LQC and string theory combined in one theory

[dhillonv10]

Hi all,

I have been reading a bit upon LQC and from what I understand this theory of QG makes some fundamentally different assumptions from string theory but has there any work been done that relates these two theories? I tried to look on google but no good results turned up there, ergo, I thought asking here might be a good idea. Also an extension to that question: how can one use/generalize the results proven/shown in LQC into string theory, I have something in mind for a research project but the results are shown to work out nicely in LQC, is there any possible way for me to use those over in the framework of String/M theory? From what I can tell, I don't think that's possible, but then again my knowledge is limited there might be a trick using dualities or something of that nature to do so. Thanks for your time.

- Vikram

 

[marcus]

It is not a high priority among LQG folks. LQG offers a quantum spacetime to build other stuff on, so the aim is to be able to have a more fundamental picture of space time that is not the usual Euclidean or Lorentzian framework or a fixed smooth manifold.

So as an exercise you could try to "do strings" on a LQG basis, instead of on the more usual differential manifold or flat space, but it might be more rewarding to do something else. Noncommutative Geometry has a version of the standard model and a number of people have been trying to do NCG on a LQG basis.
Also just plain field theory or QFT might be rewarding to figure out how to do.

But people have tried what you say! Thiemann had a paper doing strings with LQG in 2004. It had limited success.
http://arxiv.org/abs/hep-th/0401172
Then in 2009 three other people tried. Fairbairn, Noui, Sardelli
And there are some papers by Nieto I haven't looked at, along these lines:
http://arxiv.org/find/hep-th/1/au:+Nieto_J/0/1/0/all/0/1.

String research has not been making much progress lately and many of the researchers seem to have gotten out of the unification business and into other stringy and nonstringy research lines. This is not a criticism of stringy math. Interesting math but just hasnt been producing much new physics lately. So from the LQG standpoint it is not too interesting to implement string on LQG spacetime. There are other things with higher priority, although a few people do work on it now and then.

The most interesting recent work I know of is the 2009 paper I mentioned:
http://arxiv.org/abs/0908.0953
Canonical Analysis of Algebraic String Actions
Winston J. Fairbairn, Karim Noui, Francesco Sardelli
(Submitted on 6 Aug 2009 (v1), last revised 12 Sep 2009 (this version, v2))
"We investigate the canonical aspects of the algebraic first order formulation of strings introduced two decades ago by Balachandran and collaborators. We slightly enlarge the Lagrangian framework and show the existence of a self-dual formulation and of an Immirzi-type parameter reminiscent of four-dimensional first order gravity. We perform a full Hamiltonian analysis of the self-dual case: we extract the first class constraints and construct the Dirac bracket associated to the second class constraints. The first class constraints contain the diffeomorphisms algebra on the world-sheet, and the coordinates are shown to be non-commutative with respect to the Dirac bracket. The Hamilton equations in a particular gauge are shown to reproduce the wave equation for the string coordinates. In the general, non-self-dual case, we also explicit the first class constraints of the system and show that, unlike the self-dual formulation, the theory admits an extra propagating degree of freedom than the two degrees of freedom of conventional string theory. This prevents the general algebraic string from being strictly equivalent to the Nambu-Goto string."

==quote from introduction of Fairbairn Noui Sardelli==
A few years ago, Thiemann [15] reconsidered the Nambu-Goto string and proposed a quantisation of it using the techniques of loop quantum gravity (LQG) [16]. He showed that the LQG techniques, based on background independent quantisation, provides in particular a quantisation of the bosonic string in any dimensions, i.e., there is no need of critical dimensions for the quantum theory to be consistent. This result has sparked off some discussions [17] and certainly deserves to be understood deeper. We think that the algebraic formulation of the bosonic string is a better starting point to test the LQG techniques than the Nambu-Goto string for it admits a lot of similarities with the Ashtekar-Immirzi-Barbero-Holst formulation [18], [19] of general relativity. It is a first order formulation and possesses an Immirzi-type parameter. In fact, the main motivation of this article is to open an arena for a background independent quantisation of the bosonic string and to compare it to the standard Fock quantisation. Our goal is to pursue the line of research initiated by Thiemann in the context of the algebraic formulation of strings.
==endquote==

Ashtekar Barbero Immirzi Holst are core LQG names. This is definitely what you were talking about. Application of core fundamental LQG techniques to implement some type of strings.

 

[dhillonv10]

Thanks for your reply marcus, as I was waiting for someone to reply, I got Thiemann's paper that you had mentioned, it seems like an attractive idea. I am looking into the second paper that you mentioned but essentially what I want to be able to do is to use the work that you mentioned to make a bridge (metaphorically), first I want to show that this one concept in LQC is independent of the framework being used (strings or LQG) and then use the "Canonical Analysis of Algebraic String Actions" to construct the bridge and show that the concept I proved earlier to be independent can be written in terms of string theory, I am not explicitly going to write it in terms of string theory, just show that it can be done. Currently I am reconsidering the fact that do I even need to show the model-independence, if a concept can be translated using the bridge, doesn't that mean the concept is model-independent? or perhaps it may be that I am only looking at 2 models, other models may not be able to use what I show. Do you think this sounds plausible??

 

[marcus]

...Do you think this sounds plausible??

I am not one to judge. I don't know enough about you or just what model and bridge you have in mind.

Do you have a local newspaper or magazine where you live that might publish a piece of writing such as you plan?

If you live in India, you should realize that there are important LQG experts there, at several institutes and universities. You could get the email address and write to one or two of them asking their advice.

I can get you names, but cannot advise. To me the very idea of doing string with LQG tools seems extremely strange. There is a lot of serious work to be done with LQG. Making a side trip to teach "string tricks" to LQG seems like a waste of time. There is no single clearly formulated string theory that reproduces known particle physics, AFAIK.

But if you want, say what country, e.g. India, and I can find some names. There is also a google map of the centers of LQG research where you might get advice.

