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For gauge groups like $SU(2)$ and $SU(3)$ etc. we know that observable states such as mesons or baryons must be charge neutral. However, for a $U(1)$ gauge group we can have charged initial states in our scattering experiments. Why can we have states with an observable non-zero charge for the $U(1)$ but not for $SU(n)$?

Another way to state this question, to clarify, is the following: the global part of the $U(1)$ gauge group defines a good quantum number - the total charge - which allows us to split up our Hilbert space. Why does the global part of bigger gauge groups not supply a good quantum number? If it does supply such a number why must all states be charge neutral under it?

Having had a discussion with @user1504 and taking into account @FredericBrunner 's points I think I can answer this question to my satisfaction.

The summary is thus - it is possible in principle, regardless of the gauge group, to form localised lumps of the respective charges under the global part of the group. To me, this gives the global part of the group a meaning separate from the gauge part. For $U(1)$ the lumps can have positive and negative charge and for $SU(3)$ they can have each of the three colours. This means that in respect of the question, $U(1)$ and $SU(N)$ are not different. In a practical sense however, the $SU(N)$ groups are confining so one can only do this in principle at a temperature sufficiently high that the theory is deconfined and the notion of individual quarks makes sense. Similarly, $U(1)$ in 2+1 dimensions is confining and there the notion of lumps of positive and negative charge only make sense above the deconfinement transition.

Magician
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2 Answers2

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The premise of this question is wrong: U(1) is not special. There are conserved charges associated to the global symmetry group $G$ of any gauge theory.

In the Standard Model, for example, we have the weak isospin charge, which is the conserved Noether charge of the electroweak $SU(2)$. (At low energies, this conservation is obscured by the Higgs mechanism, but at high enough energies, it becomes more plain.) There is also a conserved color charge associated to the $SU(3)$ color symmetry.

As Frederic Brunner correctly points out, in our world, the color force is confining, so the value of this conserved color charge in normal physics is always 'zero'. But this is not true above QCD's Hagedorn temperature, where the system deconfines. Nor is this entirely a theoretical issue: RHIC has produced collisions intense enough to reach this deconfined state.

user1504
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The reason is confinement. Yang Mills theories with $SU(2)$ and $SU(3)$ gauge groups exhibit confinement, while for example $U(1)$ electrodynamics does not. Whether a theory is confining or not can be found out by studying the properties of Wilson loops.

Frederic Brünner
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