconstruction of multivector wave functions in 3D geometric algebra, revealing that standard Fierz identities are limited and depend on superselection rules, with implications for spinor theory.

Contribution

It introduces generalized Fierz identities considering right-side operators and clarifies the superselection rule's role in geometric multispinor frameworks.

Findings

Standard Fierz identities do not always hold.

Operators on the wave function's right side are essential.

Superselection rules determine the validity of identities.

Abstract

The inverse problem, to reconstruct the general multivector wave function from the observable quadratic densities, is solved for 3D geometric algebra. It is found that operators which are applied to the right side of the wave function must be considered, and the standard Fierz identities do not necessarily hold except in restricted situations, corresponding to the spin-isospin superselection rule. The Greider idempotent and Hestenes quaterionic spinors are included as extreme cases of a single superselection parameter.