# Are there rigorous constructions of the path integral for lattice QFT on an infinite lattice?

Lattice QFT on a finite lattice* is a completely well defined mathematical object. This is because the path integral is an ordinary finite-dimensional integral. However, if the lattice is infinite, the mathematical definition of the path integral is no longer obvious. Intuitively, I still expect the construction of this path integral to be much easier than the construction of path integrals for continuum QFT. This is because on the lattice we have an explicit UV cutoff and in particular the bare parameters of the action are finite so the action is an honest function on the field space.

Naturally, I'm interested in interacting theories (e.g. lattice Yang-Mills) since free theories lead to Gaussian integrals which are relatively easy to define even in the infinite-dimensional case.

*Usually periodic boundary conditions are used so the finite lattice is a product of cyclic finite groups

There are rigorous constructions of QFTs in infinite volume. Glimm & Jaffe's book does this for interacting 2d scalars (with the assumption that the interactions are not too strong). I'm sure you can find other examples in the literature (or perhaps someone else will point you to them).

However, restricting yourself to a lattice doesn't really buy you much. For one thing, if the lattice action you're using is approximately local and a good approximation to the true effective action, you're probably not too far from the continuum limit anyways.

One of constructive QFT's little surprises -- at least if you were brought up on Peskin & Schroder -- is that removing IR cutoffs is a considerably harder problem than removing UV cutoffs. For one thing, you usually can't just take a limit of the finite-volume measures. Instead, you have to find some collection of observables whose expectation values remain well-defined in the IR limit, and then use something like Minlos' Theorem to infer the existence of a measure. Finding the right observables isn't easy. You want to show that the expection values obey some form of cluster decomposition, so that you can safely ignore things that are happening far away. This is somewhat difficult just in massive 2d scalar field theory, where the correlation functions of the basic observables decay exponentially. (It takes Glimm & Jaffe only a few pages to show the existence of a finite volume continuum limit, but it takes them a few chapters to show the infinite volume limit exists.) It's even harder if your correlators only decay like polynomials. And in theories like Yang-Mills theory or massless 2d scalars, where the correlation functions of basic observables can actually grow with distance, it can become monstrously hard. You have to find just the right observables --e.g., exponentiated fields in the 2d scalar case -- and show that the divergences cancel out. (The Millenium Prize for Yang-Mills theory really amounts to solving the IR problem. The UV problem in finite volume is believed to be basically tractable.)

• The IR being the hard part is quite surprising, since this problem is not unique to QFT. It arises already in "plain" statistical physics in the thermodynamic limit e.g. an infinite crystal lattice. I didn't think that disciplines such as condensed matter also suffer from lack of mathematical rigour.
– Squark
Commented Oct 19, 2011 at 16:23
• Also, It'd be nice if you point out which lattice QFTs are constructed in Glimm & Jaffe and even nicer if you comment on the methodology they use to construct it. Thx
– Squark
Commented Oct 20, 2011 at 14:14

Constructing the infinite volume lattice theory is not that difficult usually. One typically uses the so-called Griffiths' inequalities. See this article by Sokal (p. 327), as well as this article by Guerra, Rosen and Simon (Section V). In the case of Yang-Mills on a lattice, the infinite volume limit is more difficult. However, it has been done rigorously at large coupling, see this article by Osterwalder and Seiler. I should add that the results I mentioned are for any number of dimensions $d\ge 2$. This is because one is just talking about the infinite volume limit on a fixed lattice. One needs to worry about dimension when one wants to also take the lattice mesh to zero.

One of the most complete books on the lattice infinite volume limit from a mathematically rigorous standpoint is "Gibbs Measures and Phase Transitions" 2nd Edition by H.-O. Georgii. See in particular the notes on sections 4.3 and 4.4 on page 458 which discusses various techniques one can use for unbounded spin systems.

• Can you pls explain which QFTs each article covers? Sokal does phi4 (in any number of dimensions?), GRS do P(phi) in 2D (only?) and OS do Yang-Mills in how many dimensions?
– Squark
Commented Oct 20, 2011 at 20:42
• The book by Georgii is nice but it handles only classical field theories on a lattice, while the question was about quantum fields. Commented Jan 24, 2018 at 19:25
• @ArnoldNeumaier: the question is about the path integral. If you allow me to switch to Euclidean formulation with $-S$ instead of $iS$ then this path integral is the integral over classical field configurations on the lattice, and that's exactly what Georgii covers. Commented Jan 24, 2018 at 20:12
• Does this suffice to produce definite Wightman functions? I haven't seen a discrete version of the Osterwalder-Schrader reconstruction theorem. Commented Jan 24, 2018 at 21:54
• Even in finite dimensions, limits of oscillating intehrals are much harder to establish than of decaying ones. So the equivalence you seem to claim seems to be not obvious Commented Jan 24, 2018 at 22:02