Spacetime

[Metric]

If you directly quantize the gravitational field via loop quantum gravity, you find that the operators corresponding to area and volume have a discrete spectrum. String theory, on the other hand, assumes a continuous manifold with local Lorentz symmetry on which to define the world-sheets of strings. So a case for the reasonableness of either possibility can be made.

[yukoncpa]

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If you directly quantize the gravitational field via loop quantum gravity, you find that the operators corresponding to area and volume have a discrete spectrum. String theory, on the other hand, assumes a continuous manifold with local Lorentz symmetry on which to define the world-sheets of strings. So a case for the reasonableness of either possibility can be made.



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In general relativity, space time is continuous if I recall. So does string theory correspond with general relativity? I can envision a curvature of space ( sort of, although I don’t exactly know what space time really means in physical terms ). But how does sharing of virtual particles effect gravity? Is there any way this could be explained to a layman? If anyone can do it, I know you can Metric. I’m counting on you.

[thylacine]

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If you directly quantize the gravitational field via loop quantum gravity, you find that the operators corresponding to area and volume have a discrete spectrum. String theory, on the other hand, assumes a continuous manifold with local Lorentz symmetry on which to define the world-sheets of strings. So a case for the reasonableness of either possibility can be made.

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When I put the survey up I was definitely associating String theory with a continuous spacetime and associating loop quantum gravity with a discrete spacetime. Actually I wanted to instead do a survey saying choose one from a list of purported theories of quantum gravity, but I couldn't be bothered figuring out the list (String theory, loop quantum gravity, and a bunch of other ones) so I did the above survey instead.

[Metric]

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In general relativity, space time is continuous if I recall. So does string theory correspond with general relativity?

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At low energy/long distances, string theory reproduces GR, though at high energy/short distances there are corrections.

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I can envision a curvature of space ( sort of, although I don’t exactly know what space time really means in physical terms ). But how does sharing of virtual particles effect gravity? Is there any way this could be explained to a layman? If anyone can do it, I know you can Metric. I’m counting on you.

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"Virtual particles" are a particle physics trick for implementing one of the main principles of quantum mechanics -- that everything that can possibly happen, contributes to the final outcome. So particle physicists sum over virtual particles... Gravity, though, is more than just particles running around on spacetime -- gravity IS spacetime. So if you're in a regime where you can do perturbation theory around a fixed "background" spacetime, you can use virtual gravitons, but if you really want to find out how things are built from the ground up, you've got to get rid of the background -- and thus the concept of virtual gravitons along with it. At least, this is the LQG approach -- a lot of string theorists may deny that this is necessary, but the LQG theorists will say that its the ultimate manifestation of the "background free" nature of physics -- build the theory non-perturbatively with no background at all.

[thylacine]

Metric says:

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"Virtual particles" are a particle physics trick for implementing one of the main principles of quantum mechanics -- that everything that can possibly happen, contributes to the final outcome. So particle physicists sum over virtual particles... Gravity, though, is more than just particles running around on spacetime -- gravity IS spacetime. So if you're in a regime where you can do perturbation theory around a fixed "background" spacetime, you can use virtual gravitons, but if you really want to find out how things are built from the ground up, you've got to get rid of the background -- and thus the concept of virtual gravitons along with it. At least, this is the LQG approach -- a lot of string theorists may deny that this is necessary, but the LQG theorists will say that its the ultimate manifestation of the "background free" nature of physics -- build the theory non-perturbatively with no background at all.

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I am a mathematician, not a physicist, so I only vaguely understand these issues. But isn't it obvious that this (the bolface part of the quote) is true. My question is, how could it be that the string theorists don't see this fundamental objection to their entire enterprise. It just seems so obvious to me that string theory could not possibly lead to a fundamental theory (unless they address this objection). What are these people doing? Any insights Metric.

[Metric]

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I am a mathematician, not a physicist, so I only vaguely understand these issues. But isn't it obvious that this (the bolface part of the quote) is true. My question is, how could it be that the string theorists don't see this fundamental objection to their entire enterprise. It just seems so obvious to me that string theory could not possibly lead to a fundamental theory (unless they address this objection). What are these people doing? Any insights Metric.

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This is a key part of the contention between string theorists and the LQG people. String theorists claim that it's enough to be able to go back and forth between (smooth) backgrounds -- and indeed, they can do this. So in a sense, the theory doesn't depend on which background is chosen -- you can do the calculation in background "A" or background "B" (your choice) and get a physical result. String theorists will argue that this is all that is needed to arrive at a "background independent" theory.

The point, though, is that all these backgrounds look the same locally -- they are all smooth manifolds with local Lorentz symmetry. This is a requirement of string theory, as currently formulated, thus LQG people tend to think this represents a kind of background.

LQG takes the view that Lorentz symmetry only needs to be emergent on large enough distance scales. At the Planck scale, there need not be any smooth Lorentz manifold geometry -- only discrete "atoms of spacetime." This is certainly a prediction of LQG, but the problem with LQG is getting from this level up to the macroscopic spacetime that we observe -- it's very computationally difficult. So this is where a lot of work in LQG is focused these days -- trying to get the correct large scale symmetries to show up in the relevant limit. Some string theorists think this is nuts -- that you should start with the manifold structure that we know and love, making connection to real-world observations much more likely. String theorists have their own problems, of course, but getting the right large-scale spacetime symmetries is not one of them, since it's assumed from the beginning.

