The problems in Quantum Geometry

[marcus]

What are the main problems in non-string QG? Who is working on them? What progress has been made in the past couple of years?

 

[marcus]

People are invited to contribute their notions of what the main problems are that need to be worked on in (non-string) QG.

Several years back we were always hearing about the low-energy limit, the semi-classical limit, contact with classical gravity. Apparently in the past couple of years there has been some progress---it's an area where people are working. Some of us may have noticed the August 2006 quote from Carlo Rovelli in NewSci. I'll give the quote in context:

===
For loop quantum gravity to succeed as a fundamental theory of gravity, it should at the very least predict that apples fall to Earth. In other words, Newton's law of gravity should naturally arise from it. It is a tall order for a theory that generates space and time from scratch to describe what happens in the everyday world, but Carlo Rovelli at the University of the Mediterranean in Marseille, France, and his team have succeeded in doing just that. "Essentially we have calculated Newton's law starting from a world with no space and no time," he says (www.arxiv.org/abs/gr-qc/0604044)[/URL].
===
[url]http://space.newscientist.com/article/mg19125645.800[/url]

In case someone wishes to check it out, here is the journal article to which Rovelli referred in the NewSci interview:
[url]http://www.arxiv.org/abs/gr-qc/0604044[/url]
Graviton propagator in loop quantum gravity
Eugenio Bianchi, Leonardo Modesto, Carlo Rovelli, Simone Speziale
41 pages, 6 figures
Class.Quant.Grav. 23 (2006) 6989-7028

IMHO this is still an active research area despite some progress being made---undoubtably there's lots more to do.

 

[marcus]

One reason the question interests me about what are Quantum Geometry's current problems is that JB in TWF #246 said something that I wasn't sure how to interpret.

...Loop quantum gravity, in particular, has problems that are just as serious as string theory.
But, the "only game in town" argument is still flawed...

It sounded like having problems was bad, but I don't think that having problems (if they are rapidly changing because you are solving the old ones and turning up new ones) is necessarily bad. And also this runs into the familiar conundrum "what does he mean by LQG?"
Does he mean the static vintage 1990s version or the spinfoam approach that Rovelli was using last year, or the evolving versions that, say, Ashtekar, Thiemann, Bojowald are currently working on?

In the latter three cases, the evolving versions seem to have been invented in 2006 or perhaps 2005. All three people (A, T, B) were at Kitp last month and what they talked about was, I guess you could call it, AQG and QC, algebraic quantum gravity and Ashtekar's latest quantum cosmology. Bojowald wrote recently that there is a convergence between how Thiemann is treating the full theory and how Ashtekar and Bojowald have independently found it useful to treat the specialization to cosmology.

Maybe we can take the latest (vintage 2006) re-invention of Quantum Geometry to be "AQG" including both the full and the convergent application to cosmology. In that case, I'd like to know what are the serious problems with AQG.

Probably it's inevitable that we continue to refer to all the non-string QG approaches as "Loop Quantum Gravity" because that's customary. It is important to keep in mind that we are talking about a mix of approaches that continues to re-invent itself, that addresses its old problems and outgrows its old limitations and develops new problems.

In that spirit I'd be very interested to learn what anybody, especially John Baez but also others, thinks are the serious problems presented by the current mix of approaches called "Loop Quantum Gravity".

 

[josh1]

IMHO this [recovering general relativity] is still an active research area despite some progress being made---undoubtably there's lots more to do.

This is misleading. The accurate statement is that despite years of trying, nothing has been accomplished that unambiguously constitutes meaningful progress on the central problem with these sorts of approaches. Further, since this is the central problem, obviously it should always be an active area of research. So when was the last serious attempt made to solve this problem? How many papers have been dedicated to this problem recently? I`ve heard that very little attention is being paid to it these days.

 

[marcus]

IMHO this [recovering general relativity] is still an active research area despite some progress being made---undoubtably there's lots more to do.

