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king vitamin said:If you use this as your definition of symmetry, then gauge transformations count, but so do anomalous symmetries. So it's not a good definition for quantum systems. E.g. pure Yang Mills is classically scale invariant but it isn't a symmetry for the quantized theory.
I believe saying that gauge transformations aren't symmetries or are "redundancies" makes sense because they do not transform distinct states into each other. If you take an arbitrary state and apply a gauge transformation to it, you end up with the same physical state. Whereas global symmetries do transform between states, for example it transforms between states in an irreducible multiplet. Or as another example, maybe every energy eigenstate is in a 1D irrep of a symmetry, but a linear combination of two states in different irreps will transform to a different linear combination.
Ok, to have anomalies included, just substitute "effective action" instead of action, and everything is fine. In the classical theory it's by the way sufficient that the variation of the action functional obeys the symmetry. I'm not so sure about the quantum case. For the validity of the Ward-Takahashi identities of proper vertex functions, excluding vacuum diagrams, it should be sufficient, but I'm not sure concerning the effective action itself, which gets a physical observable in the equilibrium case, where it is a thermodynamical potential.