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To understand anomalies consider the ground state of fermions. One useful way to imagine the ground state of fermions is as a Dirac sea, as shown on the right-hand side.
In the ground state configuration all negative energy states are filled and no positive energy states are filled, i.e. no electrons or positrons exist. A positron would be a hole in the filled up Dirac sea of negative energy states.
Now, the anomaly comes about when we consider what happens when we investigate what happens to this ground state as soon as the fermions start interacting with a gauge field.
As a result of the presence of the gauge field the energy levels shift. Some states get lifted up from the Dirac sea and become positive energy states. Other empty states with former positive energy become now negative energy states and thus holes get pushed down the Dirac sea.
The ground state then is no longer empty. Instead, we have electrons and positrons (=holes in the Dirac sea). This is what we call an anomaly.
A bit more precise, the anomaly is that we would expect that a quantity is conserved, but upon closer inspection we find that it isn't. Instead we can observe the production of the quantity.
”we must assign physical reality to Dirac’s negative energy sea,because it produces the chiral anomaly, whose effects areexperimentally observed, principally in the decay of the neutral pionto two photons, but there are other physical consequences as well.”R. Jackiw
A classical theory possesses a symmetry if the action $S(\phi)$ is unchanged by a transformation $\phi \to \delta \phi$. In a quantum theory, however, we have a symmetry if the path integral $\int D \phi e^{iS(\phi)}$ is invariant under a given transformation $\phi \to \delta \phi$. The key observation is now that invariance of the action $S(\phi)$ does not necessarily imply invariance of the path integral since the measure $D \phi$ can be non-invariant too. In more technical terms, the reason for this is that whenever we change the integration variables, we need to remember that the Jacobian can be non-trivial.
An anomaly is an obstruction to the construction of a quantum theory that has the same symmetry group as its action.
For example, the standard method to construct quantum theories ensures that if there is a symmetry group of the system, the Hilbert space should also be a representation of this group.
However, through the renormalization techniques extra $U(1)$ phases are introduced that ruin the standard argument that the Hilbert space is a representation of the system's symmetry group.
Given a Lagrangian or a Hamiltonian we can search for symmetries of the action and then use Noether's theorem to calculate the conserved charges.
Each charge corresponds $Q_a$ to one generator $G_a$ of the symmetry of the action. These Noether charges represent the generators on our Hilbert space in a quantum theory or on our phase space in a classical theory.
We can then use the charges and put them into the corresponding Lie bracket. In the classical theory, this is the Poisson bracket, in a quantum theory the commutator.
In most cases the Noether charges form a closed algebraic structure which is exactly the same as the algebra of the symmetry of the action.
$$ [Q_a,Q_b] = f_{ab}^c G_c = [G_a,G_b] $$
However, for some systems this is not the case. Instead of the original symmetry algebra the Noether charges contains an additional term on the right-hand side.
$$ [Q_a,Q_b] = f_{ab}^c G_c + X_c \neq [G_a,G_b] $$
Therefore, the Noether charges in such systems generate a different symmetry. We say they form a centrally extended version of the original algebra. The additional term $X_c$ on the right-hand side is known as the Schwinger term.
Take note that people usually talk anomalies only in a quantum context and anomalies are described as quantum mechanical symmetry breaking. However it can also happen in a classical system that we can't realize our symmetry on the phase space and hence classical anomalies also exist.
Non-conserved currents
An important implications of the situation discussed above is that currents where we would think they are conserved since they correspond to symmetries via Noether's theorem, are not really conserved. A famous example is the axial current $J_A$ in the standard model.
Purely by looking at the Lagrangian we would believe
$$ \partial_\mu J_A^\mu = 0, $$ thanks to Noether's theorem.
However, upon closer inspection (e.g. by calculation the famous Adler-Bell-Jackiw triangle diagram), we find that in the full quantum theory, we get instead
$$ \partial_\mu J_A^\mu = \frac{g^2C}{16\pi^2} G^{\mu \nu a} \tilde{G}_{\mu \nu }^a$ , $$ where $G^{\mu \nu a}$ is the field-strength tensor and $\tilde{G}_{\mu \nu }^a$ its dual.
Therefore, we find that axial rotations are not really symmetries of the Lagrangian. As a consequence, whenever we perform a axial rotation our Lagrangian changes by
$$ L \to L + \Delta L = L + \partial_\mu J_A^\mu . $$
This is important since we always need an axial rotation to diagonalize the quark mass matrices and this is an important part of the strong CP puzzle.
