The mathematical theory of Feynman integrals

Question

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly. Arguably, they are the most important such tool. Briefly, the question I'd like to ask is this: what are the existing proposals for defining Feynman integrals rigorously and why don't they work?

I should admit that my interest in all of this comes from attempts to understand Witten's definition of invariants of links in 3-manifolds (Comm Math Phys 121, 1989).

Here are some comments on some of the existing approaches and a few specific questions. (And please let me know if I've missed any important points!)

Here is a general framework. Suppose $E$ is a topological vector space of some kind and $L,f:E\to\mathbf{R}$ are functions. These are usually assumed continuous and moreover $L$ is (in all examples I know of) a polynomial of degree $\leq d$ ($d$ is fixed) when restricted on each finite-dimensional subspace. The Feynman integral is $$Z=\int_E e^{i L(x)}f(x) dx$$ where $dx$ is the (non-existent) translation-invariant measure on $E$ (the Feynman measure).

The way I understand it, this supposed to make sense since $L$ is assumed to increase sufficiently fast (at least, on the "most" of $E$) so that the integral becomes wildly oscillating and the contributions of most points cancel out.

The above is not the most general setup. In some of the most interesting applications there is a group $G$ (the gauge group) acting on $E$. The action preserves $L$ and $f$ and then what one really integrates along is the orbit space $E/G$, but still the "integration measure" on $E/G$ is supposed to come from the Feynman measure on $E$. Moreover, there is a way (called Faddeev-Popov gauge fixing) to write the resulting integral as an integral over $E'$ where $E'$ is a $\mathbf{Z}/2$-graded vector space (a super vector space), which is the direct sum $E\oplus E_{odd}$ with $E$ sitting in degree $0$ and $E_{odd}$ in degree 1.

There are two naive approaches to Feynman integrals -- finite-dimensional approximations and analytic continuation. Both are discussed in S. Albeverio, R. Hoegh-Krohn, S. Mazzucchi, Mathematical theory of Feynman path integrals, Springer LNM 523, 2 ed. As far as I understand, both work fine as long as $L$ is a non-degenerate quadratic function plus a bounded one. Moreover, in the 1st edition of the above-mentioned book Albeverio and Hoegh-Krohn give yet another definition, which works under the same hypotheses.

However, one can try to do e.g. finite dimensional approximations for any $L$ (and probably renormalize as divergencies occur). So here is the first naive question: has anyone tried to find out what are the functions $L$ for which this procedure can be carried out and leads to sensible results?

Since it is so difficult to define Feynman integrals directly, people have resorted to various tricks. For example, suppose $0$ is a non-denenerate critical point of $L$, so $L$ can be written $L(x)=Q(x)+U(x)$ where $Q$ is a non-degenerate quadratic function and $U$ is formed by the higher order terms. Introducing a parameter $h$ one can write (in finite dimensions and for a sufficiently nice $U$) the Taylor series for $\int_E e^{i (Q(x)+hU(x))}f(x) dx$ at $h=0$ in terms of Feynman diagrams. This is explained in e.g. in section 2 of the paper "Feynman integrals for pedestrians" by M. Polyak, arXiv:math/0406251. When one tries to mimic this in infinite dimensions, one runs into difficulties: the coefficients of Feynman diagrams are given by finite-dimensional but divergent integrals (e.g. if $E$ is the space of functions of some kind on some manifold, then the integrals are taken along the configuration spaces of that manifold).

A. Connes and D. Kreimer have proposed a systematic way to get rid of the divergencies (Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem, arXiv:hep-th/9912092). I think I understand how it works in the particular case of the Chern-Simons theory, but I was never able to understand the details of the above paper. In particular, I'd like to ask: what are precisely the conditions on $Q$ and $U$ in order for this procedure to be applicable? What happens when we take the resulting series in $h$? If it diverges, is it still possible to deduce the value of the Feynman integral from it when the latter can be computed by other means (as it is the case e.g. with the Chern-Simons theory on a 3-manifold)?

Finally, let me mention yet another approach to Feynman integrals: the "white noise analysis" (see e.g. Lectures on White noise functionals by T. Hida and S. Si). It was used by S. Albeverio and A. Sengupta (Comm Math Phys, 186, 1997) to define rigorously the Chern-Simons integrals when the ambient manifold is $\mathbf{R}^3$ (they have to use gauge-fixing though). Since I don't know much about this I'd like to ask a couple of naive questions: Is there a way this could help one handle the cases other approaches can't? Can this help eliminate the necessity to start with a non-degenerate quadratic function?

Answer 1

Most of this is standard theory of path integrals known to mathematical physicists so I will try to address all of your questions.

First let me say that the hypothesis you list for the action $S$ to make the path integral well defined, ie that $S=Q+V$ where $Q$ is quadratic and non-degenerate and $V$ is bounded are extremely restrictive. One should think of $V$ as defining the potential energy for interactions of the physical system and while it certainly true that one expects this to be bounded below, there are very few physical systems where this is also bounded above (this is also true for interesting mathematical applications...). Essentially requiring that the potential be bounded implies that the asymptotic behavior of $S$ in the configuration space is totally controlled by the quadratic piece. Since path integrals with quadratic actions are trivial to define and evaluate, it is not really that surprising or interesting that by bounding the potential one can make the integral well behaved.