 

[dhillonv10]

Just curious, how did you know I am from India? actually I am living in the US right now, but anyhow I understand that there's no strict paper that one can refer to and say that's string theory. I would like to present this idea to you in its entirety so do you think we can talk over email? I don't want to just state what I have in mind on an online forum as anyone with access to google can see what it is. What do you think? I think its rather rude to just send someone a pm without asking them :)

 

[marcus]

Vikram, I can be of no use to you. I do all my QG discussing in this open forum. Shortage of time makes it necessary to compartmentalize like that. I do hardly anything with PM. As a retired person with notsogood eyesight there is only time to keep up with the QG and cosmology research literature and talk about it here.

All I can think of to do is write email to an Indian LQG I have in mind, let me get the name. Here is a list and map, maybe someone is close to you:

A list of QG researchers worldwide, mostly LQG but some other:
http://en.wikipedia.org/wiki/List_of_loop_quantum_gravity_researchers
Francesca's LQG world map:
http://maps.google.com/maps/ms?ie=U...985216139270436.0004843830d27f3e6c50e&t=h&z=0

Here are selected outreach articles at Abhay Ashtekar's Institute for Gravitation and the Cosmos, he is one of the founders of LQG but he would be too important to write email to as a amateur, I fear.
http://gravity.psu.edu/outreach/index.shtml

There is a man at Chennai named Ganashyam Date who has held a LQG school and written up his lectures:

Chennai basic LQG/LQC Lectures http://arxiv.org/pdf/1004.2952
(Ganashyam Date gave a series of beginning lectures on LQG which explain certain things most clearly--only 72 pages, not a textbook but could be used as supplemental reading.)

India is strong in QG. they even have an international QG conference there every 3 years. There are many experts in various places. If you write to G. Date and he does not respond. tell me and I will think of someone else. I think they should be excited that someone who is from India but who lives abroad would still write back to them for advice about a QG idea.

Madhavan Varadarajan at the Raman Research Institute at Bangalore also comes to mind. Ashtekar just co-authored a paper with him, if I remember right. But the top are not always the best to write for advice---they may be too busy.

 

[suprised]

I have been reading a bit upon LQC and from what I understand this theory of QG makes some fundamentally different assumptions from string theory but has there any work been done that relates these two theories?

It is a bit like asking: "since lattice gauge theory is a way to regularize gauge theories, and string theory describes gauge theories as well, incl their dual relationship with branes and gravity, so why does one not combine lattice gauge theory and string theory?"

The answer is that the goals are different; lattice gauge theory is more of a non-perturbative discretizing technique for studying certain features like confinement and hadron masses, on the other hand it obscures other issues, so it is rarely used in a broader context. The situation of LQG is somewhat analogous, one difference being that lattice gauge theory supposedly provides a more rigorous non-perturbative definition of the theory, while LQG seems still far from being well-defined.

 

[tom.stoer]

... while LQG seems still far from being well-defined.

Why?

I mean I would agree with "far from unique", but it seems to provide a rather consistent framework. Perhaps it's like "gauge theory" (there is not one single gauge theory and perhaps one should not expect that there is one LQG)

 

[suprised]

Why?

I mean I would agree with "far from unique", but it seems to provide a rather consistent framework. Perhaps it's like "gauge theory" (there is not one single gauge theory and perhaps one should not expect that there is one LQG)

Well read the article of Alexandrov and Roche, for example. It's really very far from the rigor by which lattice gauge theory can be defined. But I would expect this to improve over time. With regard to uniqueness, indeed there are plenty of different schools that partly contradict each other, so it seems even the experts cannot agree with each other as to what the right approach is...thats quite different to gauge theory, which is uniquely defined once the gauge group is given, isn't it?

 

[tom.stoer]

Looking at (e.g.) Thiemann's papers I would say that they are rather close to mathematical rigour; of course this is not standard (it wasn't for gauge theory, either - when it comes to quantization). Well, yes, it's still work in progress.

 

[suprised]

Looking at (e.g.) Thiemann's papers I would say that they are rather close to mathematical rigour; of course this is not standard (it wasn't for gauge theory, either - when it comes to quantization). Well, yes, it's still work in progress.

Yes work in progress, like any other approach. But one shouldn't turn things around and make LQG appear as if it would be close to an established theory. As to the quantization, this approach does not even correctly reproduce the harmonic oscillator (whose correct quantization is pretty well experimentally established, to say the least). So that's why few people believe in this non-standard quantization.

 

[marcus]

... With regard to uniqueness, indeed there are plenty of different schools that partly contradict each other, ...

Let's try to be more precise about this. Maybe we can all agree at least on the number and kind of schools

I would be interested to know how Tom sees it, I see currently TWO main schools. Or rather it is not schools but two separate theories being developed. The LQG researchers can jump around freely and work on this or that idea--postdocs have been bouncing back and forth between Marseille and Erlangen and PennState and Potsdam and Nottingham. They cross-collaborate on papers. I don't think you can put the people into strict camps.

But there are definitely two main theories. Incidentally one of them is not discussed in the Alexandrov-Roche paper cited (I looked in vain for the most basic definitions and equations). I guess Alexandrov's "CLQG" theory (which we hear a lot about in the A-R paper) might be considered a third theory. But since it is a solo effort at this point I will just count the main two theories.

One theory is most recently presented here by Livine:
1101.5061 "The Spinfoam Framework for Quantum Gravity"
and in a definitive review article 1012.4707 by Rovelli "Loop Quantum Gravity..."

It has a clear formulation in three equations. (1, 2, 3) in the definitive December review.
It appears to be empirically falsifiable, at least some phenomenologists think so, because the bounce should produce features in CMB polarization one can find or not find. There are still plenty of things to work out in this new LQG formulation---Livine discusses several at length, Rovelli lists many open questions.

