Bilinear Forms and Fierz Identities for Real Spin Representations

TL;DR

This paper explores bilinear forms on real spinor spaces associated with Clifford algebras, deriving identities that generalize Fierz identities and reveal connections to involutory matrices similar to Krawtchouk matrices.

Contribution

It introduces two natural bilinear forms on real spinors and derives a general class of Fierz identities, linking algebraic structures to involutory matrices.

Findings

Derived general Fierz identities from spinorial tensors.

Connected identities to matrices resembling Krawtchouk matrices.

Applied identities to signatures (1,3) and (10,1).

Abstract

Given a real representation of the Clifford algebra corresponding to $R^{p+q}$ with metric of signature $(p,q)$, we demonstrate the existence of two natural bilinear forms on the space of spinors. With the Clifford action of $k$-forms on spinors, the bilinear forms allow us to relate two spinors with elements of the exterior algebra. From manipulations of a rank four spinorial tensor, we are able to find a general class of identities which, upon specializing from four spinors to two spinors and one spinor in signatures (1,3) and (10,1), yield some well-known Fierz identities. We will see, surprisingly, that the identities we construct are partly encoded in certain involutory real matrices that resemble the Krawtchouk matrices.