This is misleading. The accurate statement is that despite years of trying, nothing has been accomplished that unambiguously constitutes meaningful progress on the central problem with these sorts of approaches. Further, since this is the central problem, obviously it should always be an active area of research. So when was the last serious attempt made to solve this problem? How many papers have been dedicated to this problem recently? I`ve heard that very little attention is being paid to it these days.

Thanks for the clear statement. What you express here is a view which I think is mistaken. Perhaps as time permits i can mention some recent papers showing signs of meaningful progress on this problem, or others may.

 

[ccdantas]

Hi Marcus,

I think Baez meant that there are still important internal, technical problems to be dealt with in LQG, as well as the problem of how to falsify its various models (and interpretations -- as far as I could grasp, we will not be able to test DSR with GLAST if Bee is correct). So, in that overall sense, it would not be unreasonable to say that both fields (LQG and ST) have their serious problems. (But I hope Baez will elaborate on that, I could be wrong).

But, yes, I'd add a note saying that LQG is progressing rapidly in the past few years and the contact with experiments seems more and more promissing (or not... again, it seems that one can interpret the models differently, *but* the field is changing rapidly). In that sense LQG seems more promissing, a very active field *now*. (The problems in the case of LQG are conceptually interesting and challenging and worthy of a serious investigation, in my opinion).

ST, on the other hand, apparently has conceptual difficulties and dead-ends (like the landscape), which seems to indicate that the field is not as healthy. **I could be completely wrong here!** One thing that I am certain is that there are lots of interesting and useful mathematics to learn from ST, and who knows, some of it can end up as being useful one way or another for a consistent quantum gravity theory, yet to come. So I don't think ST should be discouraged at all, but other approaches should receive the same amount of attention and support. (Also, I think they should separate fields into "String mathematics" and "String physics" or whatever. Like there is mathematical topology and cosmological topology. Perhaps things would clean up a little).

Christine

 

[jal]

But, yes, I'd add a note saying that LQG is progressing rapidly in the past few years and the contact with experiments seems more and more promissing (or not... again, it seems that one can interpret the models differently, *but* the field is changing rapidly). In that sense LQG seems more promissing, a very active field *now*. (The problems in the case of LQG are conceptually interesting and challenging and worthy of a serious investigation, in my opinion).

I agree!
Small step are happening regularly.
I am seeing new papers regularly (thanks to Marcus).
What I feel that is missing is a structured model that can handle the dynamics. (a Quantum Minimum Length Structure)
Other people will have different ideas.
jal

 

[marcus]

I agree!
Small step are happening regularly.
...

Hi jal, I remember you from when we were discussing some of the same issues back in January. If I remember right, you referred to commentary in your PF blog. It was in this thread:
https://www.physicsforums.com/showthread.php?p=1228523#post1228523
which was motivated by Bojowald noting how his work, Thiemann's work, and Ashtekar's all seemed to be converging from independent directions on the same structures. I had just posted this:

The three papers he cites are
http://arxiv.org/gr-qc/0607039
Quantum Nature of the Big Bang: Improved dynamics
Abhay Ashtekar, Tomasz Pawlowski, Parampreet Singh
Revised version to appear in Physical Review D
Phys.Rev. D74 (2006) 084003

"An improved Hamiltonian constraint operator is introduced in loop quantum cosmology. Quantum dynamics of the spatially flat, isotropic model with a massless scalar field is then studied in detail using analytical and numerical methods. The scalar field continues to serve as `emergent time', the big bang is again replaced by a quantum bounce, and quantum evolution remains deterministic across the deep Planck regime. However, while with the Hamiltonian constraint used so far in loop quantum cosmology the quantum bounce can occur even at low matter densities, with the new Hamiltonian constraint it occurs only at a Planck-scale density. Thus, the new quantum dynamics retains the attractive features of current evolutions in loop quantum cosmology but, at the same time, cures their main weakness."

http://arxiv.org/gr-qc/0607099
Algebraic Quantum Gravity (AQG) I. Conceptual Setup
Kristina Giesel, Thomas Thiemann
AEI-2006-058

