Mathematics of path integral: state of the art

Question

I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynman path integral", which, as far as I understand, means "integrating" a functional (action) on some infinite-dimentional space of configurations (fields) of a system.

Unfortunately, it seems that, except for some few instances like Gaussian-type integrals, the quotation marks cannot be eliminated in the term "integration", cause a mathematically sound integration theory on infinite-dimensional spaces — I was told — has not been invented yet.

I would like to know the state of the art of the attempts to make this "path integral" into a well-defined mathematical entity.

Difficulties of analytical nature are certainly present, but I read somewhere that perhaps the true nature of path integral would be hidden in some combinatorial or higher-categorical structures which are not yet understood...

Edit: I should be more precise about the kind of answer that I expected to this question. I was not asking about reference for books/articles in which the path integral is treated at length and in detail. I'd have just liked to have some "fresh", (relatively) concise and not too-specialistic account of the situation; something like: "Essentially the problems are due to this and this, and there have been approaches X, Y, Z that focus on A, B, C; some progress have been made in ... but problems remain in ...".

Answer 1

It's not accurate to say that no theory of integration on infinite-dimensional spaces exists. The Euclidean-signature Feynman measure has been constructed -- as a measure on a space of distributions -- in a number of non-trivial cases, mainly by the Constructive QFT school in the 70s.

The mathematical constructions reflect the physical ideas of effective quantum field theory: One obtains the measure on the space of field histories as the limit of a sequence/net of "regularized" integrals, which encode how the effective "long distance" degrees of freedom interact with each other after one averages out the short distance degrees of freedom in various ways. (You can imagine here that long/short distance refers to some wavelet basis, and that we get the sequence of regularized integrals by varying the way we divide the wavelet basis into short distance and long distance components.)

I don't think the main problem in the subject is that we need some new notion of integration. The Feynman measures we mathematicians can construct exhibit all the richness of the "higher categories" axioms, and moreover, the numerical computations in lattice gauge theory and in statistical physics indicates that the existing framework is at the least a very good approximation.

The problem, rather, is that we need a better way of constructing examples. At the moment, you have to guess which family of regularized integrals you ought to study when you try to construct any particular example. (In Glimm & Jaffe's book, for example, they simply replace the interaction Lagrangian with the corresponding "normally ordered" Lagrangian. In lattice gauge theory, they use short-distance continuum perturbation theory to figure out what the lattice action should be.)

Then -- and this is the really hard and physically interesting part -- you have to have enough analytic control on the family to say which observables (functions on the space of distributions) are integrable with respect to the limiting continuum measure. This is where you earn the million dollars, so to speak.

Answer 2

First, there are several rigorous definitions of integration in infinite dimensional spaces, like the Bochner integral in Banach spaces (see Wikipedia), or see the book by Parthasarathy: "Probability measures on metric spaces" (this includes the Gaussian probability measures used by constructive QFT already mentioned).

These cannot be used to make the Feynman path integral into a rigorous defined mathematical entity with finite values, i.e. the problem is to get an integral that spits out finite numbers in physically interesting models.

For starters, there cannot be a translationally invariant measure (on the Borel sigma algebra) other than the one that assigns infinite volume to every open set in an infinite dimensional metric space (hint: a ball of radius r contains infinitly many pairwise disjunct copies of the ball of radius r/2). So the path integral, as it is written by physicists, certainly has no interpretation via a translationally invariant measure, contrary to what the notation usually employed may suggest.

While there currently is no mathematically rigorous definition of a Feynman path integral applicable to an interesting subset of physical models, here are some books that give some hints at the current state of the affair:

Huang and Yan: "Introduction to Infinite Dimensional Stochastic Analysis" (this contains a description of the Feynman path integral from the viewpoint of "white noise analysis"),

Sergio Albeverio, Raphael Hoegh-Krohn; Sonia Mazzucchi:"Mathematical theory of Feynman path integrals. An introduction",

Pierre Cartier, Cecile DeWitt-Morette: "Functional integration: action and symmetries".

BTW: This is in a certain sense a "one million dollar" question because one of the millenium problems of the Clay Mathematics Institute is a rigorous construction of Yang-Mills theories.

Answer 3

Recently I have been reading Kevin Costello's book (draft) Renormalization of Quantum Field Theories, which claims to work out some foundations of perturbative quantum field theory following the "Wilsonian philosophy". I don't understand this stuff well enough to really say much, and hopefully someone else can say more; but I think the basic idea is to, instead of doing integrals over infinite dimensional spaces such as $C^\infty(M)$, do integrals over finite dimensional "approximations" of these infinite dimensional spaces, for example, the space of functions $C^\infty(M)_{\leq \Lambda}$ of energy $\leq \Lambda$, where $\Lambda$ is some constant. I think energy $\leq \Lambda$ means you take the Laplacian of $M$ (corresponding to a Riemannian metric, which is probably fixed from the beginning), and you take eigenfunctions of the Laplacian corresponding to eigenvalues $\leq \Lambda$. (Someone should correct me if I'm wrong.) Then, a low energy theory should be related in an appropriate way to (indeed it should be determined by) the higher energy theories.

I might be wrong, but my impression is that it is "impossible" to make a rigorous definition of the path integral: There are various problems with defining the appropriate measures on infinite dimensional spaces. Therefore, if we wish to make path integrals "rigorous", we must find some other means to define it, or find some alternative "roundabout" solution, such as the Wilsonian idea. But again I am not an expert on this; these are just my (very) naive impressions.

There is also the Atiyah-Segal axiomatization of (topological) quantum field theory. Perhaps this can also be viewed as a "roundabout" solution to "defining" the path integral: It avoids having to define path integrals, and instead axiomatizes the properties that should hold if the path integral could be rigorously defined. Check out Atiyah's original paper and Segal's notes. One way that higher categorical stuff arises is via the "locality" property/assumption of (T)QFTs. For more on this, see for example Jacob Lurie's paper on TFTs (available on his webpage), and the references therein.

Answer 4

There is a relatively large literature on path integrals. The best book that I know of is Johnson and Lapidus, The Feynman Integral and Feynman's Operational Calculus, 2000. See also the books and papers by Cecile DeWitt-Morette.

Answer 5

Here's a relatively recent paper by Jonathan Weitsman: http://arxiv.org/abs/math/0509104

He has more recent papers, but I'm not entirely sure that they're following the program he meant to initiate with this paper.