Fierz Identity

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.

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 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 identities for a generic scalar written as the contraction of two Dirac bilinears of the same type can be written with coefficients according the following table.

Product S V T A P
S × S = 1/4 1/4 -1/4 -1/4 1/4
V × V = 1 -1/2 0 -1/2 -1
T × T = -3/2 0 -1/2 0 -3/2
A × A = -1 -1/2 0 -1/2 1
P × P = 1/4 -1/4 -1/4 1/4 1/4

For example the V × V product can be expanded as,

\left(\chi^\dagger\gamma^0\gamma^\mu\psi\right)\left(\psi^\dagger\gamma^0\gamma_\mu \chi\right)= \left(\chi^\dagger\gamma^0\chi\right)\left(\psi^\dagger\gamma^0\psi\right)- \frac{1}{2}\left(\chi^\dagger\gamma^0\gamma^\mu\chi\right)\left(\psi^\dagger\gamma^0\gamma_\mu\psi\right)- \frac{1}{2}\left(\chi^\dagger\gamma^0\gamma^\mu\gamma_5\chi\right)\left(\psi^\dagger\gamma^0\gamma_\mu\gamma_5\psi\right) -\left(\chi^\dagger\gamma^0\gamma^5\chi\right)\left(\psi^\dagger\gamma^0\gamma_5\psi\right).

Simplifications arise when the considered spinors are chiral or Majorana spinors as some term in the expansion can be vanishing.

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