"We introduce a new top down approach to canonical quantum gravity, called Algebraic Quantum Gravity (AQG):The quantum kinematics of AQG is determined by an abstract *-algebra generated by a countable set of elementary operators labelled by an algebraic graph. The quantum dynamics of AQG is governed by a single Master Constraint operator. While AQG is inspired by Loop Quantum Gravity (LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure. A natural Hilbert space representation acquires the structure of an infinite tensor product (ITP) whose separable strong equivalence class Hilbert subspaces (sectors) are left invariant by the quantum dynamics. The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states. Given such data, the corresponding coherent state defines a sector in the ITP which can be identified with a usual QFT on the given manifold and background. Thus, AQG contains QFT on all curved spacetimes at once, possibly has something to say about topology change and provides the contact with the familiar low energy physics. In particular, in two companion papers we develop semiclassical perturbation theory for AQG and LQG and thereby show that the theory admits a semiclassical limit whose infinitesimal gauge symmetry agrees with that of General Relativity. In AQG everything is computable with sufficient precision and no UV divergences arise due to the background independence of the fundamental combinatorial structure. Hence, in contrast to lattice gauge theory on a background metric, no continuum limit has to be taken, there simply is no lattice regulator that must be sent to zero."

http://arxiv.org/gr-qc/0609034
Loop quantum cosmology and inhomogeneities
Martin Bojowald
25 pages, 1 figure
Gen.Rel.Grav. 38 (2006) 1771-1795

"Inhomogeneities are introduced in loop quantum cosmology using regular lattice states, with a kinematical arena similar to that in homogeneous models considered earlier. The framework is intended to encapsulate crucial features of background independent quantizations in a setting accessible to explicit calculations of perturbations on a cosmological background. It is used here only for qualitative insights but can be extended with further more detailed input. One can thus see how several parameters occurring in homogeneous models appear from an inhomogeneous point of view. Their physical roles in several cases then become much clearer, often making previously unnatural choices of values look more natural by providing alternative physical roles. This also illustrates general properties of symmetry reduction at the quantum level and the roles played by inhomogeneities. Moreover, the constructions suggest a picture for gravitons and other metric modes as collective excitations in a discrete theory, and lead to the possibility of quantum gravity corrections in large universes."
...

================
Probably the most accessible detailed presentation is that given by Thiemann at the workshop. He was the only person who got TWO hours----he was scheduled to give a one-hour talk and they asked a lot of questions, so on the spot Horowitz scheduled another one-hour slot.
Thiemann seemed to be happy at his presentation---he was talking about MasterConstraint and also especially about AQG ...
...

IIRC Kristina Giesel has given a talk at Perimeter about AQG. ...

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

What Bojowald is saying is that he sees some convergence. He says that the approaches which he (on the one hand) and Ashtekar etal (on the other hand) have found to be workable dealing with cosmo singularities are in some way similar to Thiemann AQG.
...

I think Kristina Giesel will also be giving a talk at the QGQG school in March-April, that Francesca told us was happening. There is listed a talk by a student of Thiemann.

 

[Kea]

ST, on the other hand, apparently has conceptual difficulties and dead-ends (like the landscape), which seems to indicate that the field is not as healthy.**I could be completely wrong here!**

I'm sorry, but you guys sound a lot like the String bods who allegedly call String Theory The only Game in Town. All approaches to quantum gravity have serious conceptual problems.

 

[marcus]

I'm sorry, but you guys sound a lot like the String bods who allegedly call String Theory The only Game in Town. All approaches to quantum gravity have serious conceptual problems.

here is your chance, Kea. Explain to us about the conceptual problems of the latest incarnations of Loop :-)
I am ASKING for people to try to identify and describe problems, in this thread.
I certainly don't deny problems exist---that's what makes the game fun---and I didn't think Christine did either.

What JAL mentioned about the step-by-step doesn't contradict that either. An example is
1. LQC made good contact with semi/classical GR physics a long time ago as I guess we all know. But it was a simplified symmetry reduced model.
2. So the game has been to gradually reduce the restrictions and work out the problems little by little.

