Generalized Fierz identities

Low energy weak interactions calculations with fermions frequently involve asuperposition of quartic products of Dirac spinors, in which the order of thespinors is not the same in all the contributing terms. A common trick that isused to bring them to a uniform ordering is the Fierz transformation. We showthat the standard Fierz rearrangement formula quoted in textbooks is oneelement of a class of transformations of a quartic product amplitude, underwhich the spinors are rearranged with different orderings and, in the generalcase, some or all of the spinors are transformed to their Lorentz-invariantcomplex conjugate form. We give a pedagogical derivation of the explicit formsof all such transformation matrices. In addition to the usual Lorentz scalarquartic products, we consider pseudoscalar ones as well. Such manipulations andformulas are useful, in particular, when some of the fermions involved areMajorana particles.Comment: 18 pages, Late