Understanding the Issues of the Feynman Path Integral

Question

Let us assume we live on Euclidean $\mathbb{R}^d$ and consider the normalized partition function

\begin{equation} \begin{aligned} Z : D &\to \mathbb{R} \\ J &\mapsto \frac{\int \exp \left[ -S \left( \psi \right) + \left \langle J, \psi \right \rangle\right] \mathcal{D} \psi}{\int \exp \left[ -S \left( \psi \right) \right] \mathcal{D} \psi} \end{aligned} \end{equation}

where $D$ is taken to be the space of test functions on $\mathbb{R}^d$, $\left \langle \cdot, \cdot \right \rangle$ is the standard $L^2$ inner product and $S : D \to \mathbb{R}$ denotes some suitable classical action.

Some naive ideas to make sense of the above expressions might be:

• (M1) From a mathematical point of view, there is no Lebesgue-like measure on infinite-dimensional vector spaces. However, from my (totally inadequate) understanding of the Wiener measure, there exists a perfectly fine Gaußian measure $\mu$ (on a larger space than $D$) such that in most relevant cases $\exp \left[ - S \left( \psi \right) \right] \mathcal{D} \psi := \exp \left[ - S_{\mathrm{int}} \left( \psi \right) \right] \mathcal{D} \mu\left( \psi \right)$ where the right hand side now is well-defined.
• (M2) One may consider Hilbert space completions $H$ of $D$ with respect to some inner product and define the above quotient as a limit of integrations over finite-dimensional exhaustions of $H$ that preferably lie within $D$.

But somehow such integrals are widely considered mathematically ill-defined. What precisely is it that goes wrong? Some ideas of mine are:

• (P1) There is no unique way to embed $D$ into a larger space on which the Wiener measure might be defined
• (P2) The numerator/denominator in the above expression are infinite when considered as Lebesgue integrals with respect to the Wiener measure and taking the quotient requires some nonunique choice of limit.
• (P3) The approximation by finite-dimensional integrals is dependent on the choice of exhausting subspaces
• (P4) The approximation by finite-dimensional integrals is dependent on the choice of inner product on $D$
• (P5) The quotient in the definition of $Z$ is dependent on how the limits of exhausting subspaces is taken, i.e in analogy to $\lim_{n \to \infty} \int_{-n}^n \sin \left( n \pi x \right) \mathrm{d} x \neq \lim_{n \to \infty} \int_{-2n}^{2n + 1} \sin \left( n \pi x \right) \mathrm{d} x$.

Answer 1

I took the liberty to put some labels (M1), (M2) for the two suggested methods for making sense of the integral, as well as (P1)-(P5) for list of problems that could arise. This allows me to refer to specific items in the question.

If one replaces $J$ by $iJ$ with $J$ real-valued then the functional $Z$ is the characteristic function of the probability measure (within the totally standard framework of Lebesgue integration theory) one would like to define rigorously. The most convenient space $D$ for the test functions $J$ is Schwartz space $\mathscr{S}(\mathbb{R}^d)$. In this case the space where the wanted probability measure would live is the dual space $\mathscr{S}'(\mathbb{R}^d)$ of temperate Schwartz distributions.

About (M1): This does not work. Also, avoid the incorrect (and error prone) terminology of "Wiener measure". The latter is about $d=1$ where the measure $\mu$ is supported on a space of continuous functions and where multiplying by the in Radon-Nikodym weight $\exp(-S_{\rm int}(\psi))$ makes sense, at least in finite volume. For $d\ge 2$, the measure $\mu$, seen as a probability law for a random distribution $\psi$, is such that with probability 1, the quantity $S_{\rm int}(\psi)$ does not make sense, e.g., because it contains terms like $\psi(x)^4$. So in this sense (M1) is not a solution to the problem. However one can correct this attempt by introducing a Fourier cutoff and thus modifying $\mu$. This gives a situation similar to (M2) where one has a well defined sequence of measures on $\mathscr{S}'(\mathbb{R}^d)$ indexed by the cutoffs (frequency and volume) and the question becomes: does this sequence converge in the a suitably topology in the space of probability measures on $\mathscr{S}'(\mathbb{R}^d)$. The most natural topology here is weak convergence which corresponds to pointwise (in $J$) convergence of the corresponding functionals $Z$. This same question of convergence of the sequence of approximations (using finitely many lattice points) is the main question to be addressed by method (M2).

It is instructive to do (M2) in detail even for constructing the massive free field, i.e., the measure $\mu$. This is done in the two MathOverflow posts

https://mathoverflow.net/questions/362040/reformulation-construction-of-thermodynamic-limit-for-gff?rq=1

https://mathoverflow.net/questions/364470/a-set-of-questions-on-continuous-gaussian-free-fields-gff

What can go wrong: Let me now consider the problems that could arise and discuss (P1)-(P5). Proposing the latter as a list of potential problems, in fact amounts to putting the cart before the horse, because this tacitly presumes that existence of a limit or some outcome of method (M1) or (M2) is unproblematic, and all one has to do is worry about uniqueness of this outcome. The main problem that could occur is that there may be no limit, even if one restricts to a subsequence (loss of tightness or probability mass escaping to infinity). The other problem is that the limit obtained may be Gaussian, i.e., not interesting. See

https://mathoverflow.net/questions/260854/a-roadmap-to-hairers-theory-for-taming-infinities/260941#260941

for more on this but to make a long story short, one has to give oneself more flexibility, by allowing the couplings to depend on the cutoff, in order to get interesting limits. How to do that hinges on a renormalization group analysis. See

What is the Wilsonian definition of renormalizability?

Finally, it expected that when the limit exists, it will be unique, i.e., independent of the arbitrary choices in setting up the approximation sequence, but only few results of this type have been established rigorously. For the $\phi_3^4$ model, a recent construction is in the article by Gubinelli and Hofmanová https://arxiv.org/abs/1810.01700