The anomaly phenomenon is sometimes called quantum mechanical symmetry breaking, since the theory naively appears to have a certain symmetry, but the Hilbert space is not quite a representation of this symmetry, due to the subtleties of how quantum field theories are defined.
Not Even Wrong by P. Woit
More generally, the anomaly refers to a subtlety in quantization: a symmetry of the classical theory does not work in the expected way in the quantum theory. You already see this in the phenomenon of the one-half energy of the ground state in the harmonic oscillator. You can get rid of this by redefining the Hamiltonian, but that changes how the symmetries of the classical system are implemented in the quantum system. For a finite number of degrees of freedom, you can work with either Hamiltonian, but in QFT, with an infinite number of degrees of freedom, you don’t have a finite shift and this causes the anomaly.
There are different kinds of anomalies:
In addition to the anomaly we have been discussing, which affects the global symmetries studied in current algebra, there can also be an anomaly in the gauge symmetry of a theory. This is called a gauge anomaly. Gauge anomalies are less well understood, but definitely interfere with the standard methods for dealing with the gauge symmetry of Yang-Mills quantum field theory. If one throws out the quarks and considers the standard model with just the leptons, one finds that this theory has a gauge anomaly, and it ruins the standard renormalisation of the quantum field theory as first performed by 't Hooft and Veltman. To this day it is unknown whether or not there is some way around this problem, but it can be avoided since if one puts the quarks back in the theory, one gets an equal and opposite gauge anomaly that cancels the one coming from the leptons. The full standard model has no gauge anomaly due to this cancellation, and the principle that gauge anomalies should cancel is often insisted upon when considering any extension of the standard model.
Not Even Wrong by P. Woit
The existence of anomalies associated with global currents does not necessarily mean difficulties for the theory. On the contrary, as we saw in the case of the axial anomaly, its existence provides a solution of the Sutherland–Veltman paradox and an explanation of the electromagnetic decay of the pion. The situation is very different when we deal with local symmetries. A quantum mechanical violation of gauge symmetry leads to many problems, from lack of renormalizability to nondecoupling of negative norm states. This is because the presence of an anomaly in the theory implies that the Gauss’ law constraint $D · E_A = ρ A$ cannot be consistently implemented in the quantum theory. As a consequence, states that classically were eliminated by the gauge symmetry become propagating in the quantum theory, thus spoiling the consistency of the theory.
page 189 in Invitation to Quantum Field Theory by Alvarez-Gaume et. al.
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 argued 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.
Why is anomaly cancellation required for consistency? Two reasons are often cited. The first reason is that anomalies cause a loss of unitarity or Lorentz invariance. The point is that the gauge anomaly is a breakdown of gauge invariance at the quantum level. But we need gauge invariance to establish the equivalence of the covariant gauge and physical gauge formulations of a gauge theory, and thus to assure that the theory can be so formulated as to satisfy both unitarity and Lorentz invariance simultaneously [2]. The second reason is that anomalies cause a loss of renormalizability. The gauge anomaly causes the divergence structure of the theory, softened by gauge invariance, to become more severe, so that an infinite number of counterterms are generated.
http://www.theory.caltech.edu/~preskill/pubs/preskill-1991-anomalies.pdf
See for example, this chapter. The best explanation of this idea can be found in chapter 9 of An Invitation to Quantum Field Theory Autoren by Alvarez-Gauméand Vázquez-Mozo; and also in Intuitive understanding of anomalies: A Paradox with regularization by Holger Bech Nielsen and Masao Ninomiya.
See also, for example, here or here or Effects of Dirac's Negative Energy Sea on Quantum Numbers by R. Jackiw and also in section 4.5 in the book "Anomalies in Quantum Field Theory" by Bertlmann and the many references there.
This idea and the fact that this effect can actually be observed in solid state physics, leads, for example, Roman Jackiw to the conclusion:
It can then be shown that in the presence of a gauge field, the distinction between 'empty' positive-energy states and 'filled' negative-energy states cannot be drawn in a gauge-invariant manner, for massless, single-helicity fermions. Within this frame- work, the chiral anomaly comes from the gauge non-invariance of the infinite negative-energy sea. Since anomalies have physical consequences, we must assign physical reality to this infinite negative-energy sea. [5]
The Unreasonable Effectiveness of Quantum Field Theory by Roman Jackiw
The central result of all chiral anomaly analyses are:
\begin{align} \text{Classical Physics: }& \partial_\mu J_5^\mu =0 \notag \\ \text{Quantum Physics: }& \partial_\mu J_5^\mu =\frac{e^2}{(4\pi)^2}\epsilon^{\mu\nu\lambda\sigma} F_{\mu\nu}F_{\lambda \sigma} \notag \\ \end{align}
This means, the divergence of the axial current $J_5^\mu$ is non-zero through quantum effects, but is instead an operator that can produce two photons.
