Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish

$$\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 = -\frac{3}{2} \bar{l}_1 l_4 \bar{r}_3 r_2 - \frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2 - \frac{3}{2} \bar{l}_1 (-1) l_4 \bar{r}_3 (+1) r_2 $$$$= -\frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2 = 0$$

Why does this operator vanish? Is that true?

Why is the product of tensor currents expressed in tensor and scalar currents only, while for such combinations of Weyl spinors $l_1$, $r_2$, $r_3$ and $l_4$ in the Fierz identity all of them vanish?

  • 2
    $\begingroup$ If you use the definition of the projectors, $P_+$ and $P_-$, and their commutation with the gamma matrices, you can get the result! $\endgroup$
    – Dox
    Jul 22, 2014 at 1:33

1 Answer 1


The simplest way to see that all products of your kind vanish is to notice that one of the objects (bilinear invariant) $T_{\mu\nu}$ is a self-dual 2-form while the other $T^{\mu\nu}$ is anti-self-dual, and their contraction without a complex conjugation has to vanish.

A self-dual (anti-self-dual) antisymmetric tensor obeys $$T_{\mu\nu} = \pm \frac i2 \epsilon_{\mu\nu\kappa\lambda} T^{\kappa\lambda}$$ and in $3+1$ dimensions, it has to have complex components.

The objects $\bar \ell \sigma \ell$ are self-dual or anti-self-dual due to the chirality of the spinors. And the inner products are zero because the 6-dimensional complex space of 2-forms may be really decomposed to mutually orthogonal self-dual and anti-self-dual (3-complex-dimensional) parts.

The claims above, when translated to maths, contain lots of signs – sometimes convention-dependent signs (and extra signs and flips of chiralities from complex conjugation and so on) – that one has to be careful about. But there are also several identities of your kind one may derive.

  • $\begingroup$ I clearly see, why $\bar{l}l$ or $\bar{l}\sigma_{\mu\nu}l$ vanish, however I am not sure why does $\bar{l}_1 \sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4$ vanish. It contains tensor currents with spinors of different chiralities. $\endgroup$
    – madu
    Jul 22, 2014 at 7:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.