# Why complex reps of gauge group for chiral theory?

1. Mar 22, 2015

### Lapidus

Why must the gauge group be in a complex representation so that chirality of the fermions is respected?

thanks

2. Mar 27, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Mar 28, 2015

### Roy_1981

I think your confusion is a matter of semantics or definition/convention. By convention all standard model fermions are taken to be left handed weyl spinors i.e. massless. Then property that these massless Weyl fermions are in a complex representation of the standard model group is called or defined to be "chirality". Not to be confused with chirality matrix γ5 which determines handedness chirality of fermions. Chiral (handedness eigenstates) fermions when appear in complex representations of the gauge group makes the THEORY chiral.
Recall in QCD, we also have complex representations, e.g. fundamental of color gauge group SU(3). But this does NOT make QCD a chiral theory because the quarks/fermions being massive are not chirality eigenstates i.e. constituents are not chiral fermions.

So we need 2 things for a theory to be chiral, a) fermions must be massless, ie. chiral fermions b) they should come in complex representations of the gauge group.

4. Mar 28, 2015

### arivero

In fact there is a bit of Doctrine here, as we could equally define that QCD is non-chiral SU(3), similar to electromagnetism in this aspect.