In all our discussion of anomalies we only considered the computation of one- loop diagrams. It might happen that higher loop orders impose additional conditions. Fortunately this is not so: the Adler–Bardeen theorem [12] guarantees that the axial anomaly only receives contributions from one loop diagrams. Therefore, once anomalies are canceled (if possible) at one loop we know that there will be no new conditions coming from higher-loop diagrams in perturbation theory. The Adler–Bardeen theorem, however, only applies in perturbation theory. It is nonetheless possible that nonperturbative effects can result in the quantum violation of a gauge symmetry. This is precisely the case pointed out by Witten [13] with respect to the SU(2) gauge symmetry of the standard model. In this case the problem lies in the nontrivial topology of the gauge group SU(2). The invariance of the theory with respect to non-trivial gauge transformations requires the number of fermion doublets to be even. It is again remarkable that the family structure of the standard model makes this anomaly cancel
page 192 in "An Invitation to Quantum Field Theory" by Luis Álvarez-Gaumé et. al.
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There is perhaps a dominant perception that quantum anomalies of classical symmetries can occur only in the context of quantum field theories. Typically they arise in the course of regularizing divergent expressions in quantum fields [1, 2], causing the impression that it is these divergences that cause anomalies. It is however known that anomalies can occur in simple quantum mechanical systems such as a particle on a circle or a rigid rotor. Esteve [4, 5] explained long ago that the presence or otherwise of anomalies is a problem of domains of quantum operators. Thus while quantum state vectors span a Hilbert space H, the Hamiltonian H is seldom defined on all vectors of H. For example, the space H of square-integrable functions on R 3 contains nondifferentiable functions ψ, but the Schroedinger Hamiltonian H = − 1 2m ∇2 is not defined on such ψ. Rather H is defined only on a dense subspace DH of H. If a classical symmetry g does not preserve DH, gDH 6= DH, then Hg ψ for ψ ∈ DH is an ill-defined expression. In this case, one says that g is anomalous [4, 5]. See also [6–12].
In the present work, we explore the possibility of overcoming anomalies by using mixed states. There are excellent reasons for trying to do so, there being classical gauge symmetries like SU(3) of QCD or large diffeomorphisms (diffeos) of manifolds (see below) which can become anomalous. Color SU(3) does so in the presence of non-abelian monopoles [13–15], while “large” diffeos do so for suitable Friedman-Sorkin geon manifolds [16–18]. It is surely worthwhile to find ways to properly implement these symmetries.
[…]
While non-abelian structure groups of twisted bundles are always anomalous, abelian groups also of course can be anomalous. For instance, parity anomaly for a particle on a circle (discussed in section 2 of this work) and the axial U(1)A anomaly in the Standard Model are both abelian. The crucial issue is whether the classical symmetry preserves the domains of appropriate operators like the Hamiltonian. If they do not preserve such domains, then they are anomalous. The important feature of non-abelian structure groups of twisted bundles is that they never preserve the domain of the Hamiltonian. https://arxiv.org/pdf/1108.3898.pdf
"In field theory and in modern elementary particle theory, cocycles are used to describe anomalies."Source
When chiral fermions are coupled to gauge fields, the algebra of gauge transformations acquires a Mickelsson-Faddeev (MF) cocycle (eq. 22) - a gauge anomaly. Mickelsson, together with Rajeev and maybe others, originally tried to construct representations of the this algebra. However, it was shown in
