Spacetime

[Metric]

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So I saw that you get a Hilbert space spanned (in some sense) by some kind of labelled knotted graphs (up to diffeomorphism) in a manifold. And I see how they could be described combinatorially in some way.

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Yes, these go by various names depending on how many of the constraints they solve. Cylindrical functions, spin networks (solve the Gauss constraint), s-knots also sometimes called "abstract spin networks" (solve the diffeomorphism constraint), etc.

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But then if you stick in a surface, you Hilbert space seems to get bigger. For each knotted graph you had before, know you have many corresponding to all the different (up to whatever appropriate equivalence) ways the knotted graph can intersect the surface. Then you can stick more and more surface in, and get bigger and bigger Hilbert spaces, until you need every knotted graph (no longer up to diffeomorphism) on your original manifold.

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I am not sure what you're talking about here, but I am going to guess that you're talking about something like calculations of black hole entropy, which does involve a background spacetime and inserting a surface corresponding to the black hole horizon, and getting a Hilbert space of "surface geometry" from the spin networks. This is certainly an approximation -- the full quantum state of the spacetime isn't known, but putting in reasonable boundary conditions allows you to calculate the entropy anyway. In this case, the manifold on which you play (and surfaces you consider on it) is indeed supposed to be taken as physical spacetime -- a semiclassical approximation to the full quantum state. This is simply because it is too hard to write out the full quantum state -- this is somewhat analogous to QFT calculations of the EM field inside a conducting cavity. The full EM field, taking into account the atomic structure of the metal isn't explicitly calculated because that is too hard to write down -- instead, reasonable boundary conditions are written down, and the much simpler problem of just the fields in the cavity can be considered.

But this doesn't mean that the formalism in general depends fundamentally on a background spacetime manifold -- it does not. However, introducing one as an approximation can make certain calculations of interest many orders of magnitude easier/more practical to compute.

I am not exactly sure this is what you referring to, but hopefully it illustrates the point that you are worrying about...

[thylacine]

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So I saw that you get a Hilbert space spanned (in some sense) by some kind of labelled knotted graphs (up to diffeomorphism) in a manifold. And I see how they could be described combinatorially in some way.

[/ QUOTE ]
Yes, these go by various names depending on how many of the constraints they solve. Cylindrical functions, spin networks (solve the Gauss constraint), s-knots also sometimes called "abstract spin networks" (solve the diffeomorphism constraint), etc.

[ QUOTE ]
But then if you stick in a surface, you Hilbert space seems to get bigger. For each knotted graph you had before, know you have many corresponding to all the different (up to whatever appropriate equivalence) ways the knotted graph can intersect the surface. Then you can stick more and more surface in, and get bigger and bigger Hilbert spaces, until you need every knotted graph (no longer up to diffeomorphism) on your original manifold.

[/ QUOTE ]
I am not sure what you're talking about here, but I am going to guess that you're talking about something like calculations of black hole entropy, which does involve a background spacetime and inserting a surface corresponding to the black hole horizon, and getting a Hilbert space of "surface geometry" from the spin networks. This is certainly an approximation -- the full quantum state of the spacetime isn't known, but putting in reasonable boundary conditions allows you to calculate the entropy anyway. In this case, the manifold on which you play (and surfaces you consider on it) is indeed supposed to be taken as physical spacetime -- a semiclassical approximation to the full quantum state. This is simply because it is too hard to write out the full quantum state -- this is somewhat analogous to QFT calculations of the EM field inside a conducting cavity. The full EM field, taking into account the atomic structure of the metal isn't explicitly calculated because that is too hard to write down -- instead, reasonable boundary conditions are written down, and the much simpler problem of just the fields in the cavity can be considered.

But this doesn't mean that the formalism in general depends fundamentally on a background spacetime manifold -- it does not. However, introducing one as an approximation can make certain calculations of interest many orders of magnitude easier/more practical to compute.

I am not exactly sure this is what you referring to, but hopefully it illustrates the point that you are worrying about...

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I got interested in LQG after reading the Lee Smolin article in Scientific American in about 2003/4, and I tried reading several things (about LQG as well as GR and QFT) a couple of years ago. I only have a patchy, vague understanding of it all, as you can probably tell. (I am a mathematician, not a physicist.) Anyway ....

with what I said above, one of the things I had in mind was the consideration of an area operator A. It seems that such an operator would be associated with a surface S. If x1 and x2 are equivalent spin networks (or something of that ilk) up to diffeomorphism (or something of that ilk), then presumably they are supposed to be the same element of the Hilbert space, so |x1>=|x2>. On the other hand I thought the area calculation would depend on how x1 and x2 intersect the surface S, giving different answers for A|x1> and A|x2>, so that the introduction of surface S meant that |x1> and |x2> would have had to have been considered distinct in the presence of S, even it they could have been safely identified in the absence of any surface. This is what I meant by needing a bigger Hilbert space. Then consider what happens when you consider all area operators.

