# Tag Info

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 ...

10

There are a number of high level mathematicians who are working on giving a more mathematically precise description of perturbative QFT and the renormalization procedure. For example there is a recent paper by Borcherds http://arxiv.org/pdf/1008.0129, work of Connes and Kreimer on Hopf algebras and the work of Bloch and Kreimer on mixed Hodge structures and ...

10

Among normal books, Becker-Becker-Schwarz probably matches your summary most closely. However, you may want to look at a list of string theory books: http://motls.blogspot.com/2006/11/string-theory-textbooks.html Don't miss the "resource letter" linked at the bottom which is good for more specialized issues such as string field theory. An OK review of ...

9

Regarding your specific question: It's (2). Pertubative QFT arises from non-perturbative QFT by Taylor series approximation in the coupling coefficients. Really, non-perturbative QFT should just be called "QFT". But people often try to define a QFT by writing down a perturbative approximation, so we are stuck with this weird terminology.

7

The question here is how to organize a response. Names can be given, but I hope I can give some conceptual order. Hopefully other answers will find other ways to organize their response. The Wightman axioms are classic. I take the approach here of organizing how other approaches sit with the Wightman axioms, even though they may not be axiomatic. A useful ...

7

you'll find a lot of information on the nLab, the open online Wiki of a bunch of people working on n-categories. You should really click around and see what is there, here is the page about the "functorial" POV on QFT, the formalization of the Schrödinger picture of QFT, including TQFTs: Functorial quantum field theory there is also a page about the ...

6

I) This is discussed around eq. (23.7.1) on p. 462 in Ref. 1. The task is to perform the path integral $$\tag{1} \int_{BC} [d\phi]e^{\frac{i}{\hbar}S[\phi]} ~=~\sum_{\nu}\int\! du \int_{BC_0} [d\phi_q]e^{\frac{i}{\hbar}S[\phi_{cl}+\phi_{\nu,u}+\phi_q]}$$ over fields $\phi$ with some (possible inhomogeneous) boundary conditions $BC$. This is done by ...

6

Due to asymptotic freedom, the perturbation series expansion of QCD breaks down at low energy. The expansion parameter, the coupling constant, becomes too large and therefore we cannot rely on results anymore. The value of the coupling constant approaches the order of $1$ at an energy of several hundred MeV, a scale referred to as $\Lambda_{QCD}$ ("Lambda-...

5

I think that there may be some confusion about the term "non-perturbative solution" in the question. Perturbative methods are methods based on Taylor expansions in a small parameter, typically a coupling constant. These methods fail to capture some phenomena - e.g. instantons - which are non-perturbative phenomena. Non-perturbative methods are any methods ...

5

Here is my answer from a condensed matter physics point of view: Quantum field theory is a theory that describes the critical point and the neighbor of the critical point of a lattice model. (Lattice models do have a rigorous definition). So to rigorously define/classify quantum field theories is to classify all the possible critical points of lattice ...

5

QED is non-perturbative because it is not defined by the sum over Feynman diagrams. Standard QFT does not rely on the perturbative expansion to define scattering amplitudes, it only uses it to compute them. String theory through CFT, on the other hand, defines the string scattering amplitude through the sum over worldsheets. This is not a perturbative ...

4

Actually, Wilson (who received the Nobel for his work on renormalisation -- though that's not to say he was the first one to think of it) worked on numerical renormalisation. The starting point is to formulate (in an abstract and somewhat formal (i.e. not very computable) way) the Exact Renormalisation Group Equation. There are various ways to formulate ...

4

I think if you want a rigorous approach to QFT that incorporates the ideas of renormalization and effective field theory, you might want to take a long look at the work of the 'constructive QFT' school. Their work on rigorous Euclidean functional integration is very much in the spirit of Wilson. Two examples, chosen randomly among many: 1) Euclidean ...

4

I don't get what you're asking? What it means that non-perturbative QCD is the exploration of phases of quark matter, including the quark-gluon plasma? Because, at high energies, you're at the so called asymptotic freedom regime. At this regime, your quarks interact weekly and the perturbative calculations are possible. And quark-gluon plasma is ...