D. Pickrell, On the Mickelsson-Faddeev extensions and unitary representations, Comm. Math. Phys. 123 (1989) 617.
that the MF algebra does not possess any faithful unitary representation on a separable Hilbert space (or something like that). As a response to this disappointing result, Mickelsson developed a theory where cocycles depending on an external gauge potential are regarded as generalized representations. It is this theory which evidently is naturally formulated in terms of Gerbes.Thomas Larsson
Recently it has become clear that gauge theories with fermion display three different kinds of anomalies, all related to the global topology of the fourdimensional configuration space $C^ 4$ by the family index of the Dirac operator $D^4$ . These are the axial U(l) anomaly [the "$\pi_0(G^3)$ anomaly"], Witten's SU(2) anomaly [2] [from "$\pi_1(G^3)$" ] , and the nonabelian gauge anomaly [3] [from "$\pi_2(G^3)$"]. The diversity of the manifestations of these anomalies seems to belie their common origin, however. In the first case we find particle production in the presence of instanton fields [4], breaking of a global symmetry, and no problem with gauge invariance. In the second we find no problem with chiral charge, but instead a nonperturbative failure of gauge symmetry, while in the latter the same thing occurs even perturbatively. What is going on?
Hamiltonian Interpretation of Anomalies by Philip Nelson and Luis Alvarez-Gaume
Anomalies have been first discovered in perturbation theory by UV–regularizing a divergent diagram [1,2]. But they are not just a regularization effect, they also show up in quite different procedures. For example, in the method of dispersion relations where they occur as an IR–singularity of the transition amplitude [7,8], or within sum rules [9]. Using a quite different approach, working with path integrals, the anomalies are detected by the chiral transformation of the path integral measure [10]. In the last years a development in modern mathematical techniques attracted much attention in describing the anomalies. This was differential geometry [11–17], cohomology [18,19]and topology (Atiyah–Singer index theorem) [20–29].
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.
https://physics.stackexchange.com/questions/33195/classical-and-quantum-anomalies
The non-Abelian chiral anomaly is very well understood algebraically and topoplogically, please see the following review by R. A. Bertlmann. In the example following equation (38) in the article, all the quantities associated with the non-Abelian chiral anomaly are algebraically computed (without solving Feynman diagrams) through what is called the Stora-Zumino descent equations. These equations give on the first level the Chern-Simons term, on the second level, the anomaly (divergence of the current), on the third level, the extension in the gauge group commutation relations and on the fourth level, the associator causing the violation of the Jacobi-identity, thus resulting a non associative algebra (please see the following related physics stack exchange question).
I mentioned, the descent equations, because there is a modern concept of Gerbes trying to find geometrical realization of these equations (please see also the following Mickelsson's review). This direction of research has the potential of providing deeper understanding what are the quantum structures that we must associate to these algebras (interpreted as classical algebras of Poisson brackets) because the usual Hilbert spaces and unitary representations do not seem to work. The Mickelsson-Faddeev algebra was extensively analyzed within the theory of gerbes, please see for example this work by Hekmati, Murray, Stevenson and Vozzo (and also the above Micklsson's reference). https://physics.stackexchange.com/a/76653/37286
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. https://physics.stackexchange.com/a/34022/37286
Anomalies are often a useful first line of attack in trying to understand new systems. This is because the presence of anomalies, or the way they are canceled, can often be studied without knowing the detailed dynamics of the theory. They are in a way topological properties of the theory and thus can be studied by approximate methods.
Implications of quantum anomalies are numerous. First and foremost, their knowledge is needed in order to avoid theories which look fully “legitimate” at the classical level, but become terminally sick upon quantization. For instance, suppose one would like to build an extension of the Standard Model, with additional fermions beyond the standard three generations. If the fermion content is chosen inappropriately, such an extension may well be internally inconsistent. Second, the chiral quantum anomalies play an important role in the soft pion theory in QCD, and in the ’t Hooft matching condition, which, in turn, presents the foundation for the Seiberg duality in supersymmetric QCD. The scale quantum anomalies which are typical of asymptotically free field theories (such as QCD) can be used for establishing a number of low-energy theorems.
https://physique.cuso.ch/fileadmin/physique/document/2015_shifman_lecture_1.pdf
In four-dimensional quantum field theories, the problem of the anomaly or Schwinger term is much trickier. Current algebra in four dimensions has led to a significant amount of understanding of the physical aspects of the problem. One of the earliest physical consequences of the anomaly concerned the rate at which neutral pions decay into two photons. If one ignores the anomaly problem, current algebra predicts that this decay will be relatively slow, whereas experimentally it happens very quickly. Once one takes into account the anomaly, the current algebra calculation agrees well with experiment. This calculation depends on the number of colours in QCD, and its success was one of the earliest pieces of evidence that quarks had to come in three colours. Another successful physical prediction related to the anomaly was mentioned earlier. This is the fact that, ignoring the anomaly, there should be nine low-mass pions, the Nambu-Goldstone bosons of the spontaneously broken symmetry in current algebra. In reality, there are nine pions, but only eight of them are relatively low mass. The higher mass of the ninth one can be explained once one takes into account the effect of the anomaly.
