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.

  1. 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 $q\to q^\prime= q+\delta q$?

3 Answers 3


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.

  • $\begingroup$ I don't understand your first example. There are two types of "momentum" in this example: the momentum $P_i$ and kinematic momentum $p_i=m v_i = P_i - A_i(q)$. The momentums commute on the classical and quantum level, but kinematic momentums (that you use) do not commute already on the classical level i.e. for Poisson bracket $\{p_x, p_y\}=B$. Am I missing something? $\endgroup$
    – Alex
    Dec 8, 2018 at 14:55
  • $\begingroup$ @Alex In quantum mechanics, the non-vanishing commutator $[p_x,p_y]\neq 0$ implies that $p_x$ and $p_y$ cannot be simultaneously and consistently measured, due to the uncertainty relation. In classical mechanics, the non-vanishing Poisson bracket $\{p_x,p_y\}\neq0$, however, does not have the same physical consequence (there is no uncertainty relation in classical mechanics). This is an important difference between Poisson bracket and commutator. It is the uncertainty relation between $p_x$ and $p_y$ that gives rise to the quantum anomaly. $\endgroup$ Dec 20, 2018 at 10:10

Anomalies are not particular to quantum field theory, or even to quantum theory. An anomaly is an obstruction to representing some physically relevant group/algebra, often a symmetry group or an algebra of observables, on the state space, and means that your state space will carry not a representation of the symmetry group itself but of an extension. This notion is explained at length in this excellent answer by David Bar Moshe.

Whether the group/algebra that is obstructed is the classical Galilean group that needs the introduction of mass as a "central charge" to become the Bargmann group, the $\mathrm{SO}(3)$ of a particle as in Everett You's answer (which is a special case of a more general link between WZW models and anomalies) that needs the passage to its universal cover $\mathrm{SU}(2)$ on the spin-1/2 which is a central extension by $\mathbb{Z}_2$ or the algebra of fermionic non-Abelian charge densitites that is extended to the Mickelsson-Faddeev algebra (see again the answer by David) by the anomaly term is immaterial - it's all the same principle.

The fundamental character of an anomaly is not a non-invariance of the path integral measure, that's just a particular way to derive it.


In reference to the first reply: Alex was correct. This is not a quantum effect, since it also arises in the classical limit of the quantized Poisson bracket, which is defined as the commutator divided by iħ. The classical form of "does not commute" is that "the Hamiltonian fields associated with the quantities in question do not commute". Here, the Hamiltonian fields are X = {p,⋯}, and what we find is that

    [X₁,X₂] = X₁X₂ - X₂X₁ = {p₁,{p₂,⋯}} - {p₂,{p₁,⋯}} = {{p₁,p₂},⋯} = {B³,⋯} ≠ 0.

The "Hamilonian fields do not commute" criterion is also the quantum form of "does not commute", since the quantized Poisson brackets are, indeed, Poisson brackets over a suitably-defined Poisson manifold, and their Hamiltonian fields provide equivalent descriptions of the operators, themselves. So, the condition can be used uniformly across the board, independent of the classical versus quantum paradigm divide.

A "quantum effect" is what arises as a quantum correction, e.g. Q({p₁,p₂}) versus {Q(p₁),Q(p₂)}, where the second bracket is the quantized Poisson bracket; and where Q() denotes the quantization operation.

This follows the treatment given in Landsman (Mathematical Topics Between Classical and Quantum Mechanics (1998), Part II "Quantization and the Classical Limit"). Here, the difference Q({p₁,p₂}) - {Q(p₁),Q(p₂)} = Q(B³) - Q(B³) is 0. Discussion of projective Hilbert spaces as Poisson manifolds (actually symplectic manifolds) may be found in section I.2 in Landsman.

Anomalies are deficits that arise from an attempt to quantize a classical symmetry transform, which spoil the symmetry transform at the quantum level. Several examples, pertaining to the original question, are illustrated here https://www.physics.ncsu.edu/ntg/leegroup/library/Anomaly_QM.pdf and discussion of anomalies, in greater depth, may be found here https://en.wikipedia.org/wiki/Anomaly_%28physics%29


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