Gravitons in Loop Quantum Gravity

tom.stoer

Quote by suprised

Well I believe the story is roughly like follows ... .

I think you are right to approx. 99%, so it's unnecessary to comment on these 99%, and it's better to avoid any comments on the 1% in order not to confuse the reader with minor details.

*waterfall*

Quote by tom.stoer

In string theory and in most treatments of QFTs one starts with quantized excitations on top of a classically fixed background. The excitations are the quanta of the associated fields (photons, gravitons, ...). This approach has some limitations and LQG tries to get rid of them.

We have been focusing on the idea of spin-2 fields plus flat spacetime = curved spacetime and applying this even to String theory. But have we forgotten that the compactified dimensions in String Theory are really 6 dimensions? Unless you are saying the 6 dimensions are flat? How does one resolve this?

LQG never introduces a background and excitations living on this background, so LQG does not use gravitons as building blocks. Instead one expects that one may recover a kind of semiclassical limit or weak field limit where something like "gravitons" will show up again.

So in contrast to any other QFT where the "...ons" are the fundamental (mathematical and physical) entities in LQG the gravitons are not fundamental but only to be considered in a certain limited approximation.

tom.stoer

In the standard formulation in ST one introduces a spacetime metric. Some decades ago this was typically "4-dim. Minkowski spacetime" * "6-dim. compactified Calabi-Yau space"; in the meantime other geometries have been discovered and are studied extensively.

From ST one can derive a consistency condition for the spacetime on which strings are propagating. This consistency conditions requires Ricci-flatness and forbids arbitrary spacetimes and arbitrary compactified dimensions; Minkowski spacetime * Calabi-Yau is a typical solution, but as I said, others are possible, and in the meantime string theorists were able to relax these conditions.

The problem I see with string theory is that "spin-2 fields plus flat spacetime = curved spacetime" does not really work; you have to chose are curved spacetime in the very beginning and study propagation of strings as "weak distortions" on top of it". But these strings do never change the whole spacetime dynamically, it always stays in some fixed subsector which does not change dynamically. This holds afaik for other approaches (e.g. branes, fluxes, ...) as well.

This is what is called background dependence and is basically due to the approach "fix a background and then quantize small distortions". LQG tries to get rid of this problem and avoids to fix a classical background. But as suprised explained, one still has to introduce some kind of boundary condition or background when doing physics; it's not required for the definition of the theory (and that's a major step forward), but it's required for detailed calculations (e.g. when studying black holes in LQG one has to define an isolated horizon classically; a full dynamical setup w/o any input like boundary conditions or background is not possible).

*waterfall*

Quote by tom.stoer

In the standard formulation in ST one introduces a spacetime metric. Some decades ago this was typically "4-dim. Minkowski spacetime" * "6-dim. compactified Calabi-Yau space"; in the meantime other geometries have been discovered and are studied extensively.

From ST one can derive a consistency condition for the spacetime on which strings are propagating. This consistency conditions requires Ricci-flatness and forbids arbitrary spacetimes and arbitrary compactified dimensions; Minkowski spacetime * Calabi-Yau is a typical solution, but as I said, others are possible, and in the meantime string theorists were able to relax these conditions.

The problem I see with string theory is that "spin-2 fields plus flat spacetime = curved spacetime" does not really work; you have to chose are curved spacetime in the very beginning and study propagation of strings as "weak distortions" on top of it". But these strings do never change the whole spacetime dynamically, it always stays in some fixed subsector which does not change dynamically. This holds afaik for other approaches (e.g. branes, fluxes, ...) as well.

Well. So String Theory is about Spin-2 fields on curved spacetime? Isn't it redundant? Why produce GR from string modes in background that is already curved. Perhaps the original curved spacetime is not GR and strings are looking for modes to produce GR? Isn't this kinda contrived and redundant?