4

The standard nonperturbative way (that provided rigorous constructions in 1+1D and 1+2D QFTs) is constructing the Euclidean (imaginary time) field theory as a limit of lattice theories, and then using analytic continuation to real time via the Osterwalder--Schrader theorem. In 1+3D, there is so far no rigorous construction of an interacting QFT, but ...

4

1) As you know, $$\tag 1 \theta\epsilon^{\mu \nu \alpha \beta}F_{\mu \nu}F_{\alpha \beta} = \theta\partial_{\mu}K^{\mu},$$ where $K_{\mu}$ is the so-called Chern-Simons class. The Feynman diagrams method tells us that the term $(1)$ defines the diagram which corresponds to the two-photon (or two-three-four non-abelian bosons) vertex $V_{A}$ (where the ...

3

Instanton calculations involve integrals over collective coordinates. One of these is the instanton size $\rho$. Reliable instanton calculations are those for which the integral over instanton sizes is dominated by small sizes so that (for asymptotically free theories) the coupling constant is small and higher order corrections in the semi-classical ...

3

Leaving out numerical factors, we have that $$\mathrm{d}j_A = \mathrm{Tr}(F \wedge F)$$ This already shows that we are dealing with a topological quantity, since the RHS is the second Chern character of the gauge field (or rather, the principal bundle associated to it). Now, there is also the (3D) Chern-Simons form $$\omega = \mathrm{Tr}(F \wedge A - \... 3 Loop Quantum Gravity is an example of a non-perturbative approach to canonical quantum gravity. In fact it's mentioned in the Wiki article you linked to. Inevitably views about LQG vary, but no-one has proven it wrong in the sense that it is experimentally disproven or mathematically inconsistent. 3 Ideally, when we have a differential equation to solve, we will try to solve it analytically. Find explicit functions that encode the variables. The solutions for a harmonic oscillator, for example. The solutions of a potential in Schroedinger's equation. Those are examples of non perturbative solutions. They satisfy the differential ( or integral) ... 3 Perturbative theory is understood as an application of the Taylor expansion around a linearity g=0 (unperturbed theory) in an underlying theory:$$ f(g)=\sum_{n}A_n g^n $$The unperturbed theory could be, for example, an Hydrogen Atom for the underlying QED. Then, the QED perturbative theory is a prescription of how inserting g\neq0 corrections of the ... 3 There are many phenomenon in quantum field theory that falls outside the understanding of perturbative analysis. To understand the nonperturbative physics is to understand the full dynamics of the theory whereas perturbation theory is not reliable. The basic examples are 1) dynamical symmetry breaking (supersymmetry or chiral symmetry) which is usually ... 3 In general, for computing the scattering cross-section it is sufficient to use the pole approximation of the corresponding Green function in Heisenberg representation (details can be found in Weinberg's QFT Vol. 1, paragraph 10). This is nonperturbative approach, since poles are invisible in perturbation theory: they define bound states masses. To find them, ... 3 It is easy to make solitons in water. See this video and the video coming after it with the bubbles. They are the nonlinear solutions of the differential equations governing flow.Also I recomend the one with bubbles . Ball lightning is proposed to be a soliton solution. 2 There is indeed nothing wrong with it; people (e.g. 't Hooft) have done perturbative calculations with canoncial quantum gravity (or simply with a L=R^2 Lagrangian). In this sense it's seen as an effective field theory at lower energy scales. Google on 'Wilsonian effective field theory' for more information (I couldn't find a Wiki page). However, we would ... 2 1) On one hand, perturbative QCD means an expansion in the Yang-Mills coupling constant g_{YM} for a fixed number of colors N_c. 2) On the other hand, the 1/N_c expansion means an expansion in 1/N_c for fixed 't Hooft coupling constant$$\lambda~:=~g^2_{YM}N_c. (Here it is implicitly assumed that all dependence of $g_{YM}$ have been replaced, ...

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