# Any good idea how non-abelian gauge symmetries emerge?

• A
Gold Member
I think the story where abelian, i.e. U(1), gauge symmetry comes from is pretty straight-forward:

We describe massless spin 1 particles, which have only two physical degrees of freedom, with a spin 1 field, which is represented by a four-vector. This four-vector has 4 entries and therefore too many degrees of freedom. A description of a spin 1 particle in terms of a four-vector field is necessarily redundant and we call this redundancy "gauge symmetry". Formulated differently: particles are representations of the little groups of the Poincare group, whereas fields are representations of the complete Poincare group. This is what leads to the gauge redundancy. However, as far as I know this story only works for the familiar U(1) symmetry.

(This point of view is emphasized, for example, in Weinbergs QFT book Vol. 1 section 5.9. Someone who currently likes to emphasize this perspective is Arkani-Hamed, for example, in section 2 of his latest paper: https://arxiv.org/abs/1709.04891 or here https://arxiv.org/abs/1612.02797. I actually asked him a month ago if he knows any idea for an analogous explanation for non-abelian gauge redundancies, but unfortunately he didn't had a good answer.)

Is there any good idea where non-abelian gauge symmetries come from? The big difference, I think, is that non-abelian gauge symmetries also in some sense help us to explain the particle spectrum. For example, we have doublets and triplets of elementary particles and this is a real physical consequence and can not be regarded as an accident, because we use the "wrong" objects to describe elementary particles.

## Answers and Replies

king vitamin
Gold Member
I learned Yang-Mills theory from a course taught by Xi Yin, whose lecture notes can be found here. Beginning on page 23, he derives both U(1) and non-abelian gauge theory from some "soft theorem"-like calculations. The conceptual idea behind the proof for non-abelian gauge theory is very similar to the U(1) case, he simply asks what possibilities exist for a theory with multiple massless vector bosons, and finds the constraint that they must live in a compact Lie group.

fresh_42 and jakob1111
However, as far as I know this story only works for the familiar U(1) symmetry.

Why?

Gold Member
Why?

Simply because I've never seen a similar story for the origin of non-abelian gauge symmetries :D

If you know any reference where this is explained or have an idea how the story could go for non-abelian gauge symmetries, please let me know!

I mean nothing in this tells you that your gauge theory should be abelian, or am I missing something? You just know that some of the "apparent" states of the theory have to be unphysical, as they are in gauge theories, if you want a massless spin 1 field.