This is what is called background dependence and is basically due to the approach "fix a background and then quantize small distortions". LQG tries to get rid of this problem and avoids to fix a classical background. But as suprised explained, one still has to introduce some kind of boundary condition or background when doing physics; it's not required for the definition of the theory (and that's a major step forward), but it's required for detailed calculations (e.g. when studying black holes in LQG one has to define an isolated horizon classically; a full dynamical setup w/o any input like boundary conditions or background is not possible).

*waterfall*

Quote by tom.stoer

In the standard formulation in ST one introduces a spacetime metric. Some decades ago this was typically "4-dim. Minkowski spacetime" * "6-dim. compactified Calabi-Yau space"; in the meantime other geometries have been discovered and are studied extensively.

From ST one can derive a consistency condition for the spacetime on which strings are propagating. This consistency conditions requires Ricci-flatness and forbids arbitrary spacetimes and arbitrary compactified dimensions; Minkowski spacetime * Calabi-Yau is a typical solution, but as I said, others are possible, and in the meantime string theorists were able to relax these conditions.

The problem I see with string theory is that "spin-2 fields plus flat spacetime = curved spacetime" does not really work; you have to chose are curved spacetime in the very beginning and study propagation of strings as "weak distortions" on top of it". But these strings do never change the whole spacetime dynamically, it always stays in some fixed subsector which does not change dynamically. This holds afaik for other approaches (e.g. branes, fluxes, ...) as well.

This is what is called background dependence and is basically due to the approach "fix a background and then quantize small distortions". LQG tries to get rid of this problem and avoids to fix a classical background. But as suprised explained, one still has to introduce some kind of boundary condition or background when doing physics; it's not required for the definition of the theory (and that's a major step forward), but it's required for detailed calculations (e.g. when studying black holes in LQG one has to define an isolated horizon classically; a full dynamical setup w/o any input like boundary conditions or background is not possible).

I was reading the book "Philosophy Meets Physics at the Planck Scale" today and some stuff there opened my eyes. In the book. There are 3 main quantum gravity programmes.

1. Particle Physics approach
2. String Theory
3. Canonical Quantization or LQG

The Particle Physics approach is the one that treats spin-2 field in flat spacetime = curved spacetime. Not String Theory. For 5 years. I thought String Theory was about it. According to the above mentioned book:

"The perturbative superstrings programme involves quantizing a classical system; but the system concerned is not general relativity, but rather a system in which a one-dimensional closed string propagates in a spacetime M (whose dimension is in general not 4). More precisely, the propagation of the string is viewed as a map X : W → M from a two-dimensional ‘world-sheet’ W to spacetime M (the ‘target spacetime’). The quantization procedure quantizes X, but not the metric γ on M, which remains classical."

The above suggests string theory is not about spin-2 on flat spacetime. Can anyone confirm this (or is there any objections?). The above says string theory is about quantizing the map X : W → M. Does anyone have other ways of saying it that can make it clearer? Or web site links that specifically details this particular aspect? How do you connect it to Spin-2? Could it be possible that this world-sheet and dimensions not 4 somehow produce the predictions of spin-2 over flat spacetime. Is that what the above is saying or suggesting?

Quote by tom.stoer

From ST one can derive a consistency condition for the spacetime on which strings are propagating. This consistency conditions requires Ricci-flatness and forbids arbitrary spacetimes and arbitrary compactified dimensions; Minkowski spacetime * Calabi-Yau is a typical solution, but as I said, others are possible, and in the meantime string theorists were able to relax these conditions.

Tom, you mentioned above about the consistency conditions forbidding arbitrary compactified dimensions. Why, isn't the Calabi-Yau an arbitrary one? What is your example of "arbitrary" compactified dimensions?

Thanks.

tom.stoer

String theory is about quantization of strings; but one finds one oscillation of a closed string that corresponds to a massless spin-2 particle which is then identified with the graviton. There is a limit in which string theories reduce to quantum field theoirs of poiintlike particles; this limit is a supergravity theory containing gravitons, gravitinos (and a lot of other stuff). In that sense ST contains gravitons.

