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