I have been trying to find out the sweet middle ground of describing path integration of field theories, in between the physicist way and the mathematician way, but it seems hard to find something that is both rigorous and describes how to actually compute them.

So far what I've been able to get is this. This is in the context of Schrödinger functionals, so what I've been trying to compute is something of the form

$$\langle \Phi, \Psi \rangle = \int_{\mathcal{S}} d\gamma \Phi^*[\phi] \Psi[\phi]$$

$\Phi, \Psi$ are two wavefunctionals of the Hilbert space $L^2(\mathcal{S}(\mathbb{R}^n), d\gamma)$, so that a wavefunctional is a linear functional $\Psi \in \mathcal{S}'(\mathbb{R}^n) : \mathcal{S}(\mathbb{R}^n) \to \mathbb{C}$. That part isn't much of an issue, we can just consider the case $|0\rangle = 1$ here, that doesn't change much. So just

$$\mathcal{Z} = \int_{\mathcal{S}} d\gamma $$

The measure here is the gaussian measure on the Schwartz space. There aren't a whole lot of very readable papers on the topic (although this one is the closest I found), but as far as I can tell, given a Gaussian measure on a Banach space $X$, there is the equality

$$\int_X d\gamma e^{i f(\phi)} = e^{-\frac{1}{2} q(f, f)}$$

for $\phi \in X$, $f \in X'$, and $q$ is some non-negative, symmetric bilinear form on $X'$. Some sources write out $q$ as

$$q(f,f) = \langle f, O f\rangle_X$$

For some hermitian product $\langle ., .\rangle$ on $X$, and a positive definite, self-adjoint operator $O$. As there is also the pseudo-equality

$$d\gamma \approx e^{-\frac{1}{2} \langle \phi, O^{-1} \phi \rangle} \mathcal{D} \phi$$

I assume that $O$ is the inverse operator to $\Delta$ (since we have to work in Euclidian space), so that our integral is just something like

$$\int_{\mathcal{S}} d\gamma e^{i f(\phi)} = e^{-\frac{1}{2} \int d^nx f \Delta^{-1} f}$$

This certainly seems to work out for the case $f = 0$ as it gives us

$$\int_{\mathcal{S}} d\gamma = 1$$

I'm sure there are many issues here (for instance the hermitian product here only works if $f$ can be expressed as an actual function), but is this roughly the sort of formula we need to actually compute path integrals in a rigorous manner?

  • $\begingroup$ Great question, I've been trying for the last month to find also such middle ground, and it has been quite challenging, so I'm really looking forward to seeing an answer to this question. There's the book "Gaussian Measures" by Vladimir Bogachev that explains the Math part quite well IMO. Unfortunately, the hard part IMHO seems to be connecting to what physicists do in practice. $\endgroup$ – user1620696 Jul 29 at 14:41
  • $\begingroup$ The physical Hilbert space is not $L^2(S(\mathbb{R}^n),\gamma)$ but rather it is of the form $L^2(S'(\mathbb{R}^n),\gamma)$. In other words, the measure is on nasty distributions rather than Schwartz space. If $\mu$ is the Euclidean path integral measure on space time $\mathbb{R}^{n+1}$, i.e., $\mu$ is a probability measure on $S'(\mathbb{R}^{n+1})$, then one can in principle get $\gamma$ as the law of the restriction of Schwartz distributions, sampled according to $\mu$, to a fixed time hyperplane $\mathbb{R}^n$. $\endgroup$ – Abdelmalek Abdesselam Sep 6 at 14:11

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