Modesto shows LQG link to Loll and Reuter microstructure

[marcus]

Here is some background on Leonardo Modesto's new paper:

An outstanding puzzle in Quantum Gravity is the strange coincidence that two of the most developed approaches both produce a continuum (by different means) which looks normal 4D at large scale but at micro scale the dimensionality gradually declines to around 2D. That is the micro geometry becomes chaotic and like a fractal or a foam. In neither approach were they expecting this to happen. They just built a quantum version of General Relativity (in two different ways) and then in the process of exploring they both came across this surprising micro fractal-like geometry. Empirically, so to speak. In one case it came out of computer simulations of small quantum universes (Loll CDT Triangulations approach) and in another it came analytically using a putative fixed point of the renormalization group flow [Reuter ASQG, asymptotic safety QG.]

So we have this odd coincidence. Two very different theory approaches seem to point to the same thing. Could it actually be true about nature? And true or not, how can one explain the coincidence? In both cases the dimensionality unexpectedly declines smoothly [from 4D] to 2D at small scale.

...

http://arxiv.org/abs/0811.1396
Fractal properties of quantum spacetime
Dario Benedetti
4 pages, 2 figures
(Submitted on 10 Nov 2008)

"We show that in general a spacetime having a quantum group symmetry has also a scale dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what observed in some approaches to quantum gravity. In particular we analyze the cases of a quantum sphere and of k-Minkowski, the latter being relevant in the context of quantum gravity."

It is in this context that Leonardo Modesto has shown that LQG also has this curious fractal-like microstructure down near Planck scale. That the dimensionality declines from the usual 4D at large scale down to 2D at the microscopic level.

The measure of dimensionality he used was the diffusion or random-walk-based spectral dimension. One tells the dimensionality of the space one is in by seeing how easily a random walker gets lost in it. Dimensionality measured this way can take on fractional values.

It is interesting that all three approaches (Loll Blocks, Reuter Asymptotic, and LQG) came to this same conclusion about the chaotic fractally microstructure---all three seem to present a new idea of the continuum which is smooth at large and rough at small distances. But they come at this conclusion by very different analytical methods.

In any case, whether this new model of the continuum is correct or not, here is Modesto's December 2008 paper. It is only 5 pages!:
http://arxiv.org/abs/0812.2214
Fractal Structure of Loop Quantum Gravity
Leonardo Modesto
5 pages, 5 figures
(Submitted on 11 Dec 2008)
"In this paper we have calculated the spectral dimension of loop quantum gravity (LQG) using simple arguments coming from the area spectrum at different length scales. We have obtained that the spectral dimension of the spatial section runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decreases from high to low energy. We have calculated the spectral dimension of the space-time also using results from spin-foam models, obtaining a 2-dimensional effective manifold at high energy. Our result is consistent with two other approaches to non-perturbative quantum gravity: causal dynamical triangulation and asymptotic safety quantum gravity."

 

[ensabah6]

What does string theory have to say on this scale?

 

[marcus]

I believe string theories are inappropriate to the discussion in this thread because they do not offer a new model of the quantum spacetime continuum for us to study. On the contrary, perturbative string theories use an old-style (differential manifold) continuum of some fixed dimensionality---which provides a stage where stringy objects can act like particles and do their thing. It's basically a particle theory approach that does not emphasize the problem of finding a new mathematical model for an uncertain fluctuating spacetime geometry.

A quantum continuum model is expected to be chaotic at small scales with highly variable unsmooth microgeometry. Heisenberg-type uncertainty is expected to affect geometric observables such as area, volume, dimensionality, making for stormy weather at Planck scale. By contrast, the differential manifold space in which stringy objects live is smooth all the way down--at all levels of magnification--and keeps the same fixed dimensionality* down to vanishingly small scale. Such an old-style continuum would not enter into discussion here.

*including it's rolled-up compactified dimensions

 

[ensabah6]

I believe string theories are inappropriate to the discussion in this thread because they do not offer a new model of the quantum spacetime continuum for us to study. On the contrary, perturbative string theories use an old-style (differential manifold) continuum of some fixed dimensionality---which provides a stage where stringy objects can act like particles and do their thing. It's basically a particle theory approach that does not emphasize the problem of finding a new mathematical model for an uncertain fluctuating spacetime geometry.