Not Even Wrong by P. Woit
for me chiral and scale symmetry breaking are completely natural effects, but their description in our present language – quantum field theory – is awkward and leads us to extreme formulations, which make use of infinities. One hopes that there is a more felicitous description, in an as yet undiscovered language. It is striking that anomalies afflict precisely those symmetries that depend on absence of mass: chiral symmetry, scale symmetry. Perhaps when we have a natural language for anomalous symmetry breaking we shall also be able to speak in a comprehensible way about mass, which today remains a mystery.
THE UNREASONABLE EFFECTIVENESS OF QUANTUM FIELD THEORY by R. Jackiw
On the other hand, anomalies in physics are not always unwanted features to be eradicated. E.g. the trace anomalies associated to dilatation invariance lead to the Callan– Symanzik equations [7].
In dimensions other than the critical dimension, the action $S$ of the Bosonic string theory has a conformal anomaly.
page 267 in Topology and Quantum Field Theory by Charles Nash
It began with the pioneering work of Alvarez-Gaume and Witten [Alvarez-Gaume, Witten 1983] on gravitational anomalies, and the enthusiasm culminated in the discovery of Green and Schwarz [Green, Schwarz 1984]that gauge and gravitational anomalies may cancel each other, however, in a supersymmetric theory in 10 dimensions.
Anomalies in Quantum Field Theory by Reinhold A. Bertlmann
One of the earliest physical consequences of the anomaly concerned the rate at which neutral pions decay into two photons. If one ignores the anomaly problem, current algebra predicts that this decay will be relatively slow, whereas experimentally it happens very quickly. Once one takes into account the anomaly, the current algebra calculation agrees well with experiment. This calculation depends on the number of colours in QCD, and its success was one of the earliest pieces of evidence that quarks had to come in three colours. Another successful physical prediction related to the anomaly was mentioned earlier. This is the fact that, ignoring the anomaly, there should be nine low-mass pions, the Nambu-Goldstone bosons of the spontaneously broken symmetry in current algebra. In reality, there are nine pions, but only eight of them are relatively low mass. The higher mass of the ninth one can be explained once one takes into account the effect of the anomaly.
Not Even Wrong by P. Woit
On the other hand, anomalies in physics are not always unwanted features to be eradicated. E.g. the trace anomalies associated to dilatation invariance lead to the Callan– Symanzik equations [7].
www.atlantis-press.com/php/download_paper.php?id=754
->Are theories with gauge anomalies necessarily inconsistent?#
No! See Gauge anomalies in an effective field theory by JohnPreskill <--
The nomenclature is misleading. At its discovery, the phenomenon was unexpected and dubbed 'anomalous'. By now the surprise has worn off, and the better name today is 'quantum mechanical' symmetry breaking
The Unreasonable Effectiveness of Quantum Field Theory by R. Jackiw
You have to appreciate the frame of mind that field theorists operated in to understand their shock when they discovered in the late 1960s that quantum fluctuations can indeed break classical symmetries. Indeed, they were so shocked as to give this phenomenon the rather misleading name "anomaly", as if it were some kind of sickness of field theory. With the benefits of hindsight, we now understand the anomaly as being no less conceptually innocuous as the elementary fact that when we change integration variables in an integral we better not forget the Jacobian.
QFT in a Nutshell by A. Zee
[T]he anomaly refers to a subtlety in quantization: a symmetry of the classical theory does not work in the expected way in the quantum theory. You already see this in the phenomenon of the one-half energy of the ground state in the harmonic oscillator. You can get rid of this by redefining the Hamiltonian, but that changes how the symmetries of the classical system are implemented in the quantum system. For a finite number of degrees of freedom, you can work with either Hamiltonian, but in QFT, with an infinite number of degrees of freedom, you don’t have a finite shift and this causes the anomaly. Put differently, the anomaly is due to the fact that normal-ordering is needed in QFT, and this sometimes changes how classical symmetries appear after quantization.
It is important to avoid here the misconception that anomalies appear due to a bad choice of the way a theory is regularized in the process of quantization. When we talk about anomalies we mean a classical symmetry that cannot be realized in the quantum theory, no matter how smart we are in choosing the regularization procedure.
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.
