Gravitons in Loop Quantum Gravity

[waterfall]

How do you understand about Gravitons in Loop Quantum Gravity? All I know about it is from what I heard that "LQG hopes that its predictions for experiments occurring far below the Planck scale will be almost identical to that of gravitons on flat spacetime".

So do you consider it a pure graviton at all or some kind of pseudo-graviton (in LQG)?

I just found a paper about gravitons in LQG but may not understand it 100%

http://arxiv.org/pdf/gr-qc/0604044v2.pdf

Do you think it's correct? Please share other papers about gravitons in LQG if there is. Thanks.

 

[waterfall]

For example someday the LHC were able to detect gravitons. Can we tell whether it is a String graviton or the pseudo-graviton of Loop Quantum Gravity? What are the differences? Maybe the latter involves some kind of geometric based graviton or something?

 

[atyy]

Here's a more recent paper that references the one in the OP.

http://arxiv.org/abs/1105.0566
Euclidean three-point function in loop and perturbative gravity
Carlo Rovelli, Mingyi Zhang
"In particular, the low-energy limit of the two-point function (the “graviton propagator”) obtained in this way from the improved-Barrett-Crane spin foam dynamics [7–12] (sometime denoted the EPRL/FK model) correctly matches the graviton propagator of pure gravity in a transverse radial gauge (harmonic gauge) [13, 14]. ... The obvious next step is to compute the three-point function. In this paper we begin the three-point function analysis. We compute the three-point function from the non-perturbative theory."

BTW, Rovelli and Zhang's calculations are based on a different spin foam proposal from that used in Bianchi et al's paper referenced in the OP. Bianchi et al base their calculations on the Barrett-Crane proposal, which is now thought to be wrong, because it did not match the graviton propagator. The Rovelli and Zhang paper is based on the EPRL/FK/KKL proposal which is reviewed in Rovelli's Zakopane lectures.

 

[waterfall]

What's odd in the two papers above is they mentioned about "Graviton propagator" instead of just "Graviton". Why is that. In your own understanding, why do they call it "Graviton Propagator"? Why "Propagator"?

 

[atyy]

What's odd in the two papers above is they mentioned about "Graviton propagator" instead of just "Graviton". Why is that. In your own understanding, why do they call it "Graviton Propagator"? Why "Propagator"?

In quantum theory, a propagator contains information about the probability that a particle will take a certain path.

 

[marcus]

http://arxiv.org/abs/0905.4082
LQG propagator from the new spin foams
Eugenio Bianchi, Elena Magliaro, Claudio Perini
(Submitted on 25 May 2009)
We compute metric correlations in loop quantum gravity with the dynamics defined by the new spin foam models. The analysis is done at the lowest order in a vertex expansion and at the leading order in a large spin expansion. The result is compared to the graviton propagator of perturbative quantum gravity.
Comments: 28 pages

http://arxiv.org/abs/1109.6538
Lorentzian spinfoam propagator
Eugenio Bianchi, You Ding
(Submitted on 29 Sep 2011)
The two-point correlation function is calculated in the Lorentzian EPRL spinfoam model, and shown to match with the one in Regge calculus in a proper limit: large boundary spins, and small Barbero-Immirzi parameter, keeping the size of the quantum geometry finite and fixed. Compared to the Euclidean case, the definition of a Lorentzian boundary state involves a new feature: the notion of past- and future-pointing intertwiners. The semiclassical correlation function is obtained for a time-oriented semiclassical boundary state.
Comments: 13 pages
excerpt:

In this limit, the two-point function we obtain exactly matches the one obtained from Lorentzian Regge calculus [38]. We therefore extend to Lorentzian signature the results of [13].​

[13] E. Bianchi, E. Magliaro, and C. Perini, Nucl. Phys.
B822, 245 (2009), arXiv:0905.4082 [gr-qc].

