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In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinors. It is named after Swiss physicist Markus Fierz. The Fierz identities are also sometimes called the Fierz–Pauli–Kofink identities, as Pauli and Kofink described a general mechanism for producing such identities.
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+1 dimensions. Spinor bilinears in arbitrary dimensions are elements of a Clifford algebra; the Fierz identities can be obtained by expressing the Clifford algebra as a quotient of the exterior algebra.
When working in 4 spacetime dimensions the bivector ψ χ ¯ {\displaystyle \psi {\bar {\chi }}} may be decomposed in terms of the Dirac matrices that span the space:
The coefficients are