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 to 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,

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

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

References

A derivation of identities for rewriting any scalar contraction of Dirac bilinears can be found in 29.3.4 of L. B. Okun (1980). Leptons and Quarks. North-Holland. ISBN 978-0-444-86924-1.

See also appendix B.1.2 in T. Ortin (2004). Gravity and Strings. Cambridge University Press. ISBN 978-0-521-82475-0.