So Bianchi Ding is a sequel to Bianchi Magilaro Perini. It completes that initiative by doing the earlier analysis in the Lorentzian setting

 

[waterfall]

About LQG vs String Theory. Let me share an answer to a question I asked 2 weeks ago. I was asking if loop quantum gravity was also about spin-2 on flat spacetime. The answer is no, because LQG starts from curved spacetime and is simply a canonical quantization thing and the curvature is primary. While String Theory is actually spin-2 on flat spacetime with the spacetime curvature as secondary effect. Although one can say LQG can be modeled as spin-2 on flat spacetime by taking the GR part in LQG and doing it. But for full fledged LQG. It has spacetime curvature a priori and there is no flat spacetime underneath it. I think everybody agrees now and I assume there are no objections from anyone.

 

[atyy]

About LQG vs String Theory. Let me share an answer to a question I asked 2 weeks ago. I was asking if loop quantum gravity was also about spin-2 on flat spacetime. The answer is no, because LQG starts from curved spacetime and is simply a canonical quantization thing and the curvature is primary. While String Theory is actually spin-2 on flat spacetime with the spacetime curvature as secondary effect. Although one can say LQG can be modeled as spin-2 on flat spacetime by taking the GR part in LQG and doing it. But for full fledged LQG. It has spacetime curvature a priori and there is no flat spacetime underneath it. I think everybody agrees now and I assume there are no objections from anyone.

Although perturbative string theory has flat spacetime (or some other vacuum solution of the Einstein equation) as background, in some versions of non-perturbative string theory such as AdS/CFT the bulk space has no background and is completely free to fluctuate, so it is similar to LQG in this respect. In fact it is more radical, since LQG assumes space as a fundamental entity that fluctuates, whereas the bulk space is not a fundamental entity in AdS/CFT.

 

[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.

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.

 

[marcus]

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 of them.

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.

This is an excellent concise statement! Offhand I can't think of any place where these basic facts have been expressed more clearly.

Just to amplify, unnecessarily I think, any approach which starts with a classical fixed geometry and lays quantized excitations on top of it is called "perturbative". The primary aim of LQG, and so to speak its "claim to fame", is to strive for a "non-perturbative" quantum geometry.

In the loop approach there is no mathematical "thing" representing space or space time. There are only relationships among measurements. These are the quantum states of geometry (a state is a web of interrelated geometric measurements). It's a minimalist approach, in a sense, and makes for challenging mathematical problems.

 

[tom.stoer]

thanks!

 

[suprised]

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.

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.

Well I believe the story is roughly like follows. Strings, in the usual worldsheet formulation, assume some fixed background. Eg for the flat empty space, take this to be the Minkowski space described by a metric eta_mn, plus small fluctuations delta around it:

g_mn = eta_mn+ delta_mn

Essentially these delta describe gravitational waves whose quanta are gravitons. Other, curved backgrounds g_mn are equally possible, like eg. black holes, and one can expand around them analogously.

In LQG the “expansion point” is more like g_mn=0, so no spacetime is there. In order to recover GR as we know it, one needs to specify a backgound around which one wants to expand, essentially by putting some non-zero g_mn in by hand. Only then one can try to see what a graviton propagator etc looks like, namely by expanding around this ad-hoc background.

It is (for me) an open question what the admissible choices are, certainly flat space should be an allowed possibility. AFAIK is has not yet been proven that flat space is a solution to LQG at all. As often said, showing that that graviton propagator comes out right it is a necessary, but by no means a sufficient condition, as the 2-point function just captures the free theory.

Some ppl seem to claim the LQG ought to be background independent and thus be able to “dynamically decide itself” what the backgound is supposed to be; but how can this ever be possible without additional input. Namely in particular our whole universe should be an allowed solution, including the gravitational fields of us sitting in front of our computers, residing on Earth orbiting the sun; etc. Roughly the whole solution space of GR (plus whatever is necessary to make the theory quantum mechanically consistent) must be allowed vacua of any theory of gravity incl LQG.

The theory can’t know by itself what solution to choose, so it must be told by specifying boundary conditions or a boundary state in LQG. So roughly the necessary, fixed choice of g_mn in string theory is replaced in LQG by a choice of boundary conditions that induce the desired background g_mn in the low-energy limit. But what are then the rules that determine or constrain the admissible boundary conditions?