This is what Ashtekar and Bojowald and friends have been doing. there has been constant progress and they are now removing the homogeneity restriction.

I never would deny there could be a fundamental problem with the approach. What I like is the constant change and regular progress.

I think people who declare they know in advance that something is the RIGHT path suffer from hubris (a special kind of vanity). The proof of the pudding and time will tell :-)
Likewise people who pretend to know in advance that someone else's program has a fatal conceptual flaw that only they can see and so they know ahead of time that it is the WRONG path.

So what do you think are problems with the Quantum Cosmology that, say, Ashtekar is doing?
If you can identify any problems that will be more stuff for LQG researchers to work on.

 

[Kea]

So what do you think are problems with the Quantum Cosmology that, say, Ashtekar is doing? If you can identify any problems that will be more stuff for LQG researchers to work on.

Hah! I'm not going to spend any time at all working through a paper such as

An Introduction to Loop Quantum Gravity Through Cosmology
Abhay Ashtekar
http://www.arxiv.org/abs/gr-qc/0702030

because I see no reason to believe that I will find it particularly enlightening. Especially with phrases like 'Loop Quantum cosmology of FRW models' in the abstract. Once this theory is matched to experiment as well as existing alternatives, I might change my mind. Or if I see some conceptual clarity in their description of quantum geometry, then I might change my mind. Or if they derive the lepton masses from this framework, I might change my mind. So maybe I'm just stupid and I don't get it - but I don't believe so.

 

[marcus]

Hah! I'm not going to spend any time at all working through a paper such as

An Introduction to Loop Quantum Gravity Through Cosmology
Abhay Ashtekar
http://www.arxiv.org/abs/gr-qc/0702030
...

OK, that's fine Kea. I will go over it. it is basically a pedagogical/overview paper so it could be useful to us.

 

[jal]

Hi!
I don't think that Baez is referring to people like me. Rather, I interpret, that he is saying, "Look at what there is on the other mountains".
For instance CarlB is working on, (I think) a dynamic approach.
Also, there are people working "Casual Set" which could give some answers.
Together, ... the answer will show up.
If I see it, I will have it in my blog.
jal

 

[ccdantas]

I certainly don't deny problems exist---that's what makes the game fun---and I didn't think Christine did either.

Right!

I've just written a long post, and suddenly it went to nirwana, as Bee uses to say. Argh! :grumpy: Well, I'll not attempt to write it again. And perhaps it was for a good reason!

BTW, perhaps I could use this opportunity to announce my new blog? I've been setting it up for a few couple of months. It's somewhat different from my previous one. And I will probably not have the time to post there regularly. You'll find there just random ideas. It's a quiet place, actually (an anti-blog?) And I like it that way. So don't hope to find long polemic debates!

http://egregium.wordpress.com/

The general subjects are: physics, mathematics and philosophy. I'll specially focus on some ideas relating quantum gravity and quantum computation, specially, quantum gravity as an inherently concurrent quantum system. The ideas are very preliminary and open for exploration... Note: some posts are not self-contained and some others are quite obscure. Details will hopefully come in small steps. The core of the main idea can be found in this post:

http://egregium.wordpress.com/2007/01/16/nature-abhors-deadlocks/"

Thanks,
Christine

 

[marcus]

Yay!
lovely looking blog
I was just now reading your post about Gauss' Theorema Egregium

 

[Kea]

http://egregium.wordpress.com/

So you've become a category theorist. OK, cool.

 

[ccdantas]

So you've become a category theorist. OK, cool.

Well, that's the way you see it... Or perhaps it's a sympathetic way to see it. But evidently I'm quite far off from calling myself a category theorist. I understand that cateory theory rebuilds mathematics from the concept of processes. I would like to see the idea of Nature as an inherently (quantum mechanically) concurrent system worked out in detail.

I have a feeling this may be a missing ingredient in non-pertubative quantum gravity dynamics. Also, quantum gravity and quantum computation may have lots of common interconnections, specially in concurrent aspects of quantum information. I don't know whether I'll be able to work this out, not even the first steps. I'm trying to think further on this of course. But the idea is set for those who are interested to do this.

