# Fierz identity for Weyl spinors in tensor currents

+ 4 like - 0 dislike
2764 views

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?

This post imported from StackExchange Physics at 2014-07-22 07:50 (UCT), posted by SE-user fen
If you use the definition of the projectors, $P_+$ and $P_-$, and their commutation with the gamma matrices, you can get the result!

This post imported from StackExchange Physics at 2014-07-22 07:50 (UCT), posted by SE-user Dox

+ 3 like - 0 dislike

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.

This post imported from StackExchange Physics at 2014-07-22 07:50 (UCT), posted by SE-user Lubo Motl
answered Jul 22, 2014 by (10,278 points)
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.

This post imported from StackExchange Physics at 2014-07-22 07:50 (UCT), posted by SE-user fen

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.