In all our discussion of anomalies we only considered the computation of one- loop diagrams. It might happen that higher loop orders impose additional conditions. Fortunately this is not so: the Adler–Bardeen theorem [12] guarantees that the axial anomaly only receives contributions from one loop diagrams. Therefore, once anomalies are canceled (if possible) at one loop we know that there will be no new conditions coming from higher-loop diagrams in perturbation theory. The Adler–Bardeen theorem, however, only applies in perturbation theory. It is nonetheless possible that nonperturbative effects can result in the quantum violation of a gauge symmetry. This is precisely the case pointed out by Witten [13] with respect to the SU(2) gauge symmetry of the standard model. In this case the problem lies in the nontrivial topology of the gauge group SU(2). The invariance of the theory with respect to non-trivial gauge transformations requires the number of fermion doublets to be even. It is again remarkable that the family structure of the standard model makes this anomaly cancel
page 192 in "An Invitation to Quantum Field Theory" by Luis Álvarez-Gaumé et. al.
From the point of view of a mathematician, one aspect of the anomaly is that it is related both to the Atiyah-Singer index theorem and to a generalisation known as the index theorem for families. Whereas the original index theorem describes the number of solutions to a single equation and does this in terms of the number of solutions of a Dirac Equation, the families index theorem deals with a whole class or family of equations at once. A family of Dirac equations arises in physics because one has a different Dirac equation for every different Yang-Mills field, so the possible Yang-Mills fields parametrise a family of Dirac equations. This situation turns out to be one ideally suited to the use of general versions of the index theorem already known to mathematicians, and in turn has suggested new versions and relations to other parts of mathematics that mathematicians had not thought of before. As usual, Witten was the central figure in these interactions between mathematicians and physicists, producing a fascinating series of papers about different physical and mathematical aspects of the anomaly problem.
Not even Wrong by P. Woit
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.
Early work on current algebra during the 1960s had turned up a rather confusing problem which was dubbed an 'anomaly'. The source of the difficulty was something that had been studied by Schwinger back in 1951, and so became known as the problem of the Schwinger term appearing in certain calculations. The Schwinger term was causing the Hilbert space of the current algebra to not quite be a representation of the symmetry group of the model. The stan- dard ways of constructing quantum mechanical systems ensured that if there was a symmetry group of the system, the Hilbert space should be a representation of it. In the current algebra theory, this almost worked as expected, but the Schwinger term, or equivalently, the anomaly, indicated that there was a problem. The underlying source of the problem had to do with the neces- sity of using renormalisation techniques to define properly the current algebra quantum field theory. As in QED and most quantum field theories, these renormalisation techniques were necessary to remove some infinities that occur if one calculates things in the most straight- forward fashion. Renormalisation introduced some extra U(l) phase transformations into the problem, ruining the standard argument that shows that the Hilbert space of the quantum theory should be a representation of the symmetry group. Some way needed to be found to deal with these extra U(l) phase transformations. In two-dimensional theories, it is now well understood how to treat this problem. In this case, the anomalous U(l) phase transformations can be dealt with by just adding an extra factor of U(l) to the orig- inal infinite-dimensional symmetry group of the theory. The Hilbert space of the two-dimensional theory is a representation, but it is one of a slightly bigger symmetry group than one might naively have thought. This extra U(l) piece of the symmetry group also appears in some of the infinite dimensional Kac-Moody groups. So in two dimensions the physics leading to the anomaly and the mathemat- ics of Kac-Moody groups fit together in a consistent way. In four-dimensional quantum field theories, the problem of the anomaly or Schwinger term is much trickier. Current algebra in four dimensions has led to a significant amount of understanding of the physical aspects of the problem. One of the earliest physical conse- quences of the anomaly concerned the rate at which neutral pions decay into two photons. If one ignores the anomaly problem, current algebra predicts that this decay will be relatively slow, whereas exper- imentally it happens very quickly. Once one takes into account the anomaly, the current algebra calculation agrees well with experiment. This calculation depends on the number of colours in QCD, and its success was one of the earliest pieces of evidence that quarks had to come in three colours. Another successful physical prediction related to the anomaly was mentioned earlier. This is the fact that, ignoring the anomaly, there should be nine low-mass pions, the Nambu-Goldstone bosons of the spontaneously broken symmetry in current algebra. In reality, there are nine pions, but only eight of them are relatively low mass. The higher mass of the ninth one can be explained once one takes into account the effect of the anomaly.
page 129ff in Not Even Wrong by P. Woit
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