In standard field theory texts, a “classical symmetry” is defined to be a transformation $\phi\to\phi’$ such that the corresponding action is left invariant. The symmetry is said to survive quantization (or is “non-anomalous”) if the integration measure $\mathcal{D}\phi$ is also left invariant.
It seems, however, more natural to define a “quantum symmetry” to be a transformation such that the action-weighted measure $\mathrm{d}\mu=\mathcal{D}\phi\,e^{i S[\phi]}$ is left invariant. Indeed, non-anomalous classical symmetries satisfy this requirement. However, is it possible to have a new symmetry transformation which respects neither the action nor the measure, but still leaves $\mathrm{d}\mu$ invariant?
In other words, can there exist symmetries of the quantum theory that cannot be written in terms of the classical theory?