The other main theory is a canonical quantization of General Relativity that I believe is being pursued by Thomas Thiemann at Erlangen. There are other co-workers but I think his would be the most representative name. A former student of Thiemann named Hanno Sahlmann has written a pedagogical introduction and has several times been the main presenter at workshops/conferences (so to speak in Thiemann's place.) I don't know what to pick as a recent survey or review describing this theory. It would be nice to be able to designate a definitive status report (by Thiemann if possible) and I would welcome suggestions. Is his book still a fair account of the current state of the theory?

I don't know if this picture of there being two main LQG theories is acceptable to everybody. To me it seems a somewhat objective view of how the field actually divides up. For instance, if you go by books, there are two main books. Rovelli 2004 and Thiemann 2007. Rovelli has now reformulated, 1012.4707 is his definitive presentation, in effect replacing the 2004 book. There are already many papers based on the new formulation.

Maybe Thiemann's 2007 book is still definitive of the theory he and his associates are developing. Another major figure, Ashtekar, currently works on the cosmology application. Has recently been linking up with Rovelli's version by way of spinfoam cosmology---there's growing collaboration and two PennState PhD's are going postdoc to Marseille this year. LQG cosmology research is linked to both the current main theories.

 

[tom.stoer]

As to the quantization, this approach does not even correctly reproduce the harmonic oscillator (whose correct quantization is pretty well experimentally established, to say the least). So that's why few people believe in this non-standard quantization.

There are good reasons why in the context of GR this quantuzation makes sense (e.g. diff. inv.) I can't remember the details, so I have to find some references.

And there must be a different approach to quantization b/c standard approaches seem to fail. Any theory of QG has some new ingredient: string theory, SUGRA, LQG. As long as we have no experimental guidelines we cannot know which one is correct - unfortunately.

 

[marcus]

..., while LQG seems still far from being well-defined.

Your information seems to be out of date. LQG is mathematically welldefined. See the extensive (pedagogical) review by Livine http://arxiv.org/pdf/1101.5061
or the more condensed survey by Rovelli http://arxiv.org/pdf/1012.4707

Why?

I mean I would agree with "far from unique", but it seems to provide a rather consistent framework. Perhaps it's like "gauge theory" (there is not one single gauge theory and perhaps one should not expect that there is one LQG)

I would agree with Tom. There is no reason to say that LQG is not well-defined but one can certainly say that it is "not unique". There are several versions! It has both the combinatorial spinfoam version presented in the two recent review papers, which is exquisitely and concisely defined, and it also has an earlier canonical development which was discussed in the Alexandrov Roche paper, for example, and which is also described in Livine's January 2011 I referred to. Livine gives, in my view, quite an adequate account of the problems with the earlier canonical development.

Well read the article of Alexandrov and Roche, for example...

That does not discuss the combinatorial spinfoam formulation, which appeared in 2010, in several papers, but was not covered by A&R. I don't think you can say "not welldefined" on the basis of a reference to A&R.

... So that's why few people believe in this non-standard quantization.

The current formulation is not based on any nonstandard quantization. Various earlier approaches have served as inspiration, intuitive guides, heuristic. You seem to be quarreling with something in the past history.
It would be helpful if you could bring your perception of LQG up to date and so have a real discussion, because I am sure your insights and comments would be very interesting.

 

[dhillonv10]

Thanks for your helpful replies everyone, I do understand that LQG and string theory are addressing two totally different questions although they are both trying to define a theory of QG. I would however say that over the years LQG has made considerable progress, what I wanted to do is that through some mechanism like Thiemann's that some results that are reached in LQC *can* be translated to bosonic string theory. Its an elementary step and I have no ambition of combining the two theories, just trying to show that some results in one theory can be translated to another.

 

[tom.stoer]

Of course there are indications that both theories point towards some underlying principle of nature, especially that both theories indicate something like a holographic principle for black hole surface degrees of freedom and Bekenstein-Hawking entropy. So they are somehow related - even if I am not able to say in which way.

Then there is the notorious problem of background independence (explicitly incorporated in LQG) in string theory which according to some reasoning discussed in this forum is hidden in a complicated way in the AdS/CFT correspondence - which is unfortunately limited to AdS and should be extended to dS and perhaps other spacetime geometries. That means that there may be methods or at lwast ideas from LQG which can be transferred to string theory.

String theory seems to produce a framework for unification of all known interactions (or better: of all mathematically conceivable interactions restricted by certain consistency conditions) whereas LQG can't address this unification issue at all (most attemts I have seen so far only add matter interactions by hand).

Unfortunately I don't see how string theory can help here.

 

[suprised]

I would be glad to see how the Einstein eqs of GR come out of LQG; before this is not convincingly demonstrated, I would consider this research program as tentative. Adopting strange quantization rules to rescue the theory makes it even less concinving. I hear every year that important breakthroughs had been made, but I really fail to see that it works. I have nothing against work in progress, but am inclined against this permanent overselling of a theory that changes all the time and hasn't produced hard results apart from promises. Dismissing string theory as just "interesting math" as marcus is routinely doing turns things really on its head.

 

[marcus]

Hi Suprised, thanks for addressing me (more or less) directly. I think it would be interesting to be able to discuss LQG with you if you would learn something about it. By the general tenor of your comments over many months you give the impression that you have only second-hand and sometimes out-of-date knowledge of the current developments in the field.

I could easily be mistaken but I think of you as an actual string theorist, someone in the French/German-speaking part of the world, at a university, actually working in the general stringy areas of research. So your perspective, if it were up-to-date and well-informed would be of considerable interest and value to an observer such as myself.

It would be very easy, I think, for you to get au courant on LQG developments. At last there have appeared some good review articles presenting the theory in a concise, simple, well-defined way. It would not take you much time, I suspect, to read them and gain a basic understanding.

They also contain plenty of frank admission of shortcomings! So if you are interested in gathering reasons to dismiss what your Lqg colleagues are doing, you can find plenty of direct quotes. Direct quotes from the Lqg researchers themselves, as of circa Jan 2011, how this or that approach has encountered serious obstacles and gotten bogged down.

But at least we could be talking on the basis of actual page references to real Lqg papers and on the basis of the actual status of the field.