A similar thing maybe happens when, for spin networks x1 and x3 (or something of that ilk) you want to define <x1|x3>. Maybe it would depend on how x1 and x3 are linked together giving rise to similar issues.

e.g. An unknot may intersect a surface in many ways. Two unknots may link together in many ways.

I know you can take linear combinations of everything in sight, but I don't think that causes any extra problem.

I can't tell if I am just talking complete nonsense, or if I am driving at some fundamental issue here (or something in between).

[Metric]

I see what you're saying now -- this is a rather subtle point. Let me quote from Carlo Rovelli's excellent text "Quantum Gravity."

"Recall that in classical GR we distinguish between a metric g and a geometry [g]. A geometry is an equivalence class of metrics under diffeomorphism. For instance, in three dimensions, the euclidean metric g_a_b = delta_a_b and a flat metric g'_a_b =/ delta_a_b are different metrics, but define the same geometry [g]=[g']. The notion of geometry is diffeomorphism invariant, while the notion of metric is not. On a given manifold with coordinates x, we can define a surface by S=(sigma^1, sigma^2) --> x^a(sigma^i). Then it makes sense to ask what is the area of S in a given metric g_a_b(x), but it makes no sense to ask what is the area of S in a given geometry, because the relative location of S and the geometry is not defined.
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Now, in quantum gravity we find precisely the same situation. Above we have defined coordinate surfaces S and regions R, and their areas and volumes. Such coordinate surfaces and regions are not defined at the diffeomorphism-invariant level. However, we can nevertheless define surfaces and regions on the abstract quantum state |s> itself, and associate areas and volumes with them. A region R is simply a collection of nodes. Its boundary is an ensemble of links and defines a surface; we can say that this surface "cuts" these links. A moment of reflection will convince the reader that this is precisely the same situation in the classical theory."

Apparently this is a common source of confusion among people studying this stuff -- understanding what is and is not observable in diffeomorphism invariant physics is a very non-trivial thing.

[siegfriedandroy]

hasnt anyone seen back to the future, which discussed the space-time continuum?

[thylacine]

[ QUOTE ]
I see what you're saying now -- this is a rather subtle point. Let me quote from Carlo Rovelli's excellent text "Quantum Gravity."

"Recall that in classical GR we distinguish between a metric g and a geometry [g]. A geometry is an equivalence class of metrics under diffeomorphism. For instance, in three dimensions, the euclidean metric g_a_b = delta_a_b and a flat metric g'_a_b =/ delta_a_b are different metrics, but define the same geometry [g]=[g']. The notion of geometry is diffeomorphism invariant, while the notion of metric is not. On a given manifold with coordinates x, we can define a surface by S=(sigma^1, sigma^2) --> x^a(sigma^i). Then it makes sense to ask what is the area of S in a given metric g_a_b(x), but it makes no sense to ask what is the area of S in a given geometry, because the relative location of S and the geometry is not defined.
.
.
.
Now, in quantum gravity we find precisely the same situation. Above we have defined coordinate surfaces S and regions R, and their areas and volumes. Such coordinate surfaces and regions are not defined at the diffeomorphism-invariant level. However, we can nevertheless define surfaces and regions on the abstract quantum state |s> itself, and associate areas and volumes with them. A region R is simply a collection of nodes. Its boundary is an ensemble of links and defines a surface; we can say that this surface "cuts" these links. A moment of reflection will convince the reader that this is precisely the same situation in the classical theory."

Apparently this is a common source of confusion among people studying this stuff -- understanding what is and is not observable in diffeomorphism invariant physics is a very non-trivial thing.

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Thanks, that does clear up some confusion to some extent. At least I understood enough of what I had previously read to get confused. Where can I find `Carlo Rovelli's excellent text "Quantum Gravity."'? Is it a book, or a paper, or what? Thanks!

[Metric]

"Quantum Gravity" is a book available through amazon.com or wherever -- it is by far the best and most complete LQG text yet written. Rovelli used to have a preprint version in pdf format linked on his webpage, but I don't see it there anymore -- if you'd like the preprint version just pm me and I'll email you a copy.