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In connection with Classical and quantum anomalies, I'd like to ask for a simple explanation why some anomalies lead to valid quantum field theories while some others (happily absent in the standard model) seem to make the corresponding quantum field theory inconsistent.

Edit: More specifically, in case of an anomalous gauge symmetry: Why can't one get a valid theory by using a central extension of the gauge group as the quantum version of the gauge group? Has this been tried and not found working, giving a no-go theorem? Or would that lead to a different classical theory in the limit $\hbar\to 0$?

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In quantum field theories it is believed that anomalies in gauge symmetries (in contrast to rigid symmetries) cannot be coped with and must be canceled at the level of the elementary fields.

May be the earliest work on the subject is: C. Bouchiat, J. Iliopoulos and P. Meyer, “An Anomaly free Version of Weinberg’s Model” Phys. Lett. B38, 519 (1972). But certainly, one of the most famous ones is the Gross-Jakiw article: Effect of Anomalies on Quasi-Renormalizable Theories Phys. Rev. D 6, 477–493 (1972)

They agrgued that the 'tHooft-Veltman perturbative proof of the renormalizability of gauge theories requires the anomalous currents not to be coupled to gauge fields. In the more modern BRST quantization language, gauge anomalies give rise to anomalous terms in the Slavnov-taylor identities which cannot be canceled by local counter-terms therefore ruin the combinatorial proof of perturbative renormalizability and of the decoupling of the gauge components and ghosts which results a non-unitary S-matrix.

This phenomenon is independent of the quantum field theory dimension. In 3+1 dimension, the requirement of cancelation of the chirally coupled standard model leads to the correct particle content. In 1+1 dimensions, it leads to the dimension of string target space (Virasoro anomaly) and the optional anomaly free gauged subgroups in the WZNW models in two dimensions (Kac-Moody anomaly). In certain dimensions there is the Green-Schwrz Cancellation mechanism, but it is equivalent to an addition of a local term to the Lagrangian. This mechanism is not appropriate to 3+1 dimensions, since this term is not renormalizable.

The chiral anomaly can be corrected by adding a Wess-Zumino term to the Lagrangian, but this term is not perturbatively renormalizable, thus does not solve the nonrenormalizability problem.

However, the cancellation of anomalies does not mean that we should not seek for representations of the "anomalous" current algebras. At the contrary, in 1+1 dimensions the Virasora and the Kac-Moody algebra do not have positive energy highest weight representations unless they are not anomalous.

According to this prionciple, the spectra of each sector of the theory are determined by the anomaly of this sector, in spite of the fact that the total anomaly vanishes.

In the late eighties, R.G. Rajeev and especially Juoko Mickelsson started a project in which they seeked representations of anomalous non-centrally extended algebras.(i.e., those present in 3+1 dimensions). The presence of gauge field depended Abelian extensions (The Mickelsson-Faddeev extension), made this problem hard to solve.

One route they adopted was to consider an algebraic universal gauge theory (in the same sense as universal classifying spaces). But a no-go result by Doug Pickrell stated that the universal anomalous algebra has no nontrivial unitary representations. This result is valid for the universal model but it is discouraging for the actual (Mickelsson-Faddeev) extended algebra. There are later works on the subject by Juoko Mickelsson himself and also by Edwin Langmann; but the question remains open.


Regarding the question about the validity of an anomalously gauged theory as an effective gauge theory (a la Weinberg, Let us consider the Skyrme model for definiteness): Both are strictly perturbatively non-renormalizable, but there is a big difference in their divergence behavior:

An anomalously gauged theory requires, in each order in its loop expansion counter-terms, with arbitrarily high derivatives (or arbitrarily high external momenta). In contrast, due to the chiral perturbation theory of an effective field theory each order of the loop expansion requires only counter-terms with one order higher (by 2) in external momenta.

Chiral perturbation theory is unitary order by order in the momentum expansion. The bound on the powers of the momenta at each order makes it viable as a low energy effective theory. The current green functions respect non-anomalous Ward identities, which control the divergences.

In an anomalously gauged theory a truncation of the higher momentum power counter-terms would result S-matrix non-unitarity.

This is the main reason, why it is widely accepted that an anomalously gauged theory requires additional fields to cancel the anomalies. (This principle came into experimental demonstration in the prediction of the t-quark).

In regards of Drake's remark; there is the case of the chiral Schwinger model in two dimensions (2D-QED with chiral coupling of the electron to the photon). This model is exactly solvable. In the exact solution, the gauge boson acquires a mass which is an indication consistent with Drake's remark that more "Degrees of freedom are needed". There is the "problem" that Drake pointed out, of how the "number of degrees of freedom" jumps between the (supposedly) non-anomalous classical theory and the quantum theory. My, point of view is that the space of the Grassmann variables describing the fermions (semi-)classically is not an appropriate phase space, although it is a symplectic supermanifold (Poisson brackets can be defined between Grasmann variables). This difficulty was already encountered by: Berezin and Marinov, when they noticed that they cannot define a nontrivial Grassmann algebra valued phase space distribution, so, they stated that Grassmann variables acquire a meaning only after quantization. One way to define a fermionic phase space on which a density of states can be defined is to pick the "manifold of initial data" as in the bosonic case. In finite systems, these types of manifolds turn out to be coadjoint orbits, however I don't know of any work in which the anomaly is derived classically on "the manifold of initial data" of a fermionic field. I think that on these "honest" fermionic phase spaces, chiral anomalies are present classically; however, I don't know of any work in this direction. I consider this a very interesting problem.