A quantum continuum model is expected to be chaotic at small scales with highly variable unsmooth microgeometry. Heisenberg-type uncertainty is expected to affect geometric observables such as area, volume, dimensionality, making for stormy weather at Planck scale. By contrast, the differential manifold space in which stringy objects live is smooth all the way down--at all levels of magnification--and keeps the same fixed dimensionality* down to vanishingly small scale. Such an old-style continuum would not enter into discussion here.

*including it's rolled-up compactified dimensions

string theory -- the basic object of the theory, string, is 1 dimensional.

 

[marcus]

It's clearly possible for people to be confused about what the topic is here, so I should clarify.

To have a place in this discussion, a theory must have a mathematical object representing the geometry of the universe---a quantum state of space or spacetime geometry---and you have to be able to get inside it and run a diffusion process or random walk.

The key thing is that the dimensionality, measured at a given point within a given range of its surroundings, is a quantum observable.

The dimensionality at a certain location is measured by running an exploratory diffusion process starting at that point, like placing a drop of ink in a container of water and watching it diffuse. It is called the spectral dimension.

The topic of discussion here, in this thread, is the news that there are now three QG models that allow one to measure the spectral dimension, at any given point and any specified scale, and the curious fact that even though they are set up very differently---attempting to quantize General Relativity geometry in very different ways---they all show the same behavior. Dimensionality taking on fractional values---like 2.3 and 1.6---and changing continuously---and declining with scale.

I should emphasize that to play in this game, and to be appropriate for discussion in this thread, a theory has to show a mathematical representation of the uncertain fluctuating state of geometry---and you have to be able to go inside that mathematical object and test dimensionality, measure the spectral dimension by running a random walk. Dimension has to be a quantum observable that can assume fractional values like 1.9 and 2.4 etc.

There can be other perfectly nice theories about various other stuff which do not focus on representing the uncertain and changing geometry of the universe. That don't have dimensionality as something uncertain and variable that you can observe. That's OK, they are theories focusing on other stuff, but they are not appropriate to the discussion here. So please start a different thread to ask and discuss stuff that is off-topic here.

 

[Autochthon]

So I'm guessing the smearing out onto a 2-d spherical surface at the Schwartzschild radius of a black hole would be off limits?

 

[marcus]

You guessed right. What we are seeing is the emergence of a new (quantum) continuum model and it's good to pay some attention to what's happening. If you want a basic sample of what we are seeing arise from these three approaches, look at the Ambjorn Loll Scientific American article that I link to in my sig.

If you can't get the PDF file, please let me know and I will look for an alternative link.

This is an important development. Nearly all of physics is still constructed on a mathematical model of the continuum called a differential manifold that was invented around 1850.
A differential manifold has the same dimensionality at every point and at every scale. The dimensionality is a whole number. Moreover it is smooth at all scales.

A diff. manif. equipped with a fixed metric distance function is a metric differential manifold.
That is typical of the old continuum idea.

Quantum gravitists do not expect that nature's continuum is smooth at all scales. They take Heisenberg Uncertainty seriously. At small scale, curvature is expected to run amok. The micro-geometry of space is expected to fluctuate violently.

The old-style 1850 differential manifold model is expected not to fit. So we will need a new-style quantum continuum. It should look smooth at macro scale---at human scale. And examined at very high magnification it should be rough, chaotic, crumply, fractally.

The Loll SciAm article would be an excellent introduction to modern thinking about the continuum. But all three approaches Loll CDT, Reuter Asymptotic, and LQG are trying to develop a new quantum model of the continuum.

What interests me here, and is the topic of the thread, is how all three approaches seem to have converged on a model of space which looks 3D at human scale but where as you zoom into near Planck scale the dimensionality steadily declines down to 2 and then even below 2.

The SciAm article, for instance, shows a plot of the dimensionality as a function of scale. They got this by computer experiment, by generating hundreds of thousands of simulation universes and sampling dimensionality in them at various scales. It is an interesting result because it was not anticipated.

If you would like some peer-review journal articles about this, please let me know. That's how I learned about it. The SciAm popular account is recent, only came out this year.

 

[Hurkyl]

Is there much possibility that this is an artifact of the methods, rather than saying something deep about quantum gravity? e.g. spatial slices in LQG have loops (and spin networks); it may not be surprising that it should look roughly 1+1-dimensionish at the smallest scales simply because of that. But maybe the LQG programme could be generalized to give a "sphere quantum gravity" theory that does roughly the same things, but would look 2+1-dimensionish at the smallest scales?

