Who is Richard Borcherds and What is His Contribution to QFT?

[mitchell porter]

Richard Borcherds is a mathematician who won a Fields Medal for work involving conformal field theory. In recent years, he has been working on a rigorous formulation of perturbative QFT. http://arxiv.org/abs/1008.0129" , though somewhat technical, might interest the "foundations of QFT" people in this forum.

 

[A. Neumaier]

Richard Borcherds is a mathematician who won a Fields Medal for work involving conformal field theory. In recent years, he has been working on a rigorous formulation of perturbative QFT.

Perturbative QFT was rigorously under control already in the last century. The unsolved foundational questions are about the nonperturbative aspects, in particular the infrared behavior.

http://arxiv.org/abs/1008.0129" , though somewhat technical, might interest the "foundations of QFT" people in this forum.

''somewhat technical'' is very mild. The paper is _very_ abstract. Could you please summarize its content in a way that is intelligible on the level where we discuss here the foundations of QFT?

 

[mitchell porter]

Could you please summarize its content in a way that is intelligible on the level where we discuss here the foundations of QFT?

He covers free field theories, interacting field theories, regularization and renormalization, and anomalies of gauge theories in a few dozen pages, in the lemma/theorem/proof style of a mathematics paper. It might best be read in conjunction with Borcherds's http://arxiv.org/abs/math-ph/0204014" , which do more to physically motivate the choice of formalism.

I don't know the "constructive QFT" literature and can't say how original or unusual Borcherds's approach is, but it strikes me as exceptionally compact, and as an opportunity to learn a pure-maths perspective on some of the aspects of QFT practice which are otherwise "algorithmic" - physicists know what to do to get a number, but they may not be able to put a name to what they are doing. The paper won't teach you this maths, but it may tell you what you need to learn about (Hopf algebras, cohomology, etc).

 

[A. Neumaier]

He covers free field theories, interacting field theories, regularization and renormalization, and anomalies of gauge theories in a few dozen pages, in the lemma/theorem/proof style of a mathematics paper. It might best be read in conjunction with Borcherds's http://arxiv.org/abs/math-ph/0204014" , which do more to physically motivate the choice of formalism.

I don't know the "constructive QFT" literature and can't say how original or unusual Borcherds's approach is, but it strikes me as exceptionally compact, and as an opportunity to learn a pure-maths perspective on some of the aspects of QFT practice which are otherwise "algorithmic" - physicists know what to do to get a number, but they may not be able to put a name to what they are doing. The paper won't teach you this maths, but it may tell you what you need to learn about (Hopf algebras, cohomology, etc).

Thanks. Both his paper and his lecture notes are exclusively about perturbative QFT.

The Hopf algebra approach organizes perturbative renormalization calculations in a very efficient way, though it is also very abstract, and hard to translate into actual recipes fro proceeding with actual calculations. I can't do it, but some have done it successfully and computed some simple (nonrealistic) theories to very high order.

Hopf algebras also organize the computations needed to derive high order Runge-Kutta methods for solving ODE's, and there seems to be a connection.

On the other hand, constructive QFT is about deriving nonperturbative, non-approximate
information about a field theory. I haven't seen Hopf algebras contribute there, except that exactly solvable 2D field theories can be explained in these terms - in both cases, one encounters Yang-Baxter equations at the root of important things. But I haven't gotten around to understanding this in depth; a good exposition of the relations between Hopf algebra techniques and exactly solvable QFTs - perhaps you can point to something?

 

[mitchell porter]

Sorry, topics like: Yang-Baxter equation, R-matrix, Bethe ansatz, various solvable models... are too far from what I know about. I am studying AdS/CFT, and apparently they show up there, so maybe they will finally become meaningful to me in that context.

However, the http://maths.anu.edu.au/events/Baxter2000/abstracts.html" (never made into a paper, and the speaker is now dead) proposed to use sine-Gordon model and O(N) sigma model as examples of 2D field theories where S-matrix could be derived using Yang-Baxter. So if I were investigating this subject, I would begin with the literature on those two models.

 

[A. Neumaier]

Sorry, topics like: Yang-Baxter equation, R-matrix, Bethe ansatz, various solvable models... are too far from what I know about. I am studying AdS/CFT, and apparently they show up there, so maybe they will finally become meaningful to me in that context.

Yang-Baxter equations govern the 2-particle S-matrix in exactly solvable CFT models.

use sine-Gordon model and O(N) sigma model as examples of 2D field theories where S-matrix could be derived using Yang-Baxter.

Yes, there is a huge literature on this - mostly either messy or cryptic or both. I understand the matter superficially (only).

 

[martinbn]

Perturbative QFT was rigorously under control already in the last century.

What exactly do you mean? My impression is that it still is not.

 

[A. Neumaier]

What exactly do you mean? My impression is that it still is not.

What exactly do you mean?

The rules for perturbative QFT (including renormalization) are laid down in every QFT textbook, and their rigorous treatment gives precisely the same results, though it is derived in a less casual manner. For QED, see the QED book by Steinmann; for gauge theories, see the ghost story book by Scharf.

 

[martinbn]

What exactly do you mean?

The rules for perturbative QFT (including renormalization) are laid down in every QFT textbook, and their rigorous treatment gives precisely the same results, though it is derived in a less casual manner. For QED, see the QED book by Steinmann; for gauge theories, see the ghost story book by Scharf.

As I said it was my impression, and most probably I am wrong, that's why I am asking. May be I should have asked first what you consider rigorous. My impression was that none of the things on the list above was rigorous. Renormalization, divergent series, path integrals...

 

[A. Neumaier]

As I said it was my impression, and most probably I am wrong, that's why I am asking. May be I should have asked first what you consider rigorous. My impression was that none of the things on the list above was rigorous. Renormalization, divergent series, path integrals...

The usual presentations are far from rigorous (and don't try to be so), but there are mathematically impeccable versions in the books mentioned. The series is considered mathematically as a formal power series; this leaves the question of the convergence of the perturbation series open (it most likely isn't).

The unsolved challenges are in going rigorously beyond perturbation theory, i.e., finding a formulation where one can evaluate the full scattering expressions at a finite value of the coupling constant.

You can read more about that in Chapter B5 ''Divergences and renormalization'' of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#B5

 

[martinbn]

Thanks, I will take a look at it, also Steinamann, and Scharf.