A version of the Noether's theorem applies to local gauge symmetries. What is the Noether's charge associated with a non-abelian gauge symmetry such as the color $SU(3)$ and how is that derived? I want an expression for the color charge operator like we have an expression for the electric charge operator. Please see Eq. (9) and (11) of the answer here.
Answer
The $\mathrm{SU}(3)$ gauge symmetry is a local symmetry, and therefore it is not Noether's first, but Noether's second theorem that applies to it, which does not yield conserved quantities.
For $\mathrm{U}(1)$ gauge symmetries like the electromagnetic symmetry, there is also a global $\mathrm{U}(1)$ symmetry, and hence a conserved quantity. But the global symmetry associated to a non-Abelian gauge symmetry is just the center of the gauge group, which is discrete for $\mathrm{SU}(3)$, and hence there is no conserved quantity associated to it. This center symmetry has physical significance e.g. in models of confinement, see this question and its answer.
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