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. 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. In particular, when working in 4 spacetime dimensions the bivector ψ χ ¯ {displaystyle psi {ar {chi }}} may be decomposed in terms of the Dirac matrices that span the space: ψ χ ¯ = 1 4 ( c S 1 + c V μ γ μ + c T μ ν T μ ν + c A μ γ μ γ 5 + c P γ 5 ) {displaystyle psi {ar {chi }}={frac {1}{4}}(c_{S}mathbb {1} +c_{V}^{mu }gamma _{mu }+c_{T}^{mu u }T_{mu u }+c_{A}^{mu }gamma _{mu }gamma _{5}+c_{P}gamma _{5})} . The coefficients are c S = ( χ ¯ ψ ) , c V μ = ( χ ¯ γ μ ψ ) , c T μ ν = − ( χ ¯ T μ ν ψ ) , c A μ = − ( χ ¯ γ μ γ 5 ψ ) , c P = ( χ ¯ γ 5 ψ ) {displaystyle c_{S}=({ar {chi }}psi ),c_{V}^{mu }=({ar {chi }}gamma ^{mu }psi ),c_{T}^{mu u }=-({ar {chi }}T^{mu u }psi ),c_{A}^{mu }=-({ar {chi }}gamma ^{mu }gamma _{5}psi ),c_{P}=({ar {chi }}gamma ^{5}psi )} , and are usually determined by using the orthogonality of the basis under the trace operation. By sandwiching the above decomposition between the desired gamma structures, the identities for the contraction of two Dirac bilinears of the same type can be written with coefficients according to the following table. where S = χ ¯ ψ , ,   V = χ ¯ γ μ ψ ,   T = χ ¯ [ γ μ , γ ν ] ψ / 2 2 ,   A = χ ¯ γ 5 γ μ ψ ,   P = χ ¯ γ 5 ψ {displaystyle S={ar {chi }}psi ,,~V={ar {chi }}gamma ^{mu }psi ,~T={ar {chi }}psi /2{sqrt {2}},~A={ar {chi }}gamma _{5}gamma ^{mu }psi ,~P={ar {chi }}gamma _{5}psi } . The table is symmetric with respect to reflection across the central element. The signs in the table correspond to the case of commuting spinors, otherwise, as is the case of fermions in physics, all coefficients change signs.