The consistency condition is Ricci-flatness. CY are Ricci-flat and are in that sense not arbitrary.

*waterfall*

Quote by tom.stoer

String theory is about quantization of strings; but one finds one oscillation of a closed string that corresponds to a massless spin-2 particle which is then identified with the graviton. There is a limit in which string theories reduce to quantum field theoirs of poiintlike particles; this limit is a supergravity theory containing gravitons, gravitinos (and a lot of other stuff). In that sense ST contains gravitons.

The consistency condition is Ricci-flatness. CY are Ricci-flat and are in that sense not arbitrary.

After hours of checking at the web sites and reading old achives. I found this great site that derives it:

http://motls.blogspot.com/2007/05/wh...in-string.html

"If the worldsheet theory is consistent as a string theory, it must be scale-invariant, and the spacetime geometry must thus be Ricci-flat! We have just derived Einstein's equations from scale invariance of a two-dimensional theory."

It didn't mention about Calabi-Yau. But these are supposed to be 6 dimensional. I wonder how they could be Ricci-flat. It's like one has spike balls, how can it be flat.. unless you are saying since they are so tiny, they look flat at larger scale even though they are like spike balls?

If someone can explain too. Please do so. Thanks.

tom.stoer

look at a two-torus T²; the embedding in R³ doesn't seem to be flat, but mathematically it's possible (w/o considering an embedding) to a define an atlas consisting of flat metrics on T².

*waterfall*

Quote by tom.stoer

look at a two-torus T²; the embedding in R³ doesn't seem to be flat, but mathematically it's possible (w/o considering an embedding) to a define an atlas consisting of flat metrics on T².

Is the above an example of a two-torus? How does "one define an atlas consisting of flat metrics on T²"? In Calabi-Yau, there are 6 dimensions with 6 different axis.. how do you make it flat, maybe from looking at a distance very far? Thanks.

*waterfall*

Quote by waterfall

Is the above an example of a two-torus? How does "one define an atlas consisting of flat metrics on T²"? In Calabi-Yau, there are 6 dimensions with 6 different axis.. how do you make it flat, maybe from looking at a distance very far? Thanks.

I found the best explanation about it here.

http://universe-review.ca/R15-26-CalabiYau.htm

Thanks for pointing out that manifolds that is not flat can really be flat. It's counter-intuitive that was why I missed the concept in all the years I read Brian Greene books.

I guess the reason they make it ricci-flat is so that it can satisfy the condition "ricci-flat plus spin 2 graviton field = General Relativity. But then. Since it can't even describe the FRW Universe. What good is it I'm wondering??

atyy

The curved spacetime of a Schwarzschild black hole is Ricci flat.

String cosmology doesn't seem that well developed, but here are some recent reviews.
McAllister, Silverstein String Cosmology: A Review
Burgess, McAllister Challenges for String Cosmology

tom.stoer

unfortunately that's not a two-torus but a two-fold or double torus; in T² the "2" means two-dimensional ;-)

a two-torus T² can be defined as tupels (x,y) in an interval ]0,L_x[ * ]0,L_y[, i.e. R² with a grid where all points (x+mL_x,y+nL_y) etc. (with integers m, n) are identified; the flat metric is nothing else but the standard metric on R²

*waterfall*

Quote by atyy

The curved spacetime of a Schwarzschild black hole is Ricci flat.

String cosmology doesn't seem that well developed, but here are some recent reviews.
McAllister, Silverstein String Cosmology: A Review
Burgess, McAllister Challenges for String Cosmology

I've been trying to read the two papers for half hour already but I can't seem to get the essence. The reasons we created all those Calabi-Yau is to simulate the ricci-flatness so that the particle physics approximation of flat spacetime + spin-2 field = curved spacetime can work. But in cosmology like inflation and FRW model... it doesn't support those approximation at all. So how do strings deal with it? By doing away with Calabi-Yau and using branes? How? Maybe you or someone can just give an enlightening pointer of strings plan to resolve it. Like could strings be trying to explain them by not resorting to the ricci-flat path and directly from strings mode to cosmology spacetimes? Thanks.

