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.
may be decomposed in terms of the Dirac matrices that span the space: The coefficients are 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 The table is symmetric with respect to reflection across the central element.
For example, under the assumption of commuting spinors, the V × V product can be expanded as, Combinations of bilinears corresponding to the eigenvectors of the transpose matrix transform to the same combinations with eigenvalues ±1.