This is the landscape “problem” in disguise, since one has to specfify as extra data what semiclassical long distance limit g_mn one wants to talk about. This landscape “problem” is actually not a problem and never was. The multitude of possible solutions must be a property of any theory of gravity. So the question is what LQG can possibly add here. One thing LQG might be able to do in principle at some point (and which is not possible in the standard world-sheet formulation of string theory), is to compute transition amplitudes between certain such boundary conditions. However, not all boundary conditions, or space-times g_mn seem to be allowed, which restricts the usefulness of this kind of ideas. See the review by Rozali on background independence for further details.

 

[waterfall]

Although perturbative string theory has flat spacetime (or some other vacuum solution of the Einstein equation) as background, in some versions of non-perturbative string theory such as AdS/CFT the bulk space has no background and is completely free to fluctuate, so it is similar to LQG in this respect. In fact it is more radical, since LQG assumes space as a fundamental entity that fluctuates, whereas the bulk space is not a fundamental entity in AdS/CFT.

Are you saying the AdS/CFT is an alternative formulation of quantum gravity other than string theory and LQG, but AdS/CFT has anti-desitter spacetime with negative curvature which doesn't describe our universe at all. Unless you are saying that perhaps a future version of Ads/CFT with the right positive curvature can describe our universe?

 

[atyy]

Are you saying the AdS/CFT is an alternative formulation of quantum gravity other than string theory and LQG, but AdS/CFT has anti-desitter spacetime with negative curvature which doesn't describe our universe at all. Unless you are saying that perhaps a future version of Ads/CFT with the right positive curvature can describe our universe?

AdS/CFT is non-perturbative string theory. I don't know if a future version of something like AdS/CFT can describe our universe, but I hope there will be.

 

[waterfall]

AdS/CFT is non-perturbative string theory. I don't know if a future version of something like AdS/CFT can describe our universe, but I hope there will be.

I know AsD/CFT is non-perturbative string theory. But it describes negative curvature which doesn't describe our universe at all. So you are saying that if in the future we can't find any AsD/CFT that doesn't describe our universe, then the dual idea is refuted. Also an actual AsD/CFT would be sD/CFT but then the analogy or duality no longer holds because the antidesitter thing is done. So why is there high hopes reserved for the AsD/CFT programme?

 

[atyy]

I know AsD/CFT is non-perturbative string theory. But it describes negative curvature which doesn't describe our universe at all. So you are saying that if in the future we can't find any AsD/CFT that doesn't describe our universe, then the dual idea is refuted. Also an actual AsD/CFT would be sD/CFT but then the analogy or duality no longer holds because the antidesitter thing is done. So why is there high hopes reserved for the AsD/CFT programme?

Well, you study what you have, and learn from it. The first relativistic theory of gravity was Nordstrom's. Einstein studied Nordstrom's scalar gravity on flat spacetime, reformulated it as a curved spacetime equation, and formulated the correct tensor theory. Similarly, AdS/CFT is the first complete theory of quantum gravity, so maybe by studying it, we can learn what the correct theory is.

 

[waterfall]

Well, you study what you have, and learn from it. The first relativistic theory of gravity was Nordstrom's. Einstein studied Nordstrom's scalar gravity on flat spacetime, reformulated it as a curved spacetime equation, and formulated the correct tensor theory. Similarly, AdS/CFT is the first complete theory of quantum gravity, so maybe by studying it, we can learn what the correct theory is.

You are kinda saying that in the end, our universe may be described by something like "Michael Talbot's The Holographic Universe?

https://www.amazon.com/dp/0062014102/?tag=pfamazon01-20

It's bizarre but this seems to be what our quantum gravity programme is pointing to... because if we can find the right version of AsD/CFT, it would be "The Holographic Universe" as described Talbot.

 

[tom.stoer]

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]

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]

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]

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 website 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?

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]

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/why-are-there-gravitons-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]

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.

http://img715.imageshack.us/img715/963/doubletorusillustration.jpg

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]

http://img715.imageshack.us/img715/963/doubletorusillustration.jpg

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]

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.