Consider a classical gauge field coupled to a vector field $j^\mu$. Gauge invariance requires that $\mathcal A_\mathrm{cl}:=\partial_\mu j^\mu$ vanishes: $$ \mathcal A_\mathrm{cl}\equiv 0 $$

In other words, the source of a classical gauge field must be conserved, for otherwise the theory is inconsistent.

Let us now move on to the quantum theory, with boldface denoting operators. Even if the classical theory is gauge invariant, $\boldsymbol{\mathcal A_\mathrm{cl}}\equiv \boldsymbol 0$, we may still have a quantum anomaly, $\boldsymbol{\mathcal A_\mathrm{qm}}\neq \boldsymbol 0$, which would render the theory inconsistent.

A situation I have never seen discussed is a gauge theory coupled to a non-conserved classical source, $\boldsymbol{\mathcal A_\mathrm{cl}}\neq \boldsymbol 0$, but with a quantum anomaly that satisfies $\boldsymbol{\mathcal A_\mathrm{qm}}\equiv -\boldsymbol{\mathcal A_\mathrm{cl}}$. In such a case, the quantum source would be conserved, $$ \partial_\mu \boldsymbol j^\mu=\boldsymbol{\mathcal A_\mathrm{qm}}+\boldsymbol{\mathcal A_\mathrm{cl}}\equiv \boldsymbol 0 $$ which would mean that the quantum theory is consistent after all.

If this picture is consistent, it would open up the door to a very weird but interesting phenomenology. For one thing, the theory probably lacks a classical limit, or at least the limit is highly non-trivial.

A model like this would certainly require some meticulous tuning to ensure that the quantum anomaly precisely matches the classical one, but it seems to me that it is in principle conceivable. Or is it? Is there any obstruction to this mechanism? Is there any way to argue that this just cannot happen? Conversely, if this mechanism works, has it ever been used in the literature?

  • $\begingroup$ You must have in mind something nonanlytic in $\hbar$? A classical anomaly with no $\hbar$ dependence equaling exactly something that vanishes in the null $\hbar$ limit? Hard to illustrate schematically. $\endgroup$ – Cosmas Zachos May 5 '18 at 1:39
  • $\begingroup$ @CosmasZachos Yes, I should have been more explicit about that: the classical theory is allowed to depend on $\hbar$. Perhaps "classical" is not the best word to use here. $\endgroup$ – AccidentalFourierTransform May 5 '18 at 1:44
  • $\begingroup$ I'm not convinced that "the classical theory is allowed to depend on $\hbar$" is a meaningful thing to say. Even if your classical theory depends on a parameter which you choose to denote by that letter, I do not see a way in any of the standard quantization procedures to be allowed to claim that this parameter is, in fact, the same as the quantization parameter also denoted by $\hbar$. If you allow the "classical" theory to depend on $\hbar$ you are squarely outside standard field theory, or indeed mainstream physics, unless you can cite precedent in the literature for such a dependence. $\endgroup$ – ACuriousMind May 5 '18 at 10:53
  • $\begingroup$ @ACuriousMind The Schrodinger Lagrangian, regarded as classical field theory, depends on $\hbar$ and is perfectly mainstream... $\endgroup$ – AccidentalFourierTransform May 5 '18 at 20:17
  • $\begingroup$ But if you quantized it, would that $\hbar$ be the same as that introduced by quantizing that classical field theory? We could discuss this tomorrow in chat if you like $\endgroup$ – ACuriousMind May 5 '18 at 21:01

The suggested mechanism is one of the basic mechanisms of anomaly cancellations. Let me emphasize first, that the symmetry in the classical theory of your question is required to be anomalous and not just broken or non-existent. This requires the current divergence to satisfy the Wess-Zumino consistency condition, or equivalently, the integrated anomaly to be a one-cocycle on the gauge group.

Anomalies in gauge theories with fermions manifest their selves at the one loop level, while in bosonic theories, the anomaly exists already at the classical level due to Wess-Zumino-Witten terms (These terms depend explicitly on $\hbar$ since their integrals over closed surfaces should be multiples of $2 \pi$).

Since the net anomaly should vanish, then an anomalous theory can exist only if its anomaly is compensated by another theory with exactly the opposite anomaly. This is basically what happens with a Dirac fermion composed of two Weyl fermions of opposite chiralities.

An anomaly compensation mechanism of the kind described in the question happens, for example, in the quantum Hall effect. Here, the system is made of a bulk and an edge. The bulk theory is a Chern-Simon's theory in 2+1 D; it is anomalous at the classical level. The edge theory can be described as a theory of chiral fermions in 1+1 D. Its anomaly occurs at the one-loop level and exactly compensates the anomaly of the bulk theory. Please see the following article by Jiusi and Nair where this point is clearly explained on page 11 (This article is new, but this anomaly compensation mechanism is known long ago). (In the Abelian case, the gauge group is electromagnetism and clearly we should have this anomaly cancelation.)

Now, you could choose for the edge theory not a chiral fermion but a chiral boson. Still the anomaly gets compensated but this time, it is manifested for both theories at the classical level. This example shows that an anomaly is a real property of a system, its level of manifestation is a matter of description of the system. All descriptions are incomplete, for example, the fermionic description of QCD is by means of confined quarks, while the bosonic (low energy sigma model) description is not renormalizable. However, the anomaly can be exactly computed in both descriptions. Thus the main point is the description of anomaly as classical or quantum is not an absolute; it relies on our description of the system which is not unique.

In addition, the tuning is not very complex, because the coefficient of the Wess-Zumino-Witten (hence the anomaly) term is fixed by a quantization condition (generalization od Dirac's quantization condition of the monopole), while in the fermionic case the anomaly coefficient depends on the fermion representation, thus we have only a discrete number of cases that we need to fit between.

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    $\begingroup$ Fair enough, but calling the WZW effective term "classical" might be chosen language: it just incorporates quantum loop effects in a convenient pre-packaged format and reflects them in its coefficient. Many would think of them as quantum... $\endgroup$ – Cosmas Zachos May 7 '18 at 13:38
  • $\begingroup$ @Cosmas Zachos If you consider the single particle Wess-Zumino term (some refer to it as the Berry term) as in Duval and Horvathy arxiv.org/abs/1406.0718 or Dwivedi and Stone arxiv.org/abs/1308.4576v1; it is a part of the symplectic structure of spinning massless Poincaré orbits, except for the quantization of its coefficient. This term generates the chiral anomaly classically (in kinetic theory). This is why I think of the Wess-Zumino-Witten terms as classical except for the quantization of its coefficient which is quantum. $\endgroup$ – David Bar Moshe May 7 '18 at 14:52

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