Quantum Anomalies: Is there a way to show that we recover a classical symmetry that does not exist quantum mechanical in the classical limit? Quantum Anomalies:  Is there a way to show that we recover a classical symmetry that does not exist quantum mechanical in the classical limit?
From undergraduate quantum mechanics, I know that we recover classical physics at the limit $\hbar \to0$.
I have also read that this limit is not rigorous(correct me if I'm wrong please).
Now, since quantum anomalies are "failures of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory", can we show that at this limit we recover the classical symmetry?
If there's an proof of this using elementary QFT or fairly advanced Quantum Mechanics, I would appreciate it if somebody could provide it.
 A: A quantum anomaly is when the quantum theory violates certain classical symmetries. In particular this happens in quantum field theory when your Lagrangian (or rather Action $S$) has a (classical) symmetry. In other words, a symmetry operation leaves the Lagrangian (Action) invariant. 
The symmetry can be violated in the quantum theory if the Feynman path integral is not invariant, i.e. in $\int [d\psi] e^{iS} $ it is the $[d \psi $] that is not invariant. 
On the one hand, I guess you are right, and by taking the classical limit you can indeed recover the classical theory described by the classical Lagrangian. However, an anomaly has big implications on the theory:
If your anomalous symmetry is a gauge symmetry, then the theory will violate unitarity and is sick and useless. 
If your symmetry is a global symmetry (which due to Noether's theorem corresponds to a classically conserved charge), then anomalies imply that quantum corrections violate the classical conservation rules, and that this is simply not a good quantum number to describe your states. 