Haelfix

Quote by waterfall

I've been trying to read the two papers for half hour already but I can't seem to get the essence. The reasons we created all those Calabi-Yau is to simulate the ricci-flatness so that the particle physics approximation of flat spacetime + spin-2 field = curved spacetime can work. But in cosmology like inflation and FRW model... it doesn't support those approximation at all.

You have gotten it all mixed up (here and pretty much everywhere)!!! First of all, the reason none of this material makes any sense to you (when explained by multiple parties) is b/c you haven't heeded my advice to start from the beginning. For instance, chapter 1 of Misner, Thorne, Wheeler or some other undergraduate level book on general relativity.

What's happening is you are confusing several different english words, that are utilized in different physical and mathematical contexts. Unfortunately its often the case that these things have multiple meanings in quite different situations. Worse, you are doing this with posts on physicsforum and other places on the internet where the word choices/phrasing aren't necessarily as polished as a textbook.

When a physicist reads a paragraph written in english about some physics subject, what he has in mind is a sort of substition (this is referring to a calculation that is done on page xx of book/paper or blackboard yy). Even there it can be a little bit ambigous and another reader will ask for clarification in a very particular way where it is clear that both parties are on the same wavelength.

What you are doing is grouping together words and concepts and trying to develop an abstract intuition about the physics. Well, believe me, you can stop right there b/c it won't work. It has not worked for the greatest geniuses of our time. The only way to learn this material is the hard way. ok?

Now.. If you really, really want to talk about in what sense and how linearized gravity works, I will be happy to explain. But only if I know you have done some groundwork first (this means knowing what a metric is, what gravitational waves are, what initial value formulations of GR are, and so forth aka a mastery of the first 18 chapters of the aforementioned MTW)

*waterfall*

Quote by Haelfix

Now.. If you really, really want to talk about in what sense and how linearized gravity works, I will be happy to explain. But only if I know you have done some groundwork first (this means knowing what a metric is, what gravitational waves are, what initial value formulations of GR are, and so forth aka a mastery of the first 18 chapters of the aforementioned MTW)

I agree with you of course. As the details are what can make me comprehend 100%. But laymen just want to get a general feel of it all. And thanks to an interesting old thread where you participated intimately. I got the superficial grasp. Above I was asking about how GR came from strings, not about linearized gravity. After 3 weeks of discussing linearized gravity that spans many threads and more than a dozen people. Of course I got the basic of it. Basically the idea is that the correspondence between GR's curved manifold and a model using a flat manifold plus a spin-2 interaction has only been shown for weak gravity in a region small enough that the two manifolds can be put into 1-to-1 correspondence. This doesn't work for FRW. And I've been wondering for two days how strings can handle FRW. I'm also reading Tong paper you shared in the classic old thread:

http://www.physicsforums.com/showthread.php?t=495351
"General Relativity from String Theory"

I have read the thread twice and will do it again and again for the next few days.

You mentioned in #7 there:

After compactification, and integrating out the matter modes and taking the hbar --> 0 limit, the result is 4 dimensional Einstein Hilbert lagrangian. Solving the Euler-Lagrange equations yields Einsteins field equations in vacuum exactly.

This is completely analogous to the derivation in MTW where a spin 2 field and a weak field expansion is shown to reproduce the EFE exactly. Here though, the consisteny criteria are already staring at you in the face on the worldsheet. There is nothing else that string theory can limit too, it always must have the EFE's in the IR exactly!!!!

And you detailed it in #9:

Tong explains it perfectly. You start by fixing a background on the worldsheet, and demanding that the quantum theory be conformally invariant (eg that the beta functions vanish). After a calculation you find a set of equations or requirements that must vanish.

Up to this point, everything is perturbative to a given order and fixed.