[thylacine]

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I am a mathematician, not a physicist, so I only vaguely understand these issues. But isn't it obvious that this (the bolface part of the quote) is true. My question is, how could it be that the string theorists don't see this fundamental objection to their entire enterprise. It just seems so obvious to me that string theory could not possibly lead to a fundamental theory (unless they address this objection). What are these people doing? Any insights Metric.

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This is a key part of the contention between string theorists and the LQG people. String theorists claim that it's enough to be able to go back and forth between (smooth) backgrounds -- and indeed, they can do this. So in a sense, the theory doesn't depend on which background is chosen -- you can do the calculation in background "A" or background "B" (your choice) and get a physical result. String theorists will argue that this is all that is needed to arrive at a "background independent" theory.

The point, though, is that all these backgrounds look the same locally -- they are all smooth manifolds with local Lorentz symmetry. This is a requirement of string theory, as currently formulated, thus LQG people tend to think this represents a kind of background.

LQG takes the view that Lorentz symmetry only needs to be emergent on large enough distance scales. At the Planck scale, there need not be any smooth Lorentz manifold geometry -- only discrete "atoms of spacetime." This is certainly a prediction of LQG, but the problem with LQG is getting from this level up to the macroscopic spacetime that we observe -- it's very computationally difficult. So this is where a lot of work in LQG is focused these days -- trying to get the correct large scale symmetries to show up in the relevant limit. Some string theorists think this is nuts -- that you should start with the manifold structure that we know and love, making connection to real-world observations much more likely. String theorists have their own problems, of course, but getting the right large-scale spacetime symmetries is not one of them, since it's assumed from the beginning.

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Thanks for the insight Metric.I am fascinated by this topic, but I don't have the physics background to deeply understand it. But my instinct is that there should be an utterly discrete model of physics, which means I wouldn't even allow any kind of continuum at a fundamental level, not even the complex numbers in Hilbert spaces.

By the way the LQG papers I tried to read (with shallow comprehension or less) still seemed to have some background manifold involved. Has there been any success in getting rid of this background manifold, as far as you know?

[Metric]

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But my instinct is that there should be an utterly discrete model of physics, which means I wouldn't even allow any kind of continuum at a fundamental level, not even the complex numbers in Hilbert spaces.

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I very much sympathize with this idea -- I've seen it suggested before, but implemented in a very ad-hoc sort of way (simply discretizing the Bloch sphere, for example). It would be great if there were some kind of physical principle, perhaps phrased in information theoretic terms, that would univerally lead to something like this... But it's nearly untouched as far as I'm aware.

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By the way the LQG papers I tried to read (with shallow comprehension or less) still seemed to have some background manifold involved. Has there been any success in getting rid of this background manifold, as far as you know?

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In LQG one starts with a manifold, and defines a connection "A" on it as a canonical variable. Passing to the quantum theory, one has L^2 functions of A (with respect to some special measure) that form the "states" of the theory. However, these alone are not the "physical" states of the theory -- we want the physical states to be invariant under arbitrary diffeomorphisms of the underlying manifold (this is the expression of the "background free" nature of LQG). After modding out by all diffeomorphisms, though, the underlying manifold has essentially no physical significance -- all physically relevant information is contained on abstract spin networks. So the manifold turns out to be just a mathematical tool with which to build the various state spaces -- it is not to be interpreted as a physical background structure that has to be specified in order for the predictions of the theory to make sense.

[MoreGentilythanU]

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But my instinct is that there should be an utterly discrete model of physics, which means I wouldn't even allow any kind of continuum at a fundamental level, not even the complex numbers in Hilbert spaces.

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I very much sympathize with this idea -- I've seen it suggested before, but implemented in a very ad-hoc sort of way (simply discretizing the Bloch sphere, for example). It would be great if there were some kind of physical principle, perhaps phrased in information theoretic terms, that would univerally lead to something like this... But it's nearly untouched as far as I'm aware.

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By the way the LQG papers I tried to read (with shallow comprehension or less) still seemed to have some background manifold involved. Has there been any success in getting rid of this background manifold, as far as you know?

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In LQG one starts with a manifold, and defines a connection "A" on it as a canonical variable. Passing to the quantum theory, one has L^2 functions of A (with respect to some special measure) that form the "states" of the theory. However, these alone are not the "physical" states of the theory -- we want the physical states to be invariant under arbitrary diffeomorphisms of the underlying manifold (this is the expression of the "background free" nature of LQG). After modding out by all diffeomorphisms, though, the underlying manifold has essentially no physical significance -- all physically relevant information is contained on abstract spin networks. So the manifold turns out to be just a mathematical tool with which to build the various state spaces -- it is not to be interpreted as a physical background structure that has to be specified in order for the predictions of the theory to make sense.

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holy crap I can't wait to get my Phd in this stuff, coolest discussion ever. Nh.

[thylacine]

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In LQG one starts with a manifold, and defines a connection "A" on it as a canonical variable. Passing to the quantum theory, one has L^2 functions of A (with respect to some special measure) that form the "states" of the theory. However, these alone are not the "physical" states of the theory -- we want the physical states to be invariant under arbitrary diffeomorphisms of the underlying manifold (this is the expression of the "background free" nature of LQG). After modding out by all diffeomorphisms, though, the underlying manifold has essentially no physical significance -- all physically relevant information is contained on abstract spin networks. So the manifold turns out to be just a mathematical tool with which to build the various state spaces -- it is not to be interpreted as a physical background structure that has to be specified in order for the predictions of the theory to make sense.

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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. 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.

This may be a bit vague, but hopefully you know the issue I'm getting at. So is there really an escape from the manifold in this formulation?