• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,079 questions , 2,229 unanswered
5,348 answers , 22,758 comments
1,470 users with positive rep
819 active unimported users
More ...

  Fierz identity for Weyl spinors in tensor currents

+ 4 like - 0 dislike

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
asked Jul 21, 2014 in Theoretical Physics by fen (20 points) [ no revision ]
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

1 Answer

+ 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 Luboš Motl (10,278 points) [ no revision ]
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

Your answer

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):
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:
Then 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).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights