[Physics] Other processes than formal power series expansions in quantum field theory calculations

Question

I am not sure if this question is too naive for this site, but here it goes. In QFT calculations, it seems that everything is rooted in formal power series expansions, i.e. , what dynamical systems people would call Lindstedt series. However, from what I heard, this series (for QFT case) is known to have zero radius of convergence, and it causes tons of difficulties in theory. My question is, are there approaches that start with an iterative process that has a better chance of converging (e.g., a fixed point iteration), and build computational methods for QFT from there?

In other words, when there are so many approaches to approximate the exact solution of, say nonlinear wave (and Klein-Gordon, Yang-Mills-Higgs-Dirac etc) equations on the classical level, why do we choose, when we quantize, only a couple of approaches, such as power series, and lattice regularization (the latter essentially a finite difference method)? Note that it is milder than making QFT completely rigorous, it is just about computing things a bit differently.

Answer 1

Lack of convergence does not mean there is nothing mathematically rigorous one can extract from perturbation theory. One can use Borel summation. In fact, Borel summability of perturbation theory has been proved for some QFTs:

1. by Eckmann-Magnen-Seneor for $P(\phi)$ theories in 2d, see this article.
2. by Magnen-Seneor for $\phi^4$ in 3d, see this article.
3. by Feldman-Magnen-Rivasseau-Seneor for Gross-Neveu in 2d, see this article.

In fact these articles obtain such results by using an alternative to ordinary perturbation theory called a multiscale (or phase cell or phase space) cluster expansion. The latter is based on combinatorial structures which mimic Feynman diagrams. However, these expansions converge at small coupling.

Edit as per Timur's comment: Glimm and Jaffe's book is what you want to read in order to understand why one needs cluster expansions. It is excellent at giving the big picture: how axiomatic, Euclidean, constructive QFTs fit together, as well as with scattering theory. But for learning how to do a cluster expansion the book is outdated. The cluster expansion explained in GJ is the early one invented by Glimm, Jaffe and Spencer in their Annals of Math article. It was the first in the QFT context and as such quite a mathematical feat. However there has been many improvements and simplifications since then (around 1973). If you want to learn about cluster expansions in 2011, here is a more efficient path:

• Learn about the Mayer expansion for the polymer gas: a quick intro is in the "Additional material" at the bottom of my course webpage.
• Learn about the single scale cluster expansion, i.e., controlling the infinite volume limit when both UV and IR cut-offs are present: look at the article "Clustering bounds on n-point correlations for unbounded spin systems" on the same webpage. For a more cleaned up version see the published version, but this is not freely accessible.
• Finally the real McCoy: the multiscale cluster expansion where one tries to do all of the above and remove the cut-offs. This is somewhat like an infinite volume limit in phase space. Here there is no easy reference. All accounts of the subject are extremely difficult to read. I plan to write a pedagogical article on this in the next few months. In the meantime you could try the following: the book by Rivasseau "From Perturbative to Constructive Renormalization", the book "Wavelets and Renormalization" by Battle, and also this recent article by Unterberger (in French).