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72

Ever since the time of Newton physics is about observing nature, quantifying observations with measurements and finding a mathematical model that not only describes/maps the measurements but, most important, it is predictive. To attain this, physics uses a rigorous self-consistent mathematical model, imposing extra postulates as axioms to relate the ...


36

@annav's answer already describes well how physical theories work and how they require self-consistency. I'd like to add some comments from a different perspective to that. TL;DR Physical theories have to be self-consistent AND consistent with observation. Mathematical self-consistency Firstly if we treat a physical theory as a mathematical axiom system ...


27

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 ...


23

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 $g$. It is not possible to write any other candidate answer by dimensional analysis ...


15

If the only mathematical statements admitted in a physical theory were those having immediate empirical content (i.e. they can be tested by an unambiguous experiment), then you would have a very good case to make. Why? Because the consistency of the world of experience would guarantee the consistency of the mathematical formalism. End of story. In reality, ...


14

If theories were only used to describe what we already know and observe, maybe they would not need to be self-consistent; they could even just degenerate into big lists of observed phenomena. This is what science looked like in Sumer, 5000 years ago. If we want physical theories to be predictive, they have to be self-consistent in the sense that they have to ...


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 ...


11

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 ...


11

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 – ...


10

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 ...


10

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

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}~ 2\ell+...


8

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 ...


8

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 ...


7

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 ...


7

The main answer to the question is that the full generator $$L_n = L^{(\mathrm{m})}_n+L^{(\mathrm{g})}_n$$ in Bosonic string theory is a sum of a matter part with normal ordering constant $a=1$, $$L^{(\mathrm{m})}_n=\frac{1}{2} \eta_{\mu\nu}\sum_m:\alpha_{n-m}^{\mu}\alpha_m^{\nu}:-\hbar a\delta_n^0,$$ and a ghost part $$ L^{(\mathrm{g})}_m=\sum_n(m-n):...


7

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 ...


7

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 ...


7

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.


7

Physics is the art of compressing our knowledge of the universe. As it happens, whenever we stick two massive bodies near each other (or notice them near each other), they seem to move towards each other. Now, we could simply record the fact that every massive body (individually) is moving towards every other massive body (individually). This is a large ...


7

Mathematical theories which are not consistent prove contradictory things (this is just a statement about mathematics and what it means to be inconsistent, not to do with physics in particular). We do not want theories of physics that predict contradictory things. Ideally we don't want theories that make any wrong predictions, but if our theory makes two or ...


7

Physical theories are not a collection of mathematical axioms, they are attempts at describing nature. Not only that. Physical theories are also supposed to make predictions. This is part of the Scientific Method. One does not expect to predict new phenomena - that can later be veryfied - using a non self-consistent theory. We cannot cheat. Following ...


6

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 ...


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

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 ...


6

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 ...


6

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 $N=1$, ...


6

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 ...


5

(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 ...



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