Now you switch perspectives, and ask, what is the low energy effective lagrangian over spacetime (as opposed to the worldsheet) that gives those beta functions as equations of motion.

And you are led to the EH lagrangian. This last step is decidedly not perturbative, it is not fixed, it is simply a statement that in the hbar --> 0 limit (which takes care of all the 2+ loop corrections from the worldsheet), that the EFE's are the only possible equations of motion that reproduces that lagrangian classically. All you then need to do is show that it is unique. Which is a classical theorem by Hilbert, and you are done.

The bottomline is that there is no controversy that string theory gives GR in the low energy limit. It is basic textbook material!

(edit: The action here is indeed 26 dimensional, and strictly speaking this is the Bosonic string. The real calculation would involve compactification on the Superstring (eg 10 dimensional), and obviously it is a little more subtle with a lot more notation. But the actual proof goes through in a completely analogous manner, except there you won't derive pure GR, but rather supergravity (and then you have to worry about how to break supersymmetry))

Brilliant! Later in the thread, in msg# 50. Finbar mentioned thus:

qsa makes a good point. If we go by Tong's calculation string theory is only consistent in strictly Ricci flat space-times. Since evidence(accelerated expansion) points to us living in de-sitter space we must need to find some degrees of freedom(modes of the string), other than gravitons, which form a coherent state corresponding to de-sitter space.

Anyone know if this has been achieved??

You didn't comment on it. So maybe you agreed on the statement that something besides the gravitons modes of the string can form a coherent state corresponding to de-sitter space? This is my only question to you for this year before I started MTW for the next 2 years.

tom.stoer replied to Finbar statements above:

I guess this is a fundamental problem, namely that in a certain sense background independenca means something different in string theory. One has to prove for a certain background that a consistent quantization can be achieved. And this has to be done for each background seperately. Therefore the background (or lets say the class of backgrounds) changes the d.o.f.

I mentioned this because I can't figure the acronym "d.o.f.", what is d.o.f.? Depth of Field? I really need to know this because that old thread is a classic and it would be on my mind a lot.

tom.stoer ended it thus:

The problem is that one should somehow categorize backgrounds in terms of something like "classes" or "superselection sectors". Different sectors may or may not be "connected" by dynamics. In string theory the specific background can affect the details of the degrees of freedom living on it.

I see the following problems:
- one has to identify the correct d.o.f. for each background (sector)
- there may be backgrounds (sectors) which cannot be equipped with a viable string theory
- dynamically connected backgrounds (sectors) cannot be studied coherently if they have different string d.o.f.

Now the question is how to construct viable string theories for certain classes of backgrounds relevant in GR, especially
- dynamical collaps, e.g. pre-Schwarzschild and pre-Kerr
- FRW, dS, ...

So the question in my message previous to this was answered by tom last year. That we may need to to "construct viable string theories for certain classes of backgrounds relevant in GR, especially
- dynamical collaps, e.g. pre-Schwarzschild and pre-Kerr
- FRW, dS"

Now. Haelfix. I don't disagree with you that I read MTW book for the next one year. And I'm not looking or asking for any more details now if you don't want to ask more. I just want to know if you agree with Finbar and Tom above because in that old classic thread. You didn't respond to them. So you agreed with them? This is all I need to know at this point in time and I know I won't ask more questions before I mastered MTW book.

And Tom or others. Don't forget to tell me what is d.o.f. Many Thanks to all.

tom.stoer

d.o.f. = degree of freedom ;-)

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tom.stoer Recognitions: Science Advisor	look at a two-torus T²; the embedding in R³ doesn't seem to be flat, but mathematically it's possible (w/o considering an embedding) to a define an atlas consisting of flat metrics on T².

atyy Recognitions: Science Advisor	The curved spacetime of a Schwarzschild black hole is Ricci flat. String cosmology doesn't seem that well developed, but here are some recent reviews. McAllister, Silverstein String Cosmology: A Review Burgess, McAllister Challenges for String Cosmology