I don't know whether this could be attempted from the cateory theory point of view. It would be interesting to see this idea further developed. It is often wise to work from different perspectives.

But I don't want to further divert from the main thread, so let's get back to it!

Thanks
Christine

 

[marcus]

...
I have a feeling this may be a missing ingredient in non-pertubative quantum gravity dynamics. Also, quantum gravity and quantum computation may have lots of common interconnections,...

Since we're on feelings I'll put in my feeling too, or my hunch (for what it's worth). My hunch is that the main caterpillar-tread tractor here is "applied" Quantum Geometry and that by slowly advancing it will SHOW us what the missing math ingredients are.

What I mean by "applied" is Quantum Geometry applied to understanding the black hole singularity and the cosmological singularity. It is what Ashtekar has turned his attention to for the past three or four years (and not only he, others as well.)
This is where QG makes a good solid contact with reality (it has the correct semiclassical limit, it recovers conventional cosmology) and where it has an important TASK to perform: replace the Gen Rel singularities with something else.

So what I expect is that, like a tractor, it will keep plowing ahead and it will show us what math it needs.

If it needs the Poincaré two-group (as I suppose Laurent Freidel thinks it might, since he is studying the reprepresentations) then the tractor will point this out to us. If it needs the deSitter group and Cartan geometry (as Derek Wise suggests it might), then it will by gradual advances point this out to us as well.

Just now it has turned out that two "applied" people (Ashtekar and Bojowald) plus one more theoretical QG person (Thiemann) all discovered about the same time that they should use unembedded regular-grid states. it is a drastic change from embedded spin-nets----there is no manifold! The chances are, I suspect, that none of us would have predicted this. Not I anyway.

Who would have predicted that David Gross would decide he wanted Bojowald to come to Kitp---that is, aside from Ashtekar, the world top "applied" QG person. It must mean that he sees the challenge of getting rid of all Gen Rel singularities as an important---a kind of seminal---program. So that is why I think of it as a tractor.

and in a more general way, cosmology and high energy astrophysics as a tractor.

Well that's just personal viewpoint. As Christine suggests, let's get back to saying what are the serious problems with Quantum Geometry.

 

[rtharbaugh1]

What is the geometry of time?

My impression from the end of the concluding seminar at the singularities miniprogram at KITP UCSB in January 2007 conducted by Gary Horowitz was that the key issue is the Problem of Time. He seemed to think that a new view of time is essential to any real progress.

Why does time seem to flow in only one direction? (The Arrow of Time, Roger Highfield, 1992, and, from Wikipedia:
Halliwell, J.J. et.al. (1994). Physical Origins of Time Asymmetry. Cambridge. ISBN 0-521-56837-4. (technical).
Boltzmann, Ludwig (1964). Lectures On Gas Theory. University Of California Press. Translated from the original German by Stephen G. Brush. Originally published 1896/1898.
Peierls, R (1979). Surprises in Theoretical Physics. Princeton. Section 3.8.
Feynman, Richard (1965). The Character of Physical Law. BBC Publications. Chapter 5.
Penrose, Roger (1989). The Emperor's New Mind. Oxford University Press. ISBN 0-19-851973-7. Chapter 7.
Penrose, Roger (2004). The Road to Reality. Jonathan Cape. ISBN 0-224-04447-8. Chapter 27.
Price, Huw (1996). Time's Arrow and Archimedes' Point. ISBN 0-19-510095-6. Website
Wehrli, Hans (2006). Metaphysik - Chiralität als Grundprinzip der Physik. ISBN 3-033-00791-0.
Zeh, H. D (2001). The Physical Basis of The Direction of Time. ISBN 3-540-42081-9.
)

Nothing really new except for Wehrli, 2006, evidently not available in English translation.

I'll go look at ArXiv for recent research.


3 for time AND (quantum geometry) at
http://arxiv.org/find/hep-th,grp_math/1/abs:+AND+time+EXACT+quantum_geometry/0/1/0/past/0/1

here they are.

