I have read about anomalies in different contexts and ways. I would like to read an explanation that unified all these statements or points of view:

  1. Anomalies are due to the fact that quantum field theories (and maybe quantum mechanical theories with singular potential) must be regularized and it can be possible that none regularization procedure may respect all the symmetries of the classical theory. For instance, according to Fujikawa's viewpoint, one cannot find a regularized functional measure in the path-integral which preserves the symmetries of the classical action.

  2. Anomalies are due to the fact that quantum theory requires an ordering prescription for operators and it may happen that none ordering prescription respects the symmetries of the classical theory. Here in principle one could have anomalies in quantum mechanics with non-singular potentials.

  3. Anomalies are due to the fact that generators of the symmetry do not leave invariant the domain of definition of the Hamiltonian and thus, although the formal commutator of those generators with the Hamiltonian vanishes, the charges are not conserved due to the extra surface term that appears in the exact Heisenberg equation. It is not clear to me if in this case the anomaly is already present in the classical theory.

  4. Anomalies are due to the emergence of central charges in the algebra of conserved quantities. In this case one can have classical anomalies if it is in the algebra of Poisson brackets, or quantum anomalies if it happens at the level of quantum commutators.

I would like you to share examples and relations between the previous perspectives, and discussions about their equivalence. The only I see is the following relation:

Ordering of operators ----> different definition of path integral measure.

Ordering of operators ----> delta functions ---> regularization.

Ordering of operators ----> different algebra of conserved charges.

  • $\begingroup$ I can't think of a case of 3 where it's quantum and not classical. I always thought 1 and 2 should be the same thing fundamentally, since the operator algebra when properly defined, should tell you the divergence of every current. +1 for the question, but I think it should be at least two questions asking the relation between 1 and 2, and separately about 4 (if you know an example for 3, that too). "What's the relation between Fujikawa measure regularization and Adler failure of naive operator identities due to coinciding operators?" Is a suscinct and definite question, and less list-making. $\endgroup$
    – Ron Maimon
    Commented Jul 31, 2012 at 6:52
  • $\begingroup$ @RonMaimon Regarding the third case, as David Bar has pointed out, it seems that every quantum anomaly has a classical counterpart (or classical anomaly). The third case has a close relation with Fujikawa method. In the third case the anomaly come up because the symmetry generator take the state out the domain where the Hamiltonian is self-adjoint. $\endgroup$ Commented Jul 31, 2012 at 18:43
  • $\begingroup$ @RonMaimon In Fujikawa, different self-adjoint extensions of the Hamiltonian (corresponding to the same classical Hamiltonian) give raise different sets of eigenvectors and therefore different measures in the path-integral. So if a domain is not invariant under the action of a generator, then the measure is not keep invariant either. $\endgroup$ Commented Jul 31, 2012 at 18:46
  • $\begingroup$ Not every quantum anomaly has a classical counterpart--- central charges can appear in classical descriptions, that's all the answer is giving. There is no classical counterpart to the axial anomaly. The measure and extension idea are the same, so you just rephrased the thing. My interpretation of 1+2 is "how do you calculate the chiral anomaly just from the algebra of local fields, appropriately fixed up". I thought this was 1 and 2. I now see the relation to 3, you are thinking of the ultraviolet regulator as a boundary of sorts, which the transformation doesn't preserve. 4 still unrelated. $\endgroup$
    – Ron Maimon
    Commented Aug 1, 2012 at 7:28
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    $\begingroup$ @drake: There is a great new article (arxiv.org/abs/1305.1955v1) by Michael Stone and Vatsal Dwivedi in which they classically derive the Abelian and non-Abelian chiral anomlies using a single fermion classical model without second quantization, Grasmann variables, the path integral measure etc. $\endgroup$ Commented Jun 9, 2013 at 15:54

1 Answer 1


Nowadays there exists a more fundamental geometrical interpretation of anomalies which I think can resolve some of your questions. The basic source of anomalies is that classically and quantum-mechanically we are working with realizations and representations of the symmetry group, i.e., given a group of symmetries through a standard realization on some space we need to lift the action to the adequate geometrical objects we work with in classical and quantum theory and sometimes, this action cannot be lifted. Mathematically, this is called an obstruction to the action lifting, which is the origin of anomalies. The obstructions often lead to the possibility to the realization not of the group of symmetries itself but some extension of it by another group acting naturally on the geometrical objects defining the theory.

There are three levels of realization of a group of symmetries:

The abstract level: for example the action of the Lorentz (Galilean) group on a Minkowski (Eucledian) space. This representation, for example is not unitary, and it is not the representation we work with in quantum mechanics.

The classical level: When the group action is realized in terms of functions belonging to the Poisson algebra of some phase space. For example, the realization of the Galilean or the Lorentz groups on the phase space of a classical free particle.

The quantum level when the group action realized in terms of a linear representations of operators on some Hilbert space (or just operators belonging to some $C^*$ algebra. For example, the realization of the Galilean or the Lorentz groups on a quantum Hilbert space of a free particle.

