Can quantum effects avoid Noether's theorem? A professor said that Noether's theorem can be avoided by quantum effects, but what I understand is that a classical or quantum field under some continuous local symmetry produces a conserved charge. In the $U(1)$ example we got conservation of electric charge, and that is always found to be conserved locally and it is a crucial part in the construction of the standard model which is $SU(3)\times SU(2) \times U(1)$. So my question is if this professor is right?
 A: Suppose we are given a classical Lagrangian with a continuous group of symmetries that we can apply Noether's theorem to. There are several ways in which the corresponding quantum field theory can "avoid" Noether's theorem:

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*It's quantum: This should be somewhat obvious, but Noether's theorem is a statement about classical mechanics and "constants of motion" along classical trajectories. The quantum theory has no such trajectories - the path integral integrates over all paths regardless of whether they solve the e.o.m., it's just that the classical paths contribute the most in the steepest descent approximation - and therefore Noether's theorem just isn't a statement about the quantum theory.
The way one would argue that a classically conserved charge is also conserved in the quantum theory would be that you argue that the Noether charge Poisson-commutes with the Hamiltonian in the Hamiltonian formalism, and therefore commutes with the quantum Hamiltonian after canonical quantization. However, canonical quantization (replacing Poisson brackets by commutators) is merely a very powerful heuristic and not a well-defined map between classical and quantum physics in itself. In particular, the following can happen:


*Quantum anomalies: A symmetry which is a classical symmetry of the Lagrangian need not be a symmetry of the quantum theory, in the sense that invariance of the Lagrangian does not imply invariance of the path integral or of the quantum effective action. A standard example is the chiral anomaly of the electroweak theory, which indeed means that the classically conserved Noether current is not conserved in the quantum theory. For a more general discussion of anomalies, see this answer. Anomalies of global symmetries are interesting but non-threatening phenomena, anomalies of gauge symmetries are an obstruction to having a well-defined quantum theory, and the requirement that the total anomaly of a gauge symmetry must vanish is a powerful constraint in model building.


*Contact terms: As said above, Noether's theorem as such does not apply to quantum theories. The quantum version of it is called the Ward-Takahashi identity, which essentially states that the expectation value of the Noether current is conserved, but only up to "contact terms" in general. That is, where you would have that $(\partial_\mu j^\mu) \text{stuff} = 0$ classically, you find that
$$ \langle \partial_\mu j^\mu(x) \prod_i \phi_i(x_i)\rangle = -\mathrm{i}\sum_{j=1}^n\langle \phi_1(x_1)\dots \delta\phi_j(x_j)\dots\phi_n(x_n)\rangle,$$
where $\delta\phi_i$ is the classical infinitesimal change of the field $\phi_i$ under the symmetry in question. Note that this reduces to $\langle \partial_\mu j^\mu\rangle = 0$ in the case $n=0$.
A: It's important to distinguish between local and global symmetries.  "Noether's theorem" usually refers to her theorem that every continuous global symmetry corresponds to a conserved current.  But the proof of the theorem requires using the classical equations of motion, so it doesn't hold in the quantum case.  (More precisely, there are field configurations in which the charge is not conserved that contribute to the path integral.)  Also, as Aaron points out, quantum anomalies can break the classical symmetry and lead to non-conservation of the conserved current (e.g. the vacuum Maxwell's equations exhibit conformal symmetry which is anomalous in QED, and therefore does not hold).
But local (or "gauge") symmetries are a very different story.  These symmetries hold even without assuming the classical equations of motion (i.e. both "on-shell" and "off-shell").  All field configurations that contribute to the path integral respect these symmetries.  Gauge symmetries (like the $\text{U}(1)$ example you mention) also correspond to conserved quantities, but these are always conserved, even taking into account quantum fluctuations.  (Gauge symmetries can also be anomalous, but instead of just leading to the violation of conservation of the conserved quantity, anomalous gauge symmetries prevent you from consistently quantizing your theory in the first place, so they "break" the whole theory.)