I will give the same suggestions as last time:
Rovelli's reviews
http://arxiv.org/abs/1012.4707
http://arxiv.org/abs/1010.1939
Livine's review
http://arxiv.org/abs/1101.5061

The main review is the one by Rovelli, December 2010, but there are some points not covered there about recovering classical GR which you can find on page 5 (and half a column on page 6) of the shorter October 2010 review.

 

[atyy]

There is nothing out of date about suprised's statement that no Einstein field equations have come from LQG.

 

[marcus]

That's certainly true Atyy! I have been responding to Sup's earlier statement that the theory was not well-defined, and his citation of the Alexandrov Roche paper.

It is fine for him to move the goal-posts now, and say that his problem really is that Einstein GR has not been derived!

But it would be even better if Suprised would take a few minutes or hour to get actually directly informed, so we could all be talking about the same thing.

You, I can infer from your comments, have read both the Livine review and the Rovelli December 2010 article.

 

[atyy]

Well, maybe the theory is also not well-defined in the sense that if EPRL/FK should fail to give the right classical limit, there are other spin foam models that have not been excluded. eg. Livine's comment (p62) "But beyond this, we need to identify a family of spinfoam models, which would be stable under coarse-graining."

 

[tom.stoer]

I haven' seen LQG reproducing GR field equations, but LQC nicely reproduces the (spherical symmetric) equations (with quantum corrections), cosmological solutions and black hole solutions at low density.

 

[marcus]

Well, maybe the theory is also not well-defined in the sense that if EPRL/FK should fail...

Atyy, One can always speculate about other versions. But I am interested to know what Suprised says, about the current version that you get in Rovelli's review.

It seems to me obvious that it is both testable and (by the prevailing standards in theoretical physics) well-defined.

The mathematical definition is simple definite and concise. One can calculate. According to phenomenologists it is, moreover, falsifiable using instruments which one knows how to build (just a bit better than the currently operating ESA Planck spacecraft ).

I would hope to have some response from Super on this, although I value your comments very much too.

As far as indications of the correct classical limit go, that is also something that one can discussed based on a current reviews such as http://arxiv.org/abs/1012.4707 and http://arxiv.org/abs/1010.1939 , because those indications are extensive and gradually accumulate. The evidence is not complete but it is also not negligible---so one has to read the articles to get a sense of the current status.

The short (8 page) review http://arxiv.org/abs/1010.1939 actually has more on that. See page 5 and the first halfcolumn on page 6. It is the section called Section V: Relation with GR.

 

[Schreiberdk]

I must say that i have run upon something similar, though not involving LQG but rather String Field Theory on Causal Dynamical Triangulations - spacetime, in this paper:

http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.0719v1.pdf

 

[suprised]

marcus, I am just referring to Rovelli's recent review paper, assuming that it summarizes the current state of affairs. And there it is explictly stated that the emergence of GR out of LQG is conjectured, not proven.

There is nothing wrong with that, given that LQG is work in progress, like any other field of current active research. I am inclined to believe that for some variant or other of LQG, some reasonably convincing (nobody wants to demand rigor) argument that GR pops out will be made in the future. I am happy to assume that LQG works out for gravity much like lattice gauge theory nicely regularizes gauge theories.

What I object is to misrepresent the state of affairs, in constantly arguing that string theory would mainly be of mathematical interest, and not of physical interest. In fact, that the Einstein eqs of GR pop out almost for free from the string action has beeen known for decades, and comparing this to the situation of LQG, it is clear that string theory is decades ahead of LQG, and what you try to convey, marcus, is turning things on their head. And even if LQG would work out at some point, it is totally unclear whether any of the "problems" string theory has and is criticized for, like a landscape of solutions (which is actually desired), would be addressed or solved. That's just a hope; fine, all of this business is hope-driven, but it should be stated like that and not be oversold all the time.

 

[marcus]

marcus, I am just referring to Rovelli's recent review paper, assuming that it summarizes the current state of affairs...

I think you mean the December review paper http://arxiv.org/abs/1012.4707. I also am finding this February 2011 paper helpful http://arxiv.org/abs/1102.3660.
I think your comment indicates that you have read or at least taken a long look at the December review paper. I am encouraged and reassured by our having the same thing, something definite, in front of us.

In my posts to you I have been reacting to your statement that LQG is not well-defined.
I think by the standards of theoretical physics it is well-defined, and (judging by what phenomenologists are saying) testable.

There are variants, but the recent formulation (which you refer to) seems increasingly representative and authoritative. As such, the theory is defined in some 2 to 5 pages.
Please ask if you want references to the phenom. literature explaining the theory's testability.

Traditionally, we require of a physical theory that it be definite, concise, and testable.
There should be one well-defined compact formulation which is potentially falsifiable by available means.

You are understandably sensitive about string theory, so I would prefer not to introduce comparisons.

=====================

About GR, I have a bit more to say, but most of it can wait. Let's keep in mind that GR is structured differently from QFT. It is not constructed on a fixed geometric background, and it is not perturbative. GR is a theory about dynamic geometry.

I don't think one can honestly say that a welldefined concise testable physical theory "recovers GR" or "reproduces GR" if that theory is constructed on a fixed geometric background, or is formulated perturbatively. Simply exhibiting a graviton in a fixed metric context would not seem very convincing.

However LQG as currently formulated is not perturbative (at least in the usual sense of conventional perturabation theory) or dependent on a fixed prior metric. One can set up boundary conditions, to force approximate flatness, and in that special context analyze in graviton terms.

The review paper you cited devotes several pages to summarizing this and other circumstantial evidence that the theory's large distance limit is GR. This is summarized in the "classical limit" paragraph on page 19 (which I expect you read.)

Looking forward to your reply, Suprised. There is, I guess, a lot more to say.

 

[tom.stoer]

In fact, that the Einstein eqs of GR pop out almost for free from the string action has beeen known for decades, and comparing this to the situation of LQG, it is clear that string theory is decades ahead of LQG ...

String theory is ahead in the sense of the classical / low energy limit and regarding unification. I think this is clear and widely accepted - w/o doubt. What the LQG community claims is that the Planck scale regime is much better under control using the LQG approach b/c it avoids smooth, classical background spacetime from the every beginning (in the new formulation).

I wouldn't say that X is ahead of Y: which vehicle is better, a Ferrari or a truck? it's no so simple, is it?

And even if LQG would work out at some point, it is totally unclear whether any of the "problems" string theory has and is criticized for, like a landscape of solutions (which is actually desired), would be addressed or solved. That's just a hope; fine, all of this business is hope-driven, but it should be stated like that and not be oversold all the time.

I think the LQG community doesn't claim that. They strongly believe that all problems related to background independence - perhaps even "topogical background dependence" due to global R*M³ - simply do not exist - the latter ones may go away in the new formulation that comes w/o any manifold at all.

One must distinguish between physical problems (what is the Planck scale nature of spacetime? how do black holes look like in QG?) and artificial problems (how do we establish an anomaly-free constraint algebra in LQG?) specific to a certain approach. The latter ones are of minor interest outside the community! Background dependence is not a problem of nature but a problem of a specific approach. Therefore claiming that it is solved in LQG is of minor interest as it has not been created by nature but by string theory.

Of course LQG suffers from a "landscape problem" as well. LQG does not explain why one should start with SL(2,C)! One could start with any Spin(n), SU(n) or even Sp(2n) - but one doesn't - as one knows that Spin(4) is natural in 4-dim. But of course this uses "4" as input - that should better be produced as an output of the theory! It is still not properly understood why one gets back a 4-dim. spacetime based on graphs or spin foams on which no dimension (besides some dynamical definition based on diffusion or something like that) can be defined. It seems to be that this is not directly related to Spin(4)! Therefefore it would be interesting study other Spin(n), SU(n) or Sp(n) spin networks, determine their IR limit and check for their dimension! It is simply b/c of this explanation (why 4 dimensions?) that I don't like all the "triangulation-based" approaches. A clear starting point is an abstract spin network w/o any pre-defined or "hidden" dimension. So yes, LQG has a landscape problem, but just like ordinary gauge theory not on the level of solutions but on the level of the definition of the theory.

 

[suprised]

I don't think one can honestly say that a welldefined concise testable physical theory "recovers GR" or "reproduces GR" if that theory is constructed on a fixed geometric background, or is formulated perturbatively. Simply exhibiting a graviton in a fixed metric context would not seem very convincing.

I was talking about obtaining Einsteins Eqs of GR (or equivalently, the Lagrangian of GR), out of whatever black box. That's a minimum requirement for any theory claiming to describe gravity.

My point is that one should not talk dismissively about a theory that can do this, in favor of another one for which this remains a conjecture/hope. OK?

 

[suprised]

So yes, LQG has a landscape problem, but just like ordinary gauge theory not on the level of solutions but on the level of the definition of the theory.

Ultimately one also needs to describe the matter content (standard model) with all parameters taking the correct values. I guess few people would consider this option, namely taking the standard model simply as definition of the theory, as satisfying.

More seriously, it is clear that these questions can only be addressed after coupling LQG to matter, and extensively analyzing the options and constraints. That's still a long way to go, perhaps even outside the scope of LQG.

What I mean is that if LQG is akin to a lattice regularization of gauge theory, it might not be able to ever determine the parameters of the standard model including gravity, much like the framework of lattice gauge theory is not able to fix any structure of the standard model.

 

[Haelfix]

One of the usual confusions on this forum is that there is something wrong with formulating a theory perturbatively.

When the fact is they have it really upside down. If you can't formulate a theory perturbatively at least in principle, your theory is in quite bad shape. It means that it has no (or no known) classical solutions and therefore cannot hope to ever have a correspondance principle.

There is nothing conceptually different about GR and of course it can be formulated perturbatively (indeed it is likely utilized more that way in practise)

Now it is true that some questions and some physical phenomena are notoriously opaque in the pertubative framework (in fact sometimes almost everything can be obscured, like eg low energy QCD), which is why we like to have nonperturbative formulations as well. But the existence of the latter does not preclude the existence of the former, quite on the contrary.

The point that is often lost here, is that any nonperturbative framework must at the very least answer every question that the perturbative framework has answered and the two answers must always coincide (again at least in principle, sometimes calculating things can be difficult)!

Every LQG theorist knows this, which is why they take semiclassical calculations like Hawking radiation seriously. However it seems not to have filtered through here.

The bottomline is that many or most of the questions and problems that currently plague the string theorists, and/or the supergravity people won't simply go away, even if those theories are falsified. Whether its about the renormalization infinities or the unification paradigm, or the nontrivial vacuum structure of nature, extra dimensions, holography etc etc One way or the other, any final theory of quantum gravity will have to provide answers and explain what is physically going wrong (and right) with these 'naive' perturbative approaches.

The flip side is that we can use what we know from the perturbative formalism, to constrain any final theory.

 

[tom.stoer]

If you can't formulate a theory perturbatively at least in principle, your theory is in quite bad shape.

I think this is simply wrong.

Think about QCD. There is a scale Lambda at which perturbation expansion breaks down. But from the very beginning there is no reason why there should be a domain which allows for a non-perturbative treatment at all. We were lucky, that's all. It can very well be that a theory does not allow for a perturbative definition; gravity might be a good example as indicated by the AS approach.

 

[Haelfix]

I think you have a typo in the above post.

Now, there may or may not be a parameter that you can expand around (sometime's we simply invent one when there are no small dimensionless quantities around, like eg 1/N expansions) but in all physical theories ever studied there has always has been at least some sort of phase where such an analysis is possible (although I might have to bite my words, b/c it occurs to me that there are lower dimensional conformal theories that have no classical limit).

The question though, is what are we to make of a theory that has no saddle point solutions at all, at any scale? Particularly one that aims to describe gravity, where the classical limit better exist.

Incidentally the asymptotic safety program is entirely based on the existence of a suitable perturbation series (they typically expand around solutions that have spherical or cylindrical symmetry)..

 

[marcus]

One of the usual confusions on this forum is that there is something wrong with formulating a theory perturbatively.

When the fact is they have it really upside down. If you can't formulate a theory perturbatively at least in principle,...

Haelfix I think you may be the one who has it upside down. I don't know who you meant, your post is a bit vague and innuendo-ish, but I don't argue that there is something wrong with perturbative expansions. It can be the right tool to use.

Sometimes it's possible use pert. expan. and sometimes not. By definition the perturbation approach means you have to start with a known SOLUTION (say of a simpler problem) and introduce a small change, and normally you expand in a series around the simplecase solution.

Sometimes you can do that and sometimes not. When you can it is great. What is lame or limiting is when you ONLY have a a perturbative formulation.

I suppose people with exclusive exposure to QFT might think pert. series is the only kind of expansion, the only kind of approximation you can calculate with. They may automatically think in those terms.

Have a look at page 16 of 1102.3660. It is the section on expansions .
LQG has THREE different nonperturbative expansions. Each has a different way of truncating.
1.graph expansion
2. vertex expansion
3. large distance expansion.

When you are working near the black hole singularity or the cosmological singularity
there is no solution that you can expand around so pert. series is simply not an option!
However there are special situation where, with LQG, you force an approximately flat solution to exist. Perturbative analysis would surely be welcome there, because appropriate.

The important thing is that the theory is ABLE to be formulated non-pert. and in fact is so, and that one can calculate using a variety of expansions and approximations. Spin networks and foams are tools to do just that.

 

[marcus]

A theory which can only recover GR in a perturbative setting (expanding around a fixed, e.g. flat, metric geometry) cannot properly be said to recover GR.

Because GR is a theory of fully dynamic geometry. It misses the essential thing about GR.

 

[atyy]

A theory which can only recover GR in a perturbative setting (expanding around a fixed, e.g. flat, metric geometry) does not really recover GR.

Because GR is a theory of fully dynamic geometry.

So it would be necessary for LQG to recover the dynamics of GR. But there is no indication of this yet.

 

[marcus]

But there is no indication of this yet.

Actually there are many indications, Atyy, and more continue to accumulate.

Pages 11-17 of the February review paper have some discussion relevant to this.
BTW did you see that version 2 of 1102.3660 has been posted just in the last few days.
Looks like typos corrected and some additional explanation inserted.

 

[atyy]

Actually there are many indications, Atyy, and more continue to accumulate.

Pages 11-17 of the February review paper have some discussion relevant to this.
BTW did you see that version 2 of 1102.3660 has been posted just in the last few days.
Looks like typos corrected and some additional explanation inserted.

I think there are none, the updated version of 1102.3660 notwithstanding.

http://arxiv.org/abs/1101.5061 , p54: "On the negative side, the actual setting for practical calculations of the spinfoam graviton correlations has been much too simple up to now. These basic calculations were done mainly for a single 4-simplex, which is indeed the simplest space-time triangulation. They typically do not involve summing over bulk internal associated to internal spinfoam vertices. Thus, these calculations don’t allow to truly test the quantum gravity dynamics defined by the spinfoam models and the gluing of 4-simplices (“space-time atoms”) used to construct the amplitudes. They should be considered as kinematical checks."

 

[marcus]

Thanks for the page reference! I see you chose a negative paragraph out of Livine's tutorial/survey of spinfoam LQG. Here is the rest of the page 54 section surrounding it, to give context and provide balance:

==quote Etera Livine http://arxiv.org/abs/1101.5061 ==
3.1.2 The Practical Calculations of the Graviton Propagator

This framework for the spinfoam graviton propagator is based on a very simple setting. There has been a lot of research work done on this subject. Results are, up to now, both full of promise and very restricted.
On the positive side, we are able to compute systematically at leading order the spinfoam graviton propagator at large scale (for large values of the boundary areas j∂) for all the spinfoam models which have been defined. We have even developed techniques to extract (in principle) all quantum corrections of arbitrarily higher order (interpreted as “loop corrections”). This leads to recover the proper scaling of Newton’s law for gravity, with the gravitational potential going as the inverse distance, and even the correct spin-2 tensorial structure of the graviton (correlations) for specific spinfoam models. We even understand the relation between the spinfoam path integral and Regge calculus at large scale. The short scale behavior has also been investigated. It appears that the graviton propagator is regularized (as expected) by quantum gravity effects and that we have the emergence of a dynamical minimal length scale close to the Planck scale. All this has been tested analytically and numerically.

On the negative side, the actual setting for practical calculations of the spinfoam graviton correlations has been much too simple up to now. These basic calculations were done mainly for a single 4-simplex, which is indeed the simplest space-time triangulation. They typically do not involve summing over bulk internal associated to internal spinfoam vertices. Thus, these calculations don’t allow to truly test the quantum gravity dynamics defined by the spinfoam models and the gluing of 4-simplices (“space-time atoms”) used to construct the amplitudes. They should be considered as kinematical checks. It thus remains a challenge to go beyond the single 4-simplex and work with refined space-time triangulations, which would allow local fluctuations of the curvature in the bulk.

Here is nevertheless a (almost-exhaustive) list of the works done on the programme of the spinfoam graviton propagator:

• Definition of the framework [81, 82].

• Analytical study of the asymptotic ansatz for the spinfoam vertex amplitude in order to recover at leading the correct tensorial structure for the graviton propagator [83, 84, 85].

• Group integrals techniques to compute explicitly analytically the graviton propa- gator for the Barrett-Crane model (generalizable to arbitrary spinfoam models ex- pressed in the connection representation) [88].

• Numerical investigations of the behavior of the graviton propagator for the Barrett- Crane model, both for the large scale and short scale, both at leading order and at next-to-leading order (first order quantum gravity corrections) [89, 90]

• Calculations of the asymptotics of the graviton propagator for the EPRL-FK spin- foam model [91].

• Definition of a 3d toy model using the Ponzano-Regge model [92], numerical investi- gations and development of the tools to compute the full expansion of the correlations and solve the model analytically [93, 94].

• Study of the propagation of coherent wave-packets of geometry within a 4-simplex [95].

• Analytical and numerical study of the asymptotics of the spinfoam vertex amplitude relevant to the calculations of the large scale behavior of the graviton propagator, in 3d [96, 97] and in 4d for both the Barrett-Crane model [98] and the EPRL vertex amplitude [99, 100, 101].

• Discussion of the potential use of the recursion relations satisfied by the spinfoam vertex amplitudes to the computation of the graviton correlations and to derivation of Ward-Takahashi identities for spinfoam amplitudes [97, 102].

• Tentative calculations of the 3-point correlation functions [103].

What is very nice about this framework is that it provides a physical interpretation to the correlations computed using spinfoam models and in particular shows how to recover the classical Newton’s law for gravity from our complicated and intricate model for a quantum gravity path integral. Moreover, we can actually compute analytically these correlations, plot them numerically, check that everything is consistent, and see explicit the first elements of the spinfoam dynamics with our own eyes.

However, progress in this direction is completely coupled with necessary progress that needs to be done on the coarse-graining and renormalization of spinfoam models. Indeed, we need to be able both to repeat the same graviton correlation computations for more refined and complex bulk triangulations and to say something about the non- perturbative sum over all 2-complexes. The main hope for this is put in exploiting the group field theory formalism and studying its renormalization as a quantum field theory.

3.2 From Spinfoam Amplitudes to Non-Commutative Field Theory
Besides looking at the quantum gravity corrections to the gravitational interaction, an- other way to probe the semi-classical regime of quantum gravity and extract potential...
==endquote==

 

[Haelfix]

A theory which can only recover GR in a perturbative setting (expanding around a fixed, e.g. flat, metric geometry) cannot properly be said to recover GR.

Because GR is a theory of fully dynamic geometry. It misses the essential thing about GR.

This statement is incorrect, and I believe its the root cause of most of the inordinate amount of confusion on this forum. You can always recover GR from perturbation series, it is done for instance on page 435 of MTW, or alternatively any of the classic texts by Deser et al, or by Weinberg in his Gravitation text. It has nothing to do with whether GR is dynamical either, it is simply consistency criteria. Said another way, the full nonlinearity of GR can be bootstrapped starting merely from a weak field expansion.

In other words, if historically Einstein had only known about perturbation theory and linearized gravity he would have been able to derive the full equations from a few mild observations.

"Quantum Gravity" may however be less favorable to perturbation series in that there could be regimes and objects that perturbation series cannot see. For instance, pertubation series apparently misses whatever it is that unitarizes black hole physics, even when done around a nice and 'under control' slice, far away from any violent curvature regimes (where you are not allowed to perturb around).

The important thing is that the theory is ABLE to be formulated non-pert..

The important thing is that you should be able to do both! What is also wrong, is to simply ignore whatever insights perturbation series tells you.

 

[atyy]

This statement is incorrect, and I believe its the root cause of most of the inordinate amount of confusion on this forum. You can always recover GR from perturbation series, it is done for instance on page 435 of MTW, or alternatively any of the classic texts by Deser et al, or by Weinberg in his Gravitation text. It has nothing to do with whether GR is dynamical either, it is simply consistency criteria. Said another way, the full nonlinearity of GR can be bootstrapped starting merely from a weak field expansion.

Can this really be said to be due to perturbation series? I think it is fair to say perturbation series alone does not recover GR, since one needs the consistency criteria, which are imposed in addition to the perturbation series. Eg. on perturbation series alone, can one get below the event horizon of a black hole?

 

[marcus]

Can this really be said to be due to perturbation series? I think it is fair to say perturbation series alone does not recover GR, since one needs the consistency criteria, which are imposed in addition to the perturbation series. Eg. on perturbation series alone, can one get below the event horizon of a black hole?

I certainly agree! We are talking about chapter 18 of MTW. "Weak gravitational fields" They do not say that one can recover GR solely from its weak field version. It is an interesting part of the book, if anyone is interested they can take a look at, say, pages 420-440 on line. Go here:
http://www.google.com/search?client...hl=en&tab=wp&bav=on.1,or.&fp=1d35b10d87834f0f

Or just go to google-books and search "weak gravitational fields". Click on the first hit.

Box 18.1 on page 437 is interesting: it compares the Einstein (geometric) derivation of GR with the "spin-2" derivation (weak field+bunch strong assumptions to make the bootstrapping work)

==quote page 436==
Just as one can "descend" from general relativity to linearized theory by linearizing about flat spacetime (see below), so one can "bootstrap" one's way back up from linearlized theory to general relativity by imposing consistency between the linearlized field equations and the equations of motion, or, equivalently, by asking about [a ton of stuff, five separate technical issues which they list, and then say "and so on..."]
...But because the outlook is far from geometric (see Box 18.1), the details of the derivation are not presented here...
==endquote==

 

[atyy]

In the context of perturbative GR being a limit of string theory, shouldn't the consistency conditions come from string theory itself, and lead to string theory, rather than GR?

Also, I can imagine that a non-perturbative formulation can consist of perturbative formulations, as long as the different formulations overlap uniquely, and cover the whole space. I'd imagine the various string dualities do this at least partially.

 

[Haelfix]

Can this really be said to be due to perturbation series? I think it is fair to say perturbation series alone does not recover GR, since one needs the consistency criteria, which are imposed in addition to the perturbation series.

This paper by Deser is one of the standard references for how you bootstrap up from the weak field equations to the full nonlinear form: arXiv:gr-qc/0411023 or alternatively Weinberg 1965 Phys. Rev. 138, B988 (1965).

There are of course a number of very simple assumptions (that turn out to be empirically correct) to mathematically go from linearized gravity to the full nonlinear form. However, that shouldn't be a surprise to anyone and indeed it is part of the power and beauty of General relativity that it admits multiple isomorphic mathematical formulations.

What is unclear (and likely untrue) is whether you can do this same thing in pure quantum gravity. The reason is that you most likely do need some sort of UV completion, but that is a story for another day.

 

[Haelfix]

In the context of perturbative GR being a limit of string theory, shouldn't the consistency conditions come from string theory itself, and lead to string theory, rather than GR?

To get GR from say the bosonic string, you need basically only invoke Weyl Invariance or the vanishing of the beta function. This is a consistency criteria of string theory proper.

 

[suprised]

To get GR from say the bosonic string, you need basically only invoke Weyl Invariance or the vanishing of the beta function. This is a consistency criteria of string theory proper.

That's exactly my point, that marcus apparently does not appreciate. I do not mean here just graviton scattering processes around a backround; I mean that the whole lagrangian of GR: L = Sqrt (g) R + corrections, pops out as consistency condition in string theory (by expanding around flat space, graviton scattering can be recovered if one wishes to do so).

Thus string theory acts as a black box:

input = 2d CFT plus ghosts, integrated over 2d fields, moduli
output = Einstein gravity (plus Planck scale suppressed corrections).

That LQG provides an analogous black box, once properly defined and understood, is not unlikely but not still proven after some 25 years of research.

Thus, as of today there is no point to dismiss string theory as "purely mathematical" construct, when comparing both approaches!

 

[tom.stoer]

But this is still a pure argument.

You quantize string theory on flat spacetime and get back (as a consistency condition) the Einstein equations. But you are still on flat spacetime; spacetime hasn't become fully dynamical in that setup. It's is still a consistency condition for the classical background.

I do not doubt that this is a hint towards some underlying truth but it is definately not this truth itself.

 

[suprised]

But this is still a pure argument.

You quantize string theory on flat spacetime and get back (as a consistency condition) the Einstein equations. But you are still on flat spacetime; spacetime hasn't become fully dynamical in that setup. It's is still a consistency condition for the classical background.

I do not doubt that this is a hint towards some underlying truth but it is definately not this truth itself.

It is not necessariy a flat backround; but the formalism is on-shell only and space-time is not dynamical, this is correct. Indeed no one claims that this is the final answer!

 

[atyy]

That's exactly my point, that marcus apparently does not appreciate. I do not mean here just graviton scattering processes around a backround; I mean that the whole lagrangian of GR: L = Sqrt (g) R + corrections, pops out as consistency condition in string theory (by expanding around flat space, graviton scattering can be recovered if one wishes to do so).

Thus string theory acts as a black box:

input = 2d CFT plus ghosts, integrated over 2d fields, moduli
output = Einstein gravity (plus Planck scale suppressed corrections).

That LQG provides an analogous black box, once properly defined and understood, is not unlikely but not still proven after some 25 years of research.

Thus, as of today there is no point to dismiss string theory as "purely mathematical" construct, when comparing both approaches!

The whole GR Lagrangian pops out as a a consistency requirement for the background on which perturbative string theory is done.

But isn't the claim for string theory to reproduce GR more than that, since that only reproduces the vacuum Einstein equations. Matter minimally coupled to gravity comes from the string excitations, which are perturbative, so is it still true that the whole Lagrangian of GR pops out if one discusses non-vacuum GR?

BTW, I do of course agree there is no reason to dismiss string theory as "purely mathematical". Even Smolin would agree string theory is much more than that: "it seems that any acceptable quantum theory of gravity, whatever its ultimate formulation, is likely to reduce to a perturbative string theory in the appropriate limit." http://arxiv.org/abs/gr-qc/9508064

 

[tom.stoer]

Short question: instead of repeating where theory X fails - wouldn't it be better to discuss where theory Y might help?

 

[Haelfix]

But this is still a pure argument.

You quantize string theory on flat spacetime and get back (as a consistency condition) the Einstein equations. But you are still on flat spacetime; spacetime hasn't become fully dynamical in that setup. It's is still a consistency condition for the classical background.

I do not doubt that this is a hint towards some underlying truth but it is definately not this truth itself.

There is a terminology point that I think you understand Tom, but I think that others on this board (Marcus in particular) have repeatedly struggled with over the years.

When we say that we recover GR as a consistency condition of String theory, we mean the full dynamical nonlinear classical theory. You bootstrap the theory in a completely analogous way as you do the classical linearized theory. In other words if you were a theorist in 1910 and someone handed you string theory in its current form, you would automatically read out not just a linearized version of gravity, but the whole shebang in the correct limit where you were justified in ignoring all the decidedly quantum effects. You really do limit to exactly the 1915 theory with all the associated geometry and so forth.

However, as you said and as Surprised has noted, there is still background dependence here at the level of the formalism and that may or may not be an impediment for defining the full offshell *QUANTUM* theory. It is in this sense that morally speaking one does not quite have the exact spirit of Einstein's theory, or its corresponding ability to calculate things in a simple and elegant manner. But, having said that, it is just completely wrong to suggest that they don't get the correct *classical* dynamics out (like Marcus has suggested a few posts up).

Incidentally, the word 'dynamic' is one of those loaded physics words that has evidently created some amount of confusion here for multiple years.
It is actually simpler to talk about what is fixed in string theory... What is 'fixed' is an object in an intermediate calculational stage, namely one's choice of an appropriate classical solution (like flat space or say the Schwarzschild solution) in a background field approximation.

This approximation method was initially invented to deal with quantum gravity, but it has been since utilitzed all across QFT in different situations. For instance we might do a background field split around a gluon field in QCD. In so far as this makes sense (and it is very limited), one doesn't really think of the gluon field as being 'non dynamic', it is merely approximated to ignore the backreaction effects by the small variation that is propagating. At the end of the calculation, you sum up all the contributions and you recover the full thing. I like to bring this up, b/c it demystifies the apparently peculiar position on this board whereby gravity is somehow special or mysterious in this regard.