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?