quantum field theory - Why is there no anomaly when particle mechanics is quantized?

We know that if one or more symmetries of the action of a classical field theory is violated in its quantized version the corresponding quantum theory is said to have anomaly.

1. Is this a sole feature of quantization of a field theory? If yes, why is it that anomalies appear only after quantizing a field theory but non in ordinary non-relativistic quantum mechanics?

In field theory, if under an arbitrary symmetry transformation $\phi\rightarrow \phi^\prime=\phi+\delta\phi$, the action $S[\phi]$ is left invariant, we have a symmetry in classical field theory. But we have a symmetry of quantum field theory, if the transformation leaves the path-integral $\int\mathcal{D}\phi \exp(\frac{i}{\hbar}S[\phi])$ invariant. Therefore, even if $S(\phi)$ is invariant but the measure is not, we can have an anomaly.

2. Does it mean that the path-integral measure $\int \mathcal{D}q(t) \exp(\frac{i}{\hbar}S[q(t)])$ in ordinary quantum mechanics always remains invariant under any classical symmetry $q\to q^\prime= q+\delta q$?

Answer

Quantum mechanics can also become anomalous. An example is a charged particle moving in a uniform magnetic field. On the classical level, the system is translation invariant in both x- and y-direction. Because the magnetic field is uniform, all (gauge-invariant) measurement will yield the same result at any point of the space, hence the translation symmetry is preserved. But once the system is quantized, the momentum $p_x$ and $p_y$ no longer commute with each other, i.e.

$$[p_x,p_y]=\mathrm{i}\hbar B.$$ The non-commutativity is exactly proportional to $\hbar$, implying that this is indeed a quantum effect. In this case, if one chooses to preserve the translation along x, the translation along y must be broken, as $p_x$ and $p_y$ become incompatible observables. This effect is manifested in the wave function under the Landau gauge. Therefore the system becomes anomalous under translation.

Another closely related example is a charged particle moving on a sphere with a magnetic monopole (Dirac monopole) inside the sphere. Let the unit vector $\boldsymbol{n}=(n_1,n_2,n_3)$ be the coordinate that parameterize the position of the particle on the sphere ($\boldsymbol{n}^2=1$). The classical action can be written as a Wess-Zumino-Witten model

$$S[\boldsymbol{n}(t)]=\frac{1}{4\pi}\int\mathrm{d}t\int_0^1\mathrm{d}u\;\boldsymbol{n}\cdot\partial_t\boldsymbol{n}\times\partial_u\boldsymbol{n}.$$ The action is invariant under the SO(3) transformation of $\boldsymbol{n}$. But after quantization, the eigenstates are spin-1/2 objects, which are not linear representations of the SO(3) symmetry group. So the system has an SO(3) anomaly.