Now, passing from the abstract level to either the classical or the quantum level may be accompanied with an obstruction. These obstructions exist in already quantum and classical mechanics with finite number of degrees of freedom, and not only in quantum field theories. Two very known examples are the Galilean group which cannot be realized on the Poisson algebra of the phase space of the free particle, rather, a central extension of which with a modified commutation relation:

$$[K_i, P_j] =-i \delta_{ij}m$$

, is realized. ($K_i$ are boosts and $P_i$ are translations $m$ is the mass). This extension was discovered by Bargmann, and sometimes it is called the Bargmann group. A second example, is the realization of spin systems in terms of sections of homogeneous line bundles over the two sphere $S^2$. Now, the action of the isometry group $SO(3)$ cannot be lifted to line bundles corresponding to half integer spins, rather a $\mathbb{Z}_2$ extension of which, namely $SU(2)$ can be lifted. In this case the extended group is semisimple and the issue that $SU(2)$ being a group extension of $SO(3)$ and not just a universal cover is not usually emphasized in physics texts.

The group extensions realized as a consequence of these obstructions may require:

1) Ray representations of the original group which are true representations of the extended group. This is the case of $SO(3)$, where the half integer spins can be realized through ray epresentations of SO(3), which are true representations of $SU(2)$. In this the Lie algebras of both groups are isomorphic.

2) Group extensions corresponding Lie algebra extensions. This is the more general case corresponding for example to the Galilean case.

Now, in the quantum level, one can easier understand, why the obstructions lead to group extensions. This is because, we are looking for representations satisfying two additional conditions:

1) Unitarity

2) Positive energy

Sometimes (up to $1+1$ dimensions), we can satisfy these conditions merely by normal ordering, which results central extensions of the symmetry groups. This method apply to the case of the Virasoro and the Kac-Moody algebras which are central extensions to the Witt and loop algebras respectively, and can be obtained in the quantum level after normal ordering.

The relation between normal ordering and anomalies can be explained in that the quantization operators are needed to be Toeplitz operators. A very known example is the realization of the harmonic oscillator on the Bargmann space of analytic functions, then the Toeplitz operators are exactly those operators where all derivatives are moved to the right. This is called the Wick quantization and it exactly corresponds to normal ordering in the algebraic representation. The main property of Toeplitz operators is that their composition is performed through star products, and star products of Toeplitz operators are are also Toeplitz operators thus the algebra of quantum operators is closed, but it is not closed to the original group but rather to a central extension of which. This important interpretation hasn't been extended to field theories yet.

It is worthwhile to mention that central extensions are not the most general extensions one can obtain when a symmetry is realized in terms of operators in quantum theory, there are Abelian and even non-Abelian extensions. One of the more known extensions of this type is the Mickelsson-Faddeev extension of the algebra of chiral fermion non-Abelian charge densities when coupled to an external Yang-Mills field in $3+1$ dimensions:

$$[T_{a}(x), T_{b}(y)] = if_{ab}^c T_c(x) \delta^{(3)}x-y) +id_{ab}^c\epsilon_{ijk} \partial_i\delta^{(3)}(x-y) \partial_j A_{ck}$$

This extension is an Abelian noncentral extension.

The explanation of the existence "anomalies" in the classical case, i.e., on the Poisson algebra can be understood already in the case of the simplest symplectic manifold $\mathbb{R}^2$, the Poisson algebra is not isomorphic the translations algebra. A deeper analysis for example given in: Marsden and Ratiu page 408 for the case of the Galilean group. They showed that on the free particle Hilbert space, the Galilean group lifts to a central extension (the Bargmann group) which acts unitarily on the free particle Hilbert space: $\mathcal{H} = L^2(\mathbb{R}^3)$. Now, the projective Hilbert space $\mathcal{PH}$ is a symplectic manifold (as any complex projective space) in which the particle's phase space is embedded. The restriction of the representation to the projective Hilbert space and then to the particle's phase space retains the central extension i.e., is isomorphic to the extended group, thus the extended group acts on the Poisson algebra.

As a matter of fact one should expect always that the anomaly should be realized classically on the phase space. The case of fermionic chiral anomalies seems singular, because it is customary to say that the anomaly is existent only at the quantum level. The reason is that the space of Grassmann variables is not really a phase space, and even in the case of fermions, the anomaly exists in the classical level when one represents them in terms of "Bosonic coordinates". These anomalies are given as Wess-Zumino-Witten terms. (Of course these representations are not useful in Perturbation theory).

Another reasoning why anomalies exist always on the classical (phase space) level is that in geometric quantization, anomalies can be obtained on the level of prequantization. Now, prequantization does not require any more data than the phase space (not like the quantization itself which requires a polarization).

Now, trying to respond on your specific questions. It is true that chiral anomalies were discovered in quantum field theories when no ultraviolet regulators respecting the chiral symmetry could be found. But anomaly is actually an infrared property of the theory. The signs for that is the Adler-Bardeen theorem that no higher loop (than one) correction to the axial anomaly is present and more importantly only massless particles contribute to the anomaly. In the operator approach that I tried to adopt in this answer the anomaly is a consequence of a deformation that should be performed on the symmetry generators in order to be well defined on the physical Hilbert space and not a direct consequence of regularization.

