Advances in Applied Clifford Algebras

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

Bilinear Forms and Fierz Identities for Real Spin Representations

  • Eric O. Korman
  • George Sparling


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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  1. 1.
    R. Penrose and W. Rindler, Spinors and Space-time, vol. 2. Cambridge, 1986.Google Scholar
  2. 2.
    P. Feinsilver and J. Kocik, Krawtchouk polynomials and Krawtchouk matrices. quant-ph/0702073Google Scholar
  3. 3.
    P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum random walks. quant-ph/0702173Google Scholar
  4. 4.
    P. Lounesto, Clifford Algebras and Spinors. Cambridge, 2001.Google Scholar
  5. 5.
    A. Miemiec and I. Schnakenburg, Basics of m-theory. hep-th/0509137.Google Scholar
  6. 6.
    S. Naito, K. Osada, and T. Fukui, Fierz identities and invariance of 11-dimensional supergravity action. Phys. Rev. D 34 (Jul, 1986) 536-552.Google Scholar
  7. 7.
    I. Porteous, Clifford Algebras and the Classical Groups. Cambridge, 1995.Google Scholar
  8. 8.
    H. B. Lawson and M. L. Michelsohn, Spin Geometry. Princeton, 1990.Google Scholar

Copyright information

© 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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