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15

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 ...


14

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

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 ...


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 ...


7

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 ...


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 ...


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

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 ...


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

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}~ ...


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 ...


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

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

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.


4

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 ...


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," ...


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 ...


3

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 $N=1$, ...


2

From what we understand today, p-branes are honest degrees of freedom, on equal footing with strings. Shop you have a good question. But I don't think anyone had so far managed to consistently quantize a p-brane. Loosely, a brane has much more degrees of frerdom than a string and it's difficult to get them under control. So quantizing it is a technical ...


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 ...


2

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 ...


2

In the case of an ungauged symmetry, the product of the exponents of the action functional corresponding of all sectors of the theory needs not to be a scalar, i.e., there can be still a net anomaly as far as its symmetry is not gauged. In this case, the exponent of the action functional will be a section of a line bundle, and the wave function(als) ...


2

(1) String Theory is a very mathematical theory based on some natural assumptions, and this ends up relating Quantum Mechanics and General Relativity, as we want. Some of the equations in String Theory, however, have a proportionality constant $c$ in it, called the central charge. And when we manipulate these equations and set them equal to each other, we ...


2

One can posit mathematical string theories in any dimensions of any kind. However, I do not understand why there are ten dimensions and not just any other number? The specific dimensions arise from the requirements of the known physics encapsulated in the Standard Model and other data coming from particle physics, plus the requirement of General ...


2

The beta function beyond 1-loop is scheme dependent, but the physical quantities you can extract from it are scheme independent (at least if you can compute the beta function at all order). For instance, even though the fixed point position is scheme dependent, the critical exponent are not. On the other hand, if you stop at a given order in the loop ...


2

The original reference for the derivation of this result is http://arxiv.org/abs/hep-th/9302047. A brief answer to your questions is: the Euler density in generic number of dimensions is a diffeo-invariant functional of the metric, built out of the curvature. In dimension 2d it is a polynomial of degree d in the Riemann tensor. Its defining property ...


2

Let me add a few comments to Michael Brown's answer/comment. As he mentioned, a QFT is well defined with an action $and$ a regulator. We always wish to use regulators that preserve gauge invariance, since that is a redundancy of our description and should not be removed in our quantum theory. However, any regulator that preserves gauge invariance, ...



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