General Fierz-type identities are examined and their well-known connection with completeness relations in matrix vector spaces is shown. In particular, I derive the chiral Fierz identities in a simple and systematic way by using a chiral basis for the complex matrices. Other completeness relations for the fundamental representations of algebras can be extracted using the same reasoning.
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Any operation applied to an element in the vector space must result in another element of the vector space.
may be considered as a dimensional vector space if spanned by real matrices and purely complex matrices with real expansion coefficients only, that is, over the reals .
The SU(2) and SU(3) generators are and , respectively.
I will denote as Dirac bilinears the proper bilinears containing the two spinors as well as the associated matrices alone, because the Fierz identities do not depend on the spinors involved.
The lowering of space-time indices is equivalent to the hermitian conjugation operation.
