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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.

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Also note that this asymptotic series proves to be extremely precise, e.g. the theoretical result for $g-2$ calculated in this series agrees with the experiment to 10 significant digits. See en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment –  Yuji Oct 6 '11 at 9:20
@Yuji: Yes, as you said, it is only an asymptotic series, which means that initially it may get close to the "exact value", but after some point it will diverge. A related question is if there is anything known in terms of how many terms we should calculate in order to get the most accurate result. –  timur Oct 6 '11 at 17:10
Quantum Field Theory perturbative expansions are not a Lindstedt series, they are a Taylor series. The Lindstedt series should not be mentioned. Also, when you quantize, you do not quantize a Taylor series, you take the Taylor series in the path integral. –  Ron Maimon Oct 12 '11 at 3:53

5 Answers 5

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).
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Thanks a lot! To understand and learn to use cluster expansions, would Glimm-Jaffe's Quantum physics: a functional integral point of view be a reasonable starting point? I am a mathematician with background in PDE and wavelets, and with an undergraduate degree in physics. –  timur Oct 14 '11 at 0:04
Alternately, if you want to start off somewhere easier or in a more physics-y style, David Goodstein's book 'States of Matter' has a pretty nice introduction to the cluster expansion, in its original context. –  user1504 Oct 14 '11 at 22:13

I think you raise a very important question, but I think you make it sound more trivial than it is. The point is: a lot of physicists would like to have alternative expansions, but it is very difficult to come up with one. If you've got some suggestions, don't hesitate to put it forward.

The standard expansion starts from the time evolution operator $\mathcal{U}(t,t_0)$ and a Hamiltonian $\hat{H}$, which together form the Schroedinger equation:

$$ -i\partial_t \mathcal{U} = \hat{H}\mathcal{U}$$

Integrating this gives, $$ \mathcal{U}(t,t_0) = 1 - i\int_{t_0}^t dt_1 \hat{H}(t_1)\mathcal{U}(t_1,t_0)$$ and by iterating, i.e. substituing this expression for $\mathcal{U}$ on the right hand side, you can come up with a formal power series for $\mathcal{U}$ called Dyson's series. You can modify it in some ways, like splitting the Hamiltonian into a solvable and perturbative part, and correspondingly for the time evolution operators. In the end you'll end up with expressing correlation functions that you want in terms of a series of correlation functions of some model that you know. And it's natural for this series to be an expansion in terms of the coupling constant of the perturbative part.

So can you get around this expansion? Well, sometimes there are some non-perturbative approaches available. You have, for instance, the realm of exactly solvable models. These rely on the presence of severe symmetry constraints. Examples are certain 2D conformal field theories, in which correlation functions satisfy differential equations. These equations arise due to restriction on the operator algebra (the presence of so-called null-states) and Ward identities associated with the symmetry algebra, which includes the conformal structure. Powerful stuff.

Other examples are the Bethe ansatz and the algebraic Bethe ansatz. As far as I understand these models are based on constructing a full set of eigenstates in some Hilbert space + extension, without an explicit reference to the Hamiltonian (meaning the Hamiltonian is subject to some restrictions, but need not be explicitly known). This is a very powerful technique and valid for the entire range of the coupling constant. But requiring integrability can be quite a constraint.

AdS/CFT was also mentioned, which is a marvellous weak/strong coupling duality. This makes use of the idea that correlation functions are the same for two seemingly different theories, which differ in dimensionality and the presence of gravity. Lattice regularization works also quite well, as far as I know.

An alternative expansion to Dyson's series which comes to mind is the Magnus expansion (see also here). The biggest advantage to this expansion is that it stays unitary once you cut the series off somewhere. But is it a strong alternative..?

My view on the matter is that a new expansion or approach could very well be the next best thing since sliced bread.

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To add to Olaf's answer, the specific series is called a Neumann series which is the solution to Fredholm integral equations. –  rcollyer Oct 25 '11 at 18:15

Are you essentially asking about non-perturbative approaches to QFT? Lattice QCD (based on Monte-Carlo sampling) and various strong-couplig/weak-coupling dualities (like AdS/CFT) come to mind as most prominent examples.

This is more of a hint than a real answer, of course.

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Thanks for the answer. The question was more in the spirit of 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)? –  timur Oct 6 '11 at 17:14
I can only wave my hands along the line that the structure of even the most basic QFT is much richer (e.g., infinitely dimensional Hilbert space for "every point in space") that a finite-dimensional classical theory. There could be a deeper and simpler mathematical reason, which I'd be also interested to learn. –  Slaviks Oct 6 '11 at 18:56

I think you will save yourself considerable time by looking at a reference like: N. Nagaosa QFT in Condensed Matter Physics, pg 78, section 3.4. From a computational perspective "Lattice Gauge Theory and the Confinement Problem" will explain to you why physicists, when they are interested in introducing a cutoff, in energy or length etc, will need to address how many of the infinite degrees of freedom from the original Hamiltonian remain intact locally on the lattice. In doing so you learn that modeling fermions can become problematic. It's a quick read and might help you polish your question with a minimal time investment.

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Thanks for the answer! I am afraid I cannot get much out of the book. Are you essentially saying that since lattice has so many problems, other possible approaches must have even more problems? –  timur Oct 8 '11 at 14:42
No, I just want you to be aware that as you are learning more about these approaches, each of them has their limitations. I pointed out this reference because it'll help you gauge how much familiarity you might have already with the subject matter. It will also be good for you to try to figure out what exactly a physicist means by an asymptotic series expansion, and why under rescaling of the coupling constant, the asymptotic approach fails. Without one on one tutoring, it will take too long to explain these things. –  MA Perez Oct 11 '11 at 0:28

In some sense, i understand this question of yours as regarding to more "mathematically precise" approaches to QFT: in the end of the day, your question implies "non-perturbative definitions of QFT" in a form or another — afterall, if you can use some other tool, why not turn the problem around and define your theory based on how you can use such tool?

Along those lines, there's quite a lot to be said, given that there are a few different ways to define a QFT (with different levels of "mathematical rigor"):

  • Axiomatic QFT, Constructive QFT, Algebraic QFT, and Local Quantum Physics (à la Haag);
  • Functional Integration (à la Feynman Path Integrals; White Noise Calculus; etc) and approximating expansions (which is where your question seems to be more naturally formulated);
  • Vertex Algebras (VOAs, Borcherds Algebras) and CFTs;
  • More probabilistic approaches, eg, Schramm-Loewner equation;
  • Chiral and factorization algebras;
  • TFTs and higher category theory;
  • etc.

So, the use of series expansions is but one of these efforts — and, along those lines, there are other series that are relevant, eg, Large-N expansions.

On the other hand, having said the above, it is true that there are other methods that could have something to add to the usual way that things are done — giving voice to your concern. For example, there are ways to discretize space that are "compatible" with differential geometric objects (such as $n$-forms and so on; largely under the name of "Discrete Differential Geometry" or "Geometric Discretization") which could be used in lattice formulations of QFT and are currently not. Also, there are all sorts of different "finite difference schemes" of discretization, keeping different symmetries of the original problem (diff eq), that could be used to illuminate certain properties of the theory, but a discussion of this issue does not seem to feature in the lattice community.

Thus, in the end of the day, Olaf has a point: if you've got suggestions, by all means, put them forth! ;-)

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