Advances in Applied Clifford Algebras

, Volume 22, Issue 2, pp 329–363 | Cite as

Bilinear Forms and Fierz Identities for Real Spin Representations



Given a real representation of the Clifford algebra corresponding to \({\mathbb{R}^{p+q}}\) 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 introduced in [1], 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 [2][3].


Fierz identities spinors Clifford algebras spin representation 


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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Laboratory of Axiomatics, Department of MathematicsUniversity of PittsburghPittsburghUSA

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