Hot answers tagged anomaly
11
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 ...
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 ...
8
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 ...
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
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 ...
5
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 ...
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
$$\text{Abelian chiral anomaly in}~ ...
5
In quantum field theories it is believed that anomalies in gauge symmetries (in contrast to rigid symmetries) cannot be coped with and must be canceled at the level of the elementary fields.
May be the earliest work on the subject is: C. Bouchiat, J. Iliopoulos and P. Meyer, “An Anomaly free Version of Weinberg’s Model” Phys. Lett. B38, 519 (1972).
But ...
5
Well, I hope I am not oversimplifying your question because I guest that all I am going to say is well-known for you.
There are symmetries which correspond to physical symmetries such as spatial rotational or translational symmetry. These symmetries are not necessary for the consistency of the theory and thus the quantum theory has not to respect the ...
3
1) The axial vector current $j^{\mu 5}$ is a pseudovector
$$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 ...
3
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 ...
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
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
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
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 because ...
1
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 ...
1
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 ...
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