Question

I'm wondering about current efforts to provide mathematical foundations and more solid definition for quantum field theories. I am aware of such efforts in the context of the simpler topological or conformal field theories, and of older approaches such as algebraic QFT, and the classic works of Wightman, Streater, etc. etc . I am more interested in more current approaches, in particular such approaches that incorporate the modern understanding of the subject, based on the renormalization group. I know such approaches exists and have had occasions to hear interesting things about them, I'd be interested in a brief overview of what's out there, and perhaps some references.

Edit: Thanks for all the references and the answers, lots of food for thought! As followup: it seems to me that much of that is concerned with formalizing perturbative QFT, which inherits its structure from the free theory, and looking at various interesting patterns and structures which appear in perturbation theory. All of which is interesting, but in addition I am wondering about attempts to define QFT non-perturbatively, by formalizing the way physicists think about QFT (in which the RNG is the basic object, rather than a technical tool). I appreciate this is a vague question, thanks everyone for the help.

Answer 1

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 renormalization http://www.math.uchicago.edu/~bloch/monodromy.pdf just to name a few. To be honest, I am not mathematically sophisticated enough to judge what has been accomplished in these papers, but I think there are some problems in QFT which will probably involve some rather high-powered mathematics of the type being developed in these papers. For example, the current attempt to reformulate N=4 SYM in terms of Grassmannians apparently has some connection to rather deep mathematical objects called Motives. Results on the degree of transcendentality which show up in perturbative N=4 SYM amplitudes also seem beyond what physicists really understand and I believe the presence of transcendental objects (like ?(3)) in QFT amplitudes provides some of the motivation for the work of Bloch and Kreimer. I'm not an expert on this stuff, so perhaps someone else will chime in with a more complete explanation and additional references.

Edit: One more reference which is closer to the spirit of the original question is a book in progress by Costello on perturbative quantum field theory treated from the Wilsonian, effective field theory point of view. Notes are available online at http://www.math.northwestern.edu/~costello/renormalization