Next you ask if anyone has studied the question of when an action $S$ gives rise to a well defined path integral: $ \int \mathcal{D}f \ e^{-S[f(x)]}$

The answer of course is yes. The people who come to mind first are Glimm and Jaffe who have made whole careers studying this issue. In all cases of interest $S$ is an integral $S=\int L$ where the integral is over your spacetime manifold $M$ (in the simplest case $\mathbb{R}^{n}$) and the problem is to constrain $L$. The problem remains unsolved but nevertheless there are some existence proofs. The basic example is a scalar field theory, ie we are trying to integrate over a space of maps $\phi: M \rightarrow \mathbb{R}$. We take an $L$ of the form:

$ L = -\phi\Delta \phi +P(\phi)$

Where in the above $\Delta$ is the Laplacian, and $P$ a polynomial. The main nontrivial result is then that if $M$ is three dimensional, and $P$ is bounded below with degree less than seven then the functional integral exists rigorously. Extending this analysis to the case where $M$ has dimension four is a major unsolved problem.

Moving on to your next point, you ask about another approach to path integrals called perturbation theory. The typical example here is when the action is of the form $S= Q+\lambda V$ where $Q$ is quadratic, $V$ is not, and $\lambda$ is a parameter. We attempt a series expansion in $\lambda$. The first thing to say here, and this is very important, is that in doing this expansion I am not attempting to define the functional integral by its series expansion, rather I am attempting to approximate it by a series. Let me give an example of the difference. Consider the following function $f(\lambda)$:

$f(\lambda)=\int_{-\infty}^{\infty}dx \ e^{-x^{2}-\lambda x^{4}}$

The function $f$ is manifestly non-analytic in $\lambda$ at $\lambda=0$. Indeed if $\lambda<0$ the integral diverges, while if $\lambda \geq 0$ the integral converges. Nevertheless we can still be rash and attempt to define a series expansion of $f$ in powers of $\lambda$ by expanding the exponential and then interchanging the order of summation and integration (illegal to be sure!). We arrive at a formal series:

$s(\lambda)=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{n!}\int_{-\infty}^{\infty}dx \ e^{-x^{2}}(-x^{4})^{n}$

Of course this series diverges. However this expansion was not in vain. $s(\lambda)$ is a basic example of an asymptotic series. For small $\lambda$ truncating the series at finite order less than $\frac{1}{\lambda^{2}}$ gives an excellent approximation to the function $f(\lambda)$

Returning to the example of Feynman integrals, the first point is that the perturbation expansion in $\lambda$ is an asymptotic series not a Taylor series. Thus just as for $s(\lambda)$ it is misguided to ask if the series converges...we already know that it does not! A better question is to ask for which actions $S$ this approximation scheme of perturbation theory itself exists. On this issue there is a complete and rigorous answer worked out by mathematical physicists in the late 70s and 80s called renormalization theory. A good reference is the book by Collins "Renormalization." Connes and Kreimer have not added new results here; rather they have given modern proofs of these results using Hopf algebras etc.

Finally I will hopefully answer some of your questions about Chern-Simons theory. The basic point is that Chern-Simons theory is a topological field theory. This means that it suffers from none of the difficulties of usual path integrals. In particular all quantities we want to compute can be reduced to finite dimensional integrals which are of course well defined. Of course since we lack an independent definition of the Feynman integral over the space of connections, the argument demonstrating that it reduces to a finite dimensional integral is purely formal. However we can simply take the finite dimensional integrals as the definition of the theory. A good expository account of this work can be found in the recent paper of Beasley "Localization for Wilson Loops in Chern-Simons Theory."

Overall I would say that by far the currently most developed approach to studying path integrals rigorously is that of discretization. One approximates spacetime by a lattice of points and the path integral by a regular integral at each lattice site. The hard step is to prove that the limit as the lattice spacing $ a $ goes to zero, the so-called continuum limit, exists. This is a very hard analysis problem. Glimm and Jaffe succeeded in using this method to construct the examples I mentioned above, but their arguments appear limited. Schematically when we take the limit of zero lattice size we also need to take a limit of our action, in other words the action should be a function of $ a$. We now write $S(a)=Q+\lambda V+H(a,\lambda)$ Where as usual $Q$ is quadratic $V$ is not an $\lambda$ is a parameter. Our original action is $S=Q +\lambda V$

The question is then can we find an $H(a,\lambda)$ such that a suitable $a\rightarrow 0$ limit exists? A priori one could try any $H$ however the arguments of Glimm and Jaffe are limited to the case where $H$ is polynomial in $\lambda$. Physically this means that the theory is very insensitive to short distance effects, in other words one could modify the interactions slightly at short distances and one would find essentially the same long distance physics. It seems that new methods are needed to generalize to a larger class of continuum limits.