# Tag Info

17

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

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

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 ...

8

A nonperturbative theory means a theory where all the results can be calculated in principle to arbitrary accuracy on a computer. This is really just the same as a well-defined theory, a theory which is mathematically ok, and which makes sense, which isn't incomplete in some way at some high energy or when you do measurements at some high accuracy. An ...

8

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 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 ...

6

The problem is that the perturbation series, even in the best behaved theories, is not a sufficient criteria for reconstructing the theory. In the case of QCD, you can reconstruct the non-perturbative theory by defining a path integral on a lattice, and taking the limit of a fine lattice with the coupling logarithmically going to zero as the lattice spacing ...

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 $\Lambda_{QCD}$ ...

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

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 ...

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

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) 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 ...

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

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 ...

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

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.

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

I don't know the answer to question 1, but for question 2, the close relative of the matrix models are the 2-brane approximations of deWit, Hoppe, and Nicolai, who essentially give the same 1d matrix models as an approximate description of 11-dimensional supergravity 2branes many years before BFSS. The BFSS work discovers the a physical interpretation of ...

2

Non-perturbative RG methods exist and are useful. They have been used to understand several phenomena in (nonequilibrium) statistical mechanics. NPRG of the KPZ equation Reaction-Diffusion processes Critical dynamics Nonequilibrium steady-states Voter models Branching and annihilating random walks 3D Ising model There are also nice reviews and books on ...

2

First of all, you can simplify this by rewriting it as an indefinite integral of $x = Z/F$ like so: $$\int_\epsilon^\infty \frac{ds}{s^3} \left ( \frac{s/2}{\sinh s/2} \right )^2 e^{-s x} = - \frac{1}{4} \int_\epsilon^\infty ds \int_0^x dx \frac{1}{\sinh^2 s/2} e^{-s x}$$ From here, you have roughly ...

2

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 - ...

2

If you assume separability of the wave function, i.e., $\psi(\mathbf x)=u(x)v(y)w(z)$, you can solve the individual components separately: \begin{align} -\frac{\hbar^2}{2\mu}\frac{d^2u(x)}{dx^2}+V_1(x)u(x)&=E_1u(x)\\ -\frac{\hbar^2}{2\mu}\frac{d^2v(y)}{dy^2}+V_2(y)v(y)&=E_2v(y)\tag{1}\\ ...

2

In the case of QCD there is asymptotic freedom, meaning that though the theory is strongly coupled at low energies (such that we still cannot analytically calculate how the atomic nucleus stays together) the coupling becomes less and less as we go to higher and higher energies. This means that the ultimate picture of the behavior of quarks is a weakly ...

1

Instantons are characterized by the winding number and a set of collective parameters (e.g. location of the centers of the instantons, their sizes and the inequivalent orientations in the global group space / space-time). Quantum fluctuations of a unit winding number instanton can either leave the collective parameters unchanged (non-zero modes), or change ...

1

He doesn't really "state" the result without deriving it. He derives it. See e.g. this sentence on page 2: It is further shown that since the D-brane tension arises from the disk, it scales in string units as $g^{−1}$, $g$ being the closed string coupling The tension of D-branes goes like $1/g_s$ because the tension may be calculated from the disk ...

1

No. Feynman diagrams are made by summing over the perturbative contributions of quantum amplitudes. They cannot hold non-perturbative information.

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