1. gr-qc/0607130 [abs, ps, pdf, other] :
Title: Quantum Geometry and its Implications for Black Holes
Authors: Martin Bojowald
Comments: 16 pages, Plenary talk at ``Einstein's Legacy in the New Millenium,'' Puri, India, December 2005
Journal-ref: Int.J.Mod.Phys. D15 (2006) 1545-1559

Abstract:
“General relativity successfully describes space-times at scales that we can observe and probe today, but it cannot be complete as a consequence of singularity theorems. For a long time there have been indications that quantum gravity will provide a more complete, non-singular extension which, however, was difficult to verify in the absence of a quantum theory of gravity. By now there are several candidates which show essential hints as to what a quantum theory of gravity may look like. In particular, loop quantum gravity is a non-perturbative formulation which is background independent, two properties which are essential close to a classical singularity with strong fields and a degenerate metric. In cosmological and black hole settings one can indeed see explicitly how classical singularities are removed by quantum geometry: there is a well-defined evolution all the way down to, and across, the smallest scales. As for black holes, their horizon dynamics can be studied showing characteristic modifications to the classical behavior. Conceptual and physical issues can also be addressed in this context, providing lessons for quantum gravity in general. Here, we conclude with some comments on the uniqueness issue often linked to quantum gravity in some form or another.




2. hep-th/0604120 [abs, ps, pdf, other] :
Title: Towards Gravity from the Quantum
Authors: Fotini Markopoulou
Comments: Expanded version of the contribution to "Towards Quantum Gravity", edited by D.Oriti, to be published by C.U.P.

Abstract:

"We review quantum causal histories starting with their interpretations as a quantum field theory on a causal set and a quantum geometry. We discuss the difficulties that background independent theories based on quantum geometry encounter in deriving general relativity as the low energy limit. We then suggest that general relativity should be viewed as a strictly effective theory coming from a fundamental theory with no geometric degrees of freedom. The basic idea is that an effective theory is characterized by effective coherent degrees of freedom and their interactions. Having formulated the pre-geometric background independent theory as a quantum information theoretic processor, we are able to use the method of noiseless subsystems to extract such coherent (protected) excitations. We follow the consequences, in particular, the implications of effective locality and time. "


3. gr-qc/0604013 [abs, ps, pdf, other] :
Title: Quantum Nature of the Big Bang: An Analytical and Numerical Investigation
Authors: Abhay Ashtekar, Tomasz Pawlowski, Parampreet Singh
Comments: Revised version to appear in Physical Review D. References added and typos corrected
Journal-ref: Phys.Rev. D73 (2006) 124038

Abstract:

"Analytical and numerical methods are developed to analyze the quantum nature of the big bang in the setting of loop quantum cosmology. They enable one to explore the effects of quantum geometry both on the gravitational and matter sectors and significantly extend the known results on the resolution of the big bang singularity. Specifically, the following results are established for the homogeneous isotropic model with a massless scalar field: i) the scalar field is shown to serve as an internal clock, thereby providing a detailed realization of the `emergent time' idea; ii) the physical Hilbert space, Dirac observables and semi-classical states are constructed rigorously; iii) the Hamiltonian constraint is solved numerically to show that the big bang is replaced by a big bounce. Thanks to the non-perturbative, background independent methods, unlike in other approaches the quantum evolution is deterministic across the deep Planck regime. Our constructions also provide a conceptual framework and technical tools which can be used in more general models. In this sense, they provide foundations for analyzing physical issues associated with the Planck regime of loop quantum cosmology as a whole. "

Hope this is helpful to the discussion.

R

 

[jal]

What I feel that is missing is a structured model that can handle the dynamics. (a Quantum Minimum Length Structure)
It seems that Martin Bojowald went into more details to repeat what I said.
http://arxiv.org/PS_cache/gr-qc/pdf/0702/0702144.pdf
Singularities and Quantum Gravity
I'll post the rest in the new thred that you started.
jal