I always thought of Fierz identities as a kind of completeness relation (*) for products of spinors. To use the bra-ket notation:
$$|a\rangle\langle b| = \sum k_i \langle b|M_i|a\rangle M_i$$
for some convenient trace orthogonal basis.

To find the $k_i$ for the specific basis and space that you're working in, you multiply by some $M_j$ and take the trace:
$$ tr(M_j |a\rangle\langle b|) = \langle b|M_j|a\rangle
= \sum_i k_i tr( M_i M_j ) \langle b|M_i|a\rangle
$$
since the basis is orthogonal with respect to the trace, we see that $k_i^{-1} = tr( M_i M_i )$.

This is then used to prove the Fierz identities as
$$ \langle a|U|b\rangle \langle c|V|d\rangle
= \sum_i\langle a|U (k_i \langle c|M_i|b\rangle M_i) V|d\rangle
= \sum_i k_i \langle c|M_i|b\rangle\langle a|U M_i V|d\rangle
$$

Thus you get your Fierz rearrangements. Note that depending on your definitions, if you're deriving the Fierz identities for anticommuting spinors, then you might have a sign factor in the first trace formula and definition of $k_i$ that I gave.

The standard Fierz identities are for 4-dimensional spinors, with the basis of gamma matrices
$$ 1\,,\quad \gamma_\mu\,,\quad
\Sigma_{\mu\nu}\;(\mu<\nu)\,,\quad
\gamma_5\gamma_\mu\,,\quad \gamma_5 $$
which has $1 + 4 + 6 + 4 + 1 = 16$ elements, which is what you'd expect for 4*4 matrices.
The basis elements are normally on both the left and right hand side of the Fierz identity. The traces for the above basis can be calculated from the Wikipedia Gamma matrix page.

In the previous question, Some Majorana fermion identities, I used 2-component notation to check the identities. This is convenient since the completeness relation is really simple. Using the conventions of Buchbinder and Kuzenko the spinor completeness relation is simply the decomposition into an antisymmetric and symmetric part:
$$ \psi_\alpha \chi_\beta = \frac12\varepsilon_{\alpha\beta}\psi\chi - \frac12(\sigma^{ab})_{\alpha\beta}\psi\sigma_{ab}\chi \,,
$$
and similarly for the dotted-spinors (complex conjugate representation).
This means that you don't need a table of coefficients to do Fierz rearrangements, all the steps fit easily in your memory.

I'm not sure what the best reference for Fierz identities is. Maybe have a look at Generalized Fierz identities and references within.

## Footnote (*)

You can also think of the completeness relation purely in terms of the matrices.
This is an alternate approach to derive the Fierz identities and is the one used in Generalized Fierz identities.

This post imported from StackExchange Physics at 2014-05-20 08:37 (UCT), posted by SE-user Simon