Secondly, the anomaly exists in equally on both levels quantum and classical (on the phase space). The case of fermions and regularization was addressed separately.

Update - Elaboration of the spin case:

Here is the elaboration of the $SO(3)$, $SU(2)$ case which contains all the ingredients regarding the obstruction to lifting and group extensions, except that it does not have a corresponding Lie algebra extension.

We work on $S^2$ using the stereographic projection coordinate given in terms of the polar coordinates by:

$$z = \tan \frac{\theta}{2} e^{i \phi}$$

An element of the group $SU(2)$

$$g=\begin{pmatrix} \alpha& \beta\\ -\bar{\beta} & \bar{\alpha } \end{pmatrix}$$

acts on $S^2$ according to the Möbius transformation:

$$ z \rightarrow z^g = \frac{\alpha z + \beta}{-\bar{\beta} z + \bar{\alpha } }$$

However, one observes that the action of the special element:

$$g_0=\begin{pmatrix} -1& 0\\ 0 & -1 \end{pmatrix}$$

is identical to the action of the identity. This element is an SU(2) element that projects to the unity of SO(3) (This can be seen from its three dimensional representation which is the unit matrix). Thus the group which acts nontrivially on $S^2$ is $SO(3)$

Now quantum mechanically spin systems can be realized on the sphere in Hilbert spaces of analytic functions:

$$ (\psi, \xi) = \int_{S^2} \overline{\xi(z)} \psi(z) \frac{dzd\bar{z}}{(1+\bar{z}z)^2}$$

Transforming under $SU(2)$ according to:

$$ \psi(z) \rightarrow \psi^g(z) = (-\bar{\beta} z + \bar{\alpha })^{2j} \psi(z^{g^{-1}})$$

This is a ray representation of $SO(3)$ as $SO(3)$ does not have half integer representations.

Now, the first observation (the quantum level) is that the special element does not act on the wave functions as the unit operator, for half integer spins it adds a phase of $\pi$. This is what is meant that the $SO(3)$ action cannot be lifted to the quantum Hilbert space.

Now turning to the the classical level. The symplectic form on $S^2$ is proportional to its area element. The proportionality constant has to be an integer in a prequantizable theory (Dirac quantization condition)

$$\omega = 2j \frac{dz \wedge d\bar{z}}{(1+\bar{z}z)^2}$$

The corresponding Poisson bracket between two functions on the sphere:

$$\{f, h\} =\frac{1}{2j} (1+\bar{z}z)(\partial_z f \partial_{\bar{z}} h - \partial_z h \partial_{\bar{z}} f)$$

The function generating the group action in the Poisson algebra is given by:

$$f_g= \left(\frac{\alpha \bar{z}z + \beta \bar{z} - \bar{\beta}z + \bar{\alpha}}{1+\bar{z}z}\right)^{2j}$$

Now, the function representing the unity of SU(2) in the function $f=1$, while the function representing the special element is $f=-1$ for half integer spins, which is a different function (It has to be a constant because it belongs to the center of $SU(2)$, thus it has to Poisson commute with all functions.

Thus even at the classical level, the action of $SO(3)$ does not lift to the Poisson algebra.

Now, regarding the question of classically distinguishing $SU(2)$ of $SO(3)$. If you compute the classical partition function of a spin $\frac{1}{2}$ gas interacting with a magnetic field, it will be different than say spin $1$, but spin $\frac{1}{2}$ exists in the first place only if $SU(2)$ acts because $SO(3)$ allows only integer spins.

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    $\begingroup$ Here are some examples referring to both your questions. In the WZNW model in two dimensions, one gets the Kac-Moody algebra at the classical level (Poisson brackets) with the correct central charge. In fact this computation was performed by Witten in his original article: projecteuclid.org/…. For a more clear derivation please see: citeseerx.ist.psu.edu/viewdoc/summary?doi= $\endgroup$ Commented Aug 2, 2012 at 15:20
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    $\begingroup$ (cont.) But, if one uses the Sugawara construction for the Virasoro generators from the current generators, the Virasoro central charge (in the Poisson brackets) will be zero. However, in the Liouville theory one already obtains a nonvanishing Virasoro central charge in the Poisson brackets, please see the following article by Toppan: $\endgroup$ Commented Aug 2, 2012 at 15:22
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    $\begingroup$ (cont.) webcache.googleusercontent.com/… $\endgroup$ Commented Aug 2, 2012 at 15:23
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    $\begingroup$ Actually, I have a little more information on this subject. I'll try to add an update when I can $\endgroup$ Commented Aug 2, 2012 at 15:50
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    $\begingroup$ For anyone else interested in this perspective on anomalies, I found "Nelson , Alvarez-Gaumé : Hamiltonian interpretation of anomalies" projecteuclid.org/euclid.cmp/1103942612 extremely illuminating $\endgroup$
    – jak
    Commented Jul 7, 2017 at 8:54

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