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I just aadded a sentence about Yang-Mills theory to gauge group, but there are some aspects of that article I feel we might want to discuss:
I don’t think that the statement “gauge groups encoded redundancies” of the mathematical description of the physics is correct. One hears this every now and then, and I suppose the idea is the observation that physical observables have to be in the trivial representation of the gauge group, but there is more to the gauge group than that.
Notably Yang-Mills theory is a theory of connections on G-principal bundles. No mathematician would ever say that the group G in a G-principal bundle just encodes a redundancy of our descriptins of that bundle. And the reason is because it is true only locally: the thing is that $\mathbf{B}G = \{* \stackrel{g \in G}{\to} * \}$ has a single object and hence is connected , but it has higher homotopy groups, and that’s where all the important information encoded by the gauge group sits.
So I would say that instead of being a redundancy of the description, instead the gauge group of Yang-Mills theory enocedes precisely the homotopy type of its moduli space. This is rather important.
A different matter are global gauge symmetries such as those that the DHR-theory deals with.
One hears this every now and then, and I suppose the idea is the observation that physical observables have to be in the trivial representation of the gauge group, but there is more to the gauge group than that.
Interesting, I did not expect that the entry was already of any interest, therefore I did not start a thread here…
A different matter are global gauge symmetries such as those that the DHR-theory deals with.
Yes, that’s all I had in mind. As far as I know there is no concept in AQFT of a local gauge symmetry, and e.g. Haag stresses that this is one of the central challenges and open problems of this approach.
No mathematician would ever say that the group G in a G-principal bundle just encodes a redundancy of our descriptins of that bundle.
I think I understand that, but does that contradict the description that gauge transformations are not observable? I mean “redundant” may sound like “uninteresting”, but that’s certainly not meant here, it is just “redundant” = “maybe there is a framework of QFT without gauge symmetries, this could be possible, because gauge symmetries are not observable”. (At least that’s how I always understood that statement).
My suggestion would be: let’s split the entry into two main subsections: one discussing gauge groups in the sense of Yang-Mills theory, the other global gauge groups in the sense of DHR. These are really two quite different concepts that accidentally go by the same name.
Ok, are you going to rewrite the idea section to fit the split?
(If these are really different concepts, maybe they should get their own page?).
I tried to edit gauge group a bit. But just a rough first attempt.
I’d says we’ll keep this in one entry for the time being, unless and until we have so much material for each of the two notions that a split is worthwhile.
added to gauge group a new section Examples with the list of examples that I had posted in reply to an MO-question here
I may be wrong, but my feeling is that there is also a notion of global gauge group at classical level, as the infinitedimensional Lie group of differentiable automorphisms of the principal bundle (rather than at the level of QFT).
Yes, there are also global gauge groups that should be discussed. For instance also for gravity on $(X,g)$ one should mention $Iso(X,g)$ as a global gauge group. Maybe I’ll add something about that later.
You did mention the global gauge group. The only thing is that I am emphasising the classical versiom while the entry was saying about QFT and local nets in quantum theory, not about the classical gauge theory.
Ah, that’s what you mean. Sure, there is also a notion of global gauge group in classical theory. I think the point of mentioning it especially for the AQFT case is that this is a definition of QFT that is independent of quantization from a classical theory, so there it is of special interest if one can still read off the global gauge group from just the information of a local net. And of course it’s one of the main theorems in the whole theory that one can (proved by Doplicher-Roberts reconstruction, as you know).
the gauge group of Yang-Mills theory enocedes precisely the homotopy type of its moduli space.
That is very clear to me, thanks!
Thanks for saying this. This made me expand that remark to a new subsection at gauge group: called now Properties – Not a redundancy.
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