Bilinear Forms and Fierz Identities for Real Spin Representations

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.

42 pages. v3 fixes typo defining M on page 28, adds reference to Krawtchouk matrices

References in corpus (1)

• Basics of M-Theory

Cited by in corpus (3)

• Spinor Fields Classification in Arbitrary Dimensions and New Classes of Spinor Fields on 7-Manifolds
• The twistor spinors of generic 2- and 3-distributions
• Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors