The geometric algebra of Fierz identities in arbitrary dimensions and signatures

TL;DR

This paper introduces a geometric algebra framework for efficiently deriving Fierz identities across various dimensions and signatures, clarifying their algebraic structures and enabling computational implementation.

Contribution

It generalizes previous methods by providing a unified, explicit geometric algebra approach to Fierz identities in arbitrary dimensions and signatures.

Findings

Provides a synthetic, efficient method for Fierz identities

Clarifies the real, complex, quaternionic structures involved

Demonstrates the approach with examples across different signatures

Abstract

We use geometric algebra techniques to give a synthetic and computationally efficient approach to Fierz identities in arbitrary dimensions and signatures, thus generalizing previous work. Our approach leads to a formulation which displays the underlying real, complex or quaternionic structure in an explicit and conceptually clear manner and is amenable to implementation in various symbolic computation systems. We illustrate our methods and results with a few examples which display the basic features of the three classes of pin representations governing the structure of such identities in various dimensions and signatures.