Integrals over the space of Riemannian metrics on $M$
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 $mathcal{Met}(M)$:
$$
S_{EH}(g)=int_M mathrm{Scal}_g ;dmu_g.
$$
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.
measure-theory differential-geometry riemannian-geometry mathematical-physics general-relativity
asked Dec 26 '18 at 18:30
rpf
1,055512
add a comment |
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 $mathcal{Met}(M)$:
$$
S_{EH}(g)=int_M mathrm{Scal}_g ;dmu_g.
$$
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.
measure-theory differential-geometry riemannian-geometry mathematical-physics general-relativity
asked Dec 26 '18 at 18:30
rpf
1,055512
2
I'm pretty sure the measure is not well defined.
– Jake
Dec 26 '18 at 22:00
1
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 Cohen
Dec 28 '18 at 16:50
1
That said, I suggest asking on mathoverflow: There are several very good mathematical physicists on that site, they might tell you more.
– Moishe Cohen
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
2 days ago
add a comment |
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 $mathcal{Met}(M)$:
$$
S_{EH}(g)=int_M mathrm{Scal}_g ;dmu_g.
$$
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.
measure-theory differential-geometry riemannian-geometry mathematical-physics general-relativity
asked Dec 26 '18 at 18:30
rpf
1,055512
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 $mathcal{Met}(M)$:
$$
S_{EH}(g)=int_M mathrm{Scal}_g ;dmu_g.
$$
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.
measure-theory differential-geometry riemannian-geometry mathematical-physics general-relativity
measure-theory differential-geometry riemannian-geometry mathematical-physics general-relativity
asked Dec 26 '18 at 18:30
rpf
1,055512
asked Dec 26 '18 at 18:30
rpf
1,055512
edited Dec 26 '18 at 20:40
asked Dec 26 '18 at 18:30
rpf
1,055512
asked Dec 26 '18 at 18:30
rpf
1,055512
asked Dec 26 '18 at 18:30
rpf
1,055512
1,055512
2
I'm pretty sure the measure is not well defined.
– Jake
Dec 26 '18 at 22:00
1
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 Cohen
Dec 28 '18 at 16:50
1
That said, I suggest asking on mathoverflow: There are several very good mathematical physicists on that site, they might tell you more.
– Moishe Cohen
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
2 days ago
add a comment |
2
I'm pretty sure the measure is not well defined.
– Jake
Dec 26 '18 at 22:00
1
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 Cohen
Dec 28 '18 at 16:50
1
That said, I suggest asking on mathoverflow: There are several very good mathematical physicists on that site, they might tell you more.
– Moishe Cohen
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
2 days ago
2
2
I'm pretty sure the measure is not well defined.
– Jake
Dec 26 '18 at 22:00
I'm pretty sure the measure is not well defined.
– Jake
Dec 26 '18 at 22:00
1
1
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 Cohen
Dec 28 '18 at 16:50
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 Cohen
Dec 28 '18 at 16:50
1
1
That said, I suggest asking on mathoverflow: There are several very good mathematical physicists on that site, they might tell you more.
– Moishe Cohen
Dec 28 '18 at 17:15
That said, I suggest asking on mathoverflow: There are several very good mathematical physicists on that site, they might tell you more.
– Moishe Cohen
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
2 days ago
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
2 days ago
add a comment |
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2
I'm pretty sure the measure is not well defined.
– Jake
Dec 26 '18 at 22:00
1
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 Cohen
Dec 28 '18 at 16:50
1
That said, I suggest asking on mathoverflow: There are several very good mathematical physicists on that site, they might tell you more.
– Moishe Cohen
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
2 days ago