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16

These are all good questions. Perhaps I can answer a few of them at once. The equation describing the violation of current conservation is $$\partial^\mu j_\mu=f(g)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$ where f(g) is some function of the coupling constant. It is not possible to write any other candidate answer by dimensional analysis and ...


15

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 ...


11

The two kinds of trace anomalies are related but distinct. The first one that you refer to is the anomaly in Weyl transformations that occurs when you put a CFT on a curved background. The CFT is still exactly conformally invariant in flat space, but this symmetry is broken by the background gravitational field. It's useful to think about CFTs in two ...


9

The anomalies in four dimensions are calculated from a triangular Feynman diagram with a chiral (left-right-asymmetric, when it comes to the couplings with the gauge bosons or gravitons) fermion running in the loop and three gauge bosons (and/or graviton[s]) attached at the vertices. For the Standard Model, all the gauge anomalies cancel (both leptons and ...


9

There are things called sigma model anomalies, see papers listed in a sample inspire database query here. Here, the anomaly is associated to the general coordinate invariance in the target space of the non-linear sigma model: the fields take values in a nontrivial manifold (and its associated vector bundles), rather than vector spaces. Classically, the ...


8

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 ...


7

It is very hard to visualize these homotopy classes, since they correspond to maps $S^4\rightarrow SU(2)\approx S^3$. The homotopy groups of spheres (and any other space) are typically very difficult to calculate in generality and physicists typically ask mathematicians. But there exist simple results in the so-called "stable range" where there is a regular ...


7

Anomalies (not anamolies) are a whole subject whose basics are covered by one or several chapters of almost any good enough quantum field theory textbook so it's counterproductive to retype this whole chapter here. But generally, in quantum field theory, anomalies are quantum mechanical effects breaking symmetries that exist in the classical theory – ...


6

This question has been posted also at http://mathoverflow.net/questions/115866/homotopy-pi-4su2z-2 with both geometric and algebraic (that was mine!) type of answers. The geometric answers tell of Pontjyagin's method of constructing explicit representations of maps to spheres. The algebraic methods tells of the answer from a general theorem which gives some ...


6

There is no chiral anomaly/gauge anomaly if the spacetime dimension $2\ell+1$ is odd, partly because $SO(2\ell+1)$ has real or pseudo-real representations, but no complex representations. There may instead be parity anomalies in odd spacetime dimensions. In fact, there is a dimensional ladder of related anomalies $$\text{Abelian chiral anomaly in}~ ...


6

A more general definition of anomaly: A QFT that has no UV completion in the same dimension is anomalous. In other words, a QFT that has no well defined short distance regularization in the same dimension is anomalous. Example: A 1+1D QFT with only one right moving fermion mode is anomalous.


6

Since $SU(2)$ is topologically a three-sphere $S^3$, you can begin by investigating the homotopy groups of spheres. Unfortunately, although there are some regular results, such as $\Pi_n(S^n)=\mathbb{Z}$, and $\Pi_m(S^n)=0$ for $m<n$, I don't think there is a single method to calculate $\Pi_m(S^n)$ for $m>n$. Individual results for $m>n$ are ...


6

The answer in the book is almost correct albeit oversimplified. If you want to jot them down, then there are several reasons for requiring that $d=9+1$ in superstring theory. None of them however are in any way "simple" (compared to the kind of explanations that the book seems to give). Let me jot down some of the reasons that come to mind. (There may be ...


5

I am not sure if this is the way you want to think about it, but I think it is worth pointing out that not having the central charge leads to a trivial quantum theory. The precise statement would be that a positive/unitary theory with c=0 has only one state, the vacuum. The details are demonstrated in J.F. Gomes. The triviality of representations of the ...


5

A good analogy for the difference between the two can be given in terms of two other examples of anomalies, that are possibly more familiar. Consider a field theory with a global symmetry, take $U(1)$ for simplicity. At the classical level, the equations of motion lead to the existence of a conserved current (Noether's theorem). At the quantum level, the ...


5

Weinberg's presentation is not pedagogically ideal, because the steps are too formal, and he puts emphasis on ones that can be misleading to a student. The presentation might lead you think that the determinant of the U factors are somehow not equal to 1 naively because of some phase business, and this is categorically not true. This is what is confusing ...


5

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 ...


