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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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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.

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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.

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) ...

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 ...

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 ...

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, ...

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 ...