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Let $M$ be a closed smooth $n$-manifold.

In the Hartle-Hawking proposal of Euclidean quantum gravity, the authors consider functional integrals of the form $$ I_M=\int_{\mathcal{Met}(M)}e^{-S(g)}\;\mathcal{D}g, $$ where $\mathcal{Met}(M)$ is the space of all Riemannian metrics on $M$, and $S:\mathcal{Met}(M)\to \mathbb{R}$ is some functional on the space of metrics. For example, Hartle-Hawking take $S$ to be the Einstein-Hilbert (a.k.a. total scalar curvature) functional on $\mathcal{Met}(M)$: $$ S_{EH}(g)=\int_M \mathrm{Scal}_g \;d\mu_g. $$

Questions: How is the "volume form" $\mathcal{D}g$ defined? Are there any examples where $I_M$ has been computed?

My guess is that the definition of $\mathcal{D}g$ comes from considering $\mathcal{Met}(M)$ as a smooth infinite dimensional Frechet manifold (the space of smooth sections of the bundle of positive definite symmetric $(2,0)$-tensors on $M$) and then endowing $\mathcal{Met}(M)$ with some Riemannian metric. Is this correct?

Any references would be appreciated.

rpf
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    I'm pretty sure the measure is not well defined. –  Dec 26 '18 at 22:00
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    Rigorous definitions of functional integrals of this form is a notorious open problem. All rigorous versions I am aware of involve replacement of Riemannian metrics with triangulations. There are notions of combinatorial Ricci and scalar curvature which serve as replacements of the Riemannian concepts. – Moishe Kohan Dec 28 '18 at 16:50
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    That said, I suggest asking on mathoverflow: There are several very good mathematical physicists on that site, they might tell you more. – Moishe Kohan Dec 28 '18 at 17:15
  • With regard to the general question: "How is $\int_{F(X)} e^{-S(f)} \mathcal Df$ defined?", I think the book by Barry Simon "Functional Integration and Quantum Mechanics" is the canonical textbook, that should explain what such expressions mean mathematically. Take this recommendation with a word of caution however: I haven't read it, and if you don't have a good relation with measure theory it should be very difficult reading. – s.harp Dec 29 '18 at 16:29

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