 

[marcus]

Is there much possibility that this is an artifact of the methods, rather than saying something deep about quantum gravity?...

Absolutely! That is the first thing that one should expect. I've wondered about this for some time.

The decline in dimensionality with scale was first reported by Loll et al in 2005. Reuter reported it for Asymptotic Safety QG around the same time. That is the coincidence that one should scrutinize, I think. Because it is well established. It has been reported and discussed at conferences repeatedly, for over 3 years.

The appearance of Modesto's paper bringing in LQG is still too new. I want to see how other LQG people take it, and what kind of followup ensues.

In the case of Loll CDT and Reuter ASQG, one has to see if there is some common element---superficially the approaches look totally different. But some common element could give rise to a method artifact in both cases.

Dario Benedetti recently looked at this and wrote a paper arguing that any QG theory with such and such feature would exhibit this. It could be clue to what you want. I will get a link.
Oh! I had it in post #1:
http://arxiv.org/abs/0811.1396
Fractal properties of quantum spacetime
Dario Benedetti
4 pages, 2 figures
(Submitted on 10 Nov 2008)

"We show that in general a spacetime having a quantum group symmetry has also a scale dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what observed in some approaches to quantum gravity. In particular we analyze the cases of a quantum sphere and of k-Minkowski, the latter being relevant in the context of quantum gravity."
I don't see how this would apply, but I will have another look at it.

 

[John86]

I am getting a gut feeling about the works of Reuter Loll and consortio.

They really are on tho something. The Continuüm thing with fractal properties is rightly thought provoking an beautifful at the same time.
When this al holds it blows away the BB singularity problem, because the universe ad small scales has fractal continious properties. it will go and on.

I don't know how it will fit the emergent side of the story.

John

 

[marcus]

BTW thanks to you both, John86 and MTd2, for calling attention to the Benedetti and Henson article. I spent part of the morning happily reading it. Hope they can extend some of their results up from 2D to 4D. More signs that things are coming together.

 

[MTd2]

BTW marcus,

I am trying to understand a little bit of some kinds of 3D and 4D formulation of QFTs. So, now concetrating on topological strings and geometrical langlands, just to get a vague idea of coming out of them. I really think stringy things can be useful to LQG, because, I don't know how to explain, but 4 manifolds might naturaly work as simulacrums of field theories with an arbitrary number of dimensions. I guess it is the impression that the attempt of formulation for classification of 4 diffeomorphic manifolds seems to me, given that the way it uses tree like (fractal) structures of cobordisms, infinitely densile packed. It resembles like 4-manifolds are tying together an infinite number of tensionless strings. And also, that might give birth

The target space of tensionless strings has an arbitrary dimension:

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+AUTHOR+SCHILD+and+title+classical+null+strings

There are more tidy bits of tensionless strings here:

http://golem.ph.utexas.edu/string/archives/000453.html

Other intuition of why 4-manifolds are naturaly simulacrums of theories in arbritary dimension it is that besides displaying the unique dimension in which the phenomenum of having infinite non diffeomorphic structures happens (exoctic structure), in which leads me to dream about feynmann diagrams ( 1 diffeomorphic structure for a given path), it is that the lowest dimension in which one can tie structures of that n dimension in order to be homeomorphic to of any group.

Besides, exotic structures are always bound in, so, it may work as sorces and sinks. One more reason to think of feymann diagrams. The bounds here are the points in space times where the measurment is taken.

So, this Benedetti talk about fractals arising from LQG, really seems excessively attractive to me. Other exciting thing about Benedetti it is that he discusses Quantum Groups as one of the things that describes dimensional transition. This is really something, because Quatum Groups is also one of the generic ways to get around Coleman Mandula theorem, which says conditions to talks about ways to have a realist theory in 4 dimensions. The other way is supersymmetry, and indeed, the reason that superstrings became initialy so attractive it was that it provided a natural way to go around that theorem while keeping a s-matrix realizable. There is another way, the one used by Garrett, althought I don't get the reason very well. Well, I really don't know the reason for Quatum Groups either, other than it is related to general topological considerations in 3D spaces, that is, topological string theory. Or so that's what I understood from Urs, in private communication...

 

[MTd2]

The second author of this new benedetti article, http://arxiv.org/abs/0812.4261, has 2 articles with Raphael Sorkin:

http://arxiv.org/abs/gr-qc/0311055
http://arxiv.org/abs/gr-qc/0202097

So, maybe we are seeing LQG approaching Causal Sets directly?