Fierz identity

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In theoretical physics, a Fierz identity is an identity that allows one to rewrite the product of two spinor bilinears as a linear combination of products of two spinor bilinears.

If the four fermion expressions happen to be scalars, these are basically the 6j relations for spinors. If they are not scalars, these are the 9j relations for spinors.

There is a version of the Fierz identities for Dirac spinors and there is another version for Weyl spinors. And there are versions for other dimensions besides 3+1D.

Spinor bilinears can be thought of as elements of a Clifford Algebra. Then the Fierz identity is the concrete realization of the relation to the exterior algebra.

The Fierz identity is named after Markus Fierz (1912-2006), a professor of physics at the Swiss Federal Institute of Technology in Zürich and a close collaborator of Wolfgang Pauli.