# 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,

${\displaystyle \left({\bar {\chi }}\gamma ^{\mu }\psi \right)\left({\bar {\psi }}\gamma _{\mu }\chi \right)=\left({\bar {\chi }}\chi \right)\left({\bar {\psi }}\psi \right)-{\frac {1}{2}}\left({\bar {\chi }}\gamma ^{\mu }\chi \right)\left({\bar {\psi }}\gamma _{\mu }\psi \right)-{\frac {1}{2}}\left({\bar {\chi }}\gamma ^{\mu }\gamma _{5}\chi \right)\left({\bar {\psi }}\gamma _{\mu }\gamma _{5}\psi \right)-\left({\bar {\chi }}\gamma _{5}\chi \right)\left({\bar {\psi }}\gamma _{5}\psi \right).}$

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.