In theoretical physics

Theoretical physics

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, a Fierz identity is an identity that allows one to rewrite bilinear

Bilinear

Bilinear may refer to:* Bilinear sampling, a method in computer graphics for choosing the color of a texture* Bilinear form* Bilinear interpolation* Bilinear map, a type of mathematical function between vector spaces...

s of the product of two spinor

Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s as a linear combination

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

 of products of the bilinears of the individual spinors. It is named after Swiss physicist Markus Fierz

Markus Fierz

Markus Eduard Fierz was a Swiss physicist, particularly remembered for his formulation of Spin-statistics theorem, and for his contributions to the development of quantum theory, particle physics, and statistical mechanics...

.

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

Clifford algebra

In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...

. 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,

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