5

The worldsheet Weyl anomaly in bosonic string theory is an example. More generally in any dimension you can have trace and Weyl anomalies that break scale or conformal invariance, even in systems with only bosons.


4

If gauge symmetries are not fake, but are real symmetries, then, when they are anomalous, it simply means that the theory just does not have the symmetries. The theory is still well defined, at least. If gauge symmetries are fake and representing redundancy in our description, then, when they are anomalous, it means that the theory is inconstant. So ...


4

1) The axial vector current $j^{\mu 5}$ is a pseudovector $$j^{\mu 5}~:=~\overline{\psi}\gamma^{\mu}\gamma^5\psi~=~j^{\mu}_R-j^{\mu}_L,\qquad j^{\mu}_{R,L}~:=~ \overline{\psi}_{R,L}\gamma^{\mu}\psi_{R,L}, $$ $$\psi_{R,L}~:=~P_{R,L}\psi,\qquad P_{R,L} ~:=~\frac{1\pm\gamma^5}{2} . $$ The $4$-divergence $d_{\mu}j^{\mu 5}$ is a pseudoscalar. That the axial ...


4

The existence of anomalies is almost always accompanied by an extension of the gauge group commutation relations. The case of non-Abelian axial anomaly is may be the most known case. The abstract gauge group algebra: $ [G_a(x), G_b(y)] = if_{ab}^{c} \delta^N(x-y)$ ($N$ is the number of dimensions), is not realized at the quantum field level When $N=1$, ...


4

I'm pretty sure the answer is "no." There's a conformal anomaly in any even number of dimensions. For instance, in 4 dimensions it's the statement that $T^\mu_\mu$ is nonzero for a CFT on a general curved background, equal to coefficients $a$ and $c$ times the Euler density and Weyl$^2$ terms. But there's no central extension of the conformal algebra the way ...


4

Your question is very interesting. I would like to mention something along the line of your question, but perhaps from another viewpoint. Recently there are some better understanding along the thinking between (1)"whether a theory is free from anomaly (the anomaly matching condition satisfied)," (2)"whether the symmetry of a theory is on-site symmetry," ...


3

Related Why do some anomalies (only) lead to inconsistent quantum field theories What if we work without the gauge redundancy, with just the physical polarizations, what goes wrong then if we don't make sure the would-be gauge symmetry is not anomalous? This is a very natural question I wondered some time ago. In electrodynamics, if you want to ...


3

I'd say that there is not a systematic summary of the status of symmetries on particle physics, but if any, it should be spread all over the PDG review. However, I'd like to comment on a few points. So far Lorentz symmetry is exact on all sectors.${}^\dagger$ Scaling (part of the conformal transformations) is broken once an energy scale is introduced in ...


3

In general the chiral non-Abelian anomaly does not vanish. It is proportional to the three dimensional symmetric tensor $$ d_{ABC} = \mathrm{tr}(T_A\{ T_B T_C\})$$ This tensor vanishes in the particular case of $SU(2)$, but it is nonvanishing for $SU(N)$, for $N>2$. It should be also mentioned, however, that in the special case of $SU(2)$ there is a ...


3

It is a non-perturbative effect because it is 1-loop exact. The triangle diagram is actually the least insightful method to think about this, in my opinion. The core of the matter is the anomaly of the chiral symmetry, which you can also, for example, calculate by the Fujikawa method examining the change of the path integral measure under the chiral ...


2

Descent equations for the trace anomaly in arbitrary dimensions are derived in this paper: http://arxiv.org/abs/0706.0340. Since this article is very technical, I think that in order to get a better understanding of the principle, it would be useful to go through the chapters on gravitational anomalies in Bertlmann's book "Anomalies in Quantum Field ...


2

Alright, I will try to answer why we need Dirac eigenstates in this procedure, but I am not sure if it is anything more that the tautology that the Fujikawa method is precisely defined by using the Dirac eigenstates. Let me briefly recap what the idea is: (All of this is for a Euclidean theory.) We consider the infinitesimal local transformaton $$\psi(x) ...


2

There is a very simple and enlightening explanation due to N.V.Gribov given in his following conference article and also beautifully explained by Dmitri Kharzeev in the following arxiv article(section 1). Gribov's argument doesn't involve the heavy machinery of quantum field theory. He actually proves that in the case of colinear electric and magnetic ...



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