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 $\lim_{n \to \infty} \int_{-n}^n \sin \left( n \pi x \right) \mathrm{d} x \neq \lim_{n \to \infty} \int_{-2n}^{2n + 1} \sin \left( n \pi x \right) \mathrm{d} x$.