A remark on the non-uniqueness of the mechanism of anomaly cancellation: We can cancel anomalies by adding a new family of fermions, various Wess-zumino terms (corresponding to different anomaly free subgroups), and may be masses to the gauge fields (as in the Schwinger model). This non-uniqueness, reflects the fact that when anomaly is present, the quantization is not unique, (in other words the theory is not completely defined). This phenomenon is known in many cases in quantum mechanics (inequivalent quantizations of a particle on a circle), and quantum field theory (theta vacua).

Finally, my point of view is that the anomaly cancellation does not dismiss the need of finding "representations" to the anomalous current algebras in each sector. This principle works in 1+1 dimensions. It should work in any dimension because according to Wigner quantum theory deals with representations of algebras. This is why I think that Mickelsson's project is important.

May be for higher dimensions, more general representations than representations on Hilbert spaces are needed due to the non-central extensions present in these dimensions. For me, this problem is very interesting.

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So it is mainly the question of renormalizability that spoils an anomalous quantum gauge theory. Would it be viable as an effective field theory in Weinberg's sense? – Arnold Neumaier Aug 12 '12 at 16:32
Gauge symmetries reduce the number of degrees of freedom (DOF). If one looses gauge invariance at the quantum stage, then at best, the quantum theory has more DOF than the classical one and I wonder how you can get the classical theory from the anomalous quantum taking the classical limit (does one lose DOF on the way?). At worst (and usually), the "quantum theory" instantly blows-up or it is not quantum at all (there are ghost). I think that renormalizability is neither the only problem nor the worst. – drake Aug 12 '12 at 22:28
Sorry for the late response, I added an update to the answer addressing your question and relating to Drake's remark. I have edited the account about the Mickelsson’s project which did not reflect my opinion on the importance of this approach which I think that is very valuable. – David Bar Moshe Aug 16 '12 at 10:22
I'm sure this has been said many times before, but "gauge symmetry" is absolutely a misnomer as what we really have is a gauge redundancy. Mathematically, we have a space of connections $\mathcal{A}$, and gauge theories are all about quantizing the Poisson space $T^\ast \mathcal{A} / \mathcal{G}$, rather than the symplectic space $T^\ast \mathcal{A}$ (which does have a gauge symmetry). From this point of view, of course an anomalous redundancy plays a different role, both conceptually and mathematically, than an anomalous symmetry. – Jonathan Aug 16 '12 at 23:36

Well, I hope I am not oversimplifying your question because I guest that all I am going to say is well-known for you.

There are symmetries which correspond to physical symmetries such as spatial rotational or translational symmetry. These symmetries are not necessary for the consistency of the theory and thus the quantum theory has not to respect the symmetry, they can be anomalous. They are approximate symmetries in the sense that they are symmetries as long as the classical approximation is valid.

However, there are other symmetries which does not reflect symmetries of nature, but redundancies in its description. These are gauge symmetries (that goest to the identity in the boundary). For instance, the U(1) symmetry or redundancy of electrodynamics whose elements go to the identity in the boundary. While in the classical theory one can disregard them (one can use the electric and magnetic field instead of the 4-potential), they are necessary for the consistency in the quantum theory (here, I do not mean one needs it to describe some observations like the Aharanov-Bohm effect —here you only need a Wilson loop which is gauge invariant like the Electric field—, but for self-consistency of the theory). Without this redundancy, one cannot (or we do not know yet) construct a quantum theory of massless, spin-1 particles that describe long range interactions which preserve unitarity, stability of the vacuum (Hamiltonian bounded from bellow) and Poincare invariance. One needs the redundancy for consistency. Therefore, if the symmetry (better said, redundancy) were anomalous, the theory would be inconsistent. For instance, if one tries to construct a quantum theory of electrodynamic without gauge invariance, one concludes either:

1) The theory is not Lorentz invariant at the quantum level, or

2) The theory does not reflect the $1/r$ potential, or

3) The Hamiltonian is unbounded from bellow and therefore it is unstable, or

4) The theory has ghosts (states with negative norm) and therefore cannot be a probabilistic theory.

Note that 1 and 2 do not lead to inconsistencies, but they are part of what we understand by electrodynamics and what we observe. 3 makes the theory instantly blows-up. 4 is incompatible with quantum mechanics.

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''Therefore, if the symmetry (better said, redundancy) were anomalous, the theory would be inconsistent.'' It is this part at which my question aims. I edited the question to make it more precise. – Arnold Neumaier Aug 12 '12 at 10:16
''I guest that all I am going to say is well-known for you.'' I understands things very well down to QED and 2D QFTs. But I have many unresolved issues with the standard model, though I understand superficially what people do. – Arnold Neumaier Aug 12 '12 at 10:18

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