Benedetti's insight connecting Loll and Reuter continuums

[marcus]

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 ERG exact renormalization group method also called QEG quantum einstein gravity.)

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 the dimensionality unexpectedly declines smoothly to 2D at small scale.

OK so Benedetti just got his PhD with Loll at Utrecht and went to postdoc at Perimeter. And he has proposed in this little 4 page paper an idea of what might be a kinship between the two non-classical continuums that might explain this coincidence. I put it out in case anyone can comment. I would like to see anyone's ideas about this.

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."

 

[marcus]

Maybe someone would like links to the Loll CDT papers or the Reuter ERG papers that talk about this drop-off of dimensionality at small scale. If so ask, but also Benedetti gives references.

Just informally we can talk about how dimensionality is measured around some given point. In quantum gravity, the dimensionality at a point is (like other geometric things) a quantum observable. So one has to have a practical idea of how you observe it.

One way is to see how volume increases with radius. You just experiment. If the volume increases as (radius)^D then probably the dimensionality is D.
Loll's group could do this because they produced millions of small quantum universes in the computer and they could go in and pick a point and explore around that point and measure stuff. And go around to lots of different points, and vary the scale. And then generate a new universe etc etc. So they could be very sure of the effect and plot the curve precisely, for how dimensionality declines with scale.

But there is another way! It's called the spectral dimension and it uses a RANDOM WALK, you see how likely you are to get lost and never get back to home base, versus the probability of accidentally returning home. The higher the dimensionality the more likely you are to get lost. Loll's team also measured dimensionality that way. They could becasue they produced millions of little universes, so they could go inside one and run a random walk starting from some point in it.

The building blocks (the simplexes) provide natural steps for doing a random walk, or natural radius and volume counts for doing it that way if you prefer. The whole thing of studying geometry in that context is appealingly straightforward.

As I recall while Benedetti was at Utrecht he did a paper with Saueressig on Reuter-style ERG quantum gravity. So he was getting immersed in both approaches.

Have to go. Back later.

 

[MTd2]

Just a quick remark on the methods used in this paper. Hhe speculates the relation of the minimum dimension with some kind of maximal commutative subspace, but is not sure of what are the necessary conditions for that. If one finds such conditions, maybe we have a new field of mathematics.

Anyway, besides supersymmetry, another known loop hole for the Coleman Mandula theorem involves some kinds of Quantum Groups, for example, two dimensional integrable quantum system theories. One of them is the sine gordon model. It's a model about solitons in 1+1 dimensions, that actualy propagates as a twist

http://en.wikipedia.org/wiki/Sine-Gordon

So, you can see some hints of a isomorphism with Yidun Wan's braid model, if you try to actualy imagine those twists as a result of braid interactions.

 

[marcus]

As I recall while Benedetti was at Utrecht he did a paper with Saueressig on Reuter-style ERG quantum gravity. So he was getting immersed in both approaches.

I misremembered. It was another Loll PhD student, Pedro Machado, who co-authored an Asymptotic Safety paper with Frank Saueressig.

But everybody in Loll's group at Utrecht is getting exposed to both approaches. Seminar talks on Reuter's approach. Roberto Percacci puts in sabbatical time as a visitor at Utrecht. And whenever Loll or Ambjorn (on the CDT side) or Reuter or Percacci (on the Asymptotic Safety side) gives a conference talk or writes a survey paper they always point out this strange coincidence of two very different approaches from two very different directions both arriving at the same strange result about micro spacetime dimensionality.

So Benedetti's background makes it natural for him to be interested in finding a mathematical basis for this convergence.

 

[marcus]

Today Leonardo Modesto posted a paper arguing that LQG also has this property, shared by the other two approaches, that dimensionality at small scale (probed at high energy) is less than the macroscopic dimensionality.

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 other two approach to non perturbative quantum gravity: causal dynamical triangulation and asymptotic safety quantum gravity."

 

[MTd2]

Yes, indeed, LQG MUST HAVE a fractal structure, given that LQG, at some imit, must make a smooth space emerge from very generic piecewise linear structures, and only in four dimensions, you have exotic smoothness. That is, you have infinite non diffeomorphic structures for a given topological structure. The funny thing, it is that the fractal structure might be formed by an infinite fractal tower of some kind bondery-wise(thus 2d) connected topological strings, immersed in a 4d manifold. This is just an idea of mine.

If you ignore such fractal structure, LQG will lose any hope of being a TOE.

In no moment here I think of SUSY, the critical of topological strings is a 3 - complex manifold. So, what I am thinking here it is instead of ascending to superstrings, using equivalence relations to some Calabi Yau compactification, I rather ascend to LQG, and stringify LQG, and see what happens without resourting to SUSY, but using exoctic smoothness.

 

[atyy]

http://arxiv.org/abs/0812.2214
Fractal Structure of Loop Quantum Gravity
Leonardo Modesto

CDT and Asymptotic Safety both seem to assume an action that is not the Einstein-Hilbert action, but one that approximates it at low energy, so I understand how CDT and AS are related. But LQG starts with the assumption that the classical Hamiltonian in new variables is exact, then quantizes that. Can the consistency with CDT and AS be more than coincidental given such different starting assumptions?

 

[MTd2]

Can the consistency with CDT and AS be more than coincidental given such different starting assumptions?

That's what I am proposing above. Among the infinite possible dimensions manifolds, only the 4th requires fractal like strutures (http://en.wikipedia.org/wiki/Casson_handle )in order to make any simply connected surfaces topologicaly homeomorphic to a sphere (http://en.wikipedia.org/wiki/Generalized_Poincaré_Conjecture). But no one knows what happens in the Piecewise linear case and the Diffeomorphic case, except that these fractal structurs gives birth to an infinite variety of non equivalent smooth structures in a same manifold.

The coincidence between CDT and AS happens because they, on a statistical level over those infinite non diffeomorphic structures, are linked to, respectively, PL picture and Diff picture.

Well, that's a guess of mine.

 

[marcus]

CDT and Asymptotic Safety both seem to assume an action that is not the Einstein-Hilbert action, but one that approximates it at low energy,..

Atyy,
I wanted to ask you about this quite apart from any connection with Modesto's paper.

There are several versions of AS quantum gravity but the most common is probably Reuter's Einstein-Hilbert truncation where he takes the E-H action (with cosmo constant Lambda) and let's both G and Lambda run with scale.

So to me this looks rather much like E-H action but with different G and Lambda used at high energy. I suppose the real question is how different does it have to get before you get a qualitative difference in micro behavior.

Please let me know if I am overlooking some significant detail here.

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

With Loll CDT it may be important to notice that the triangular building blocks have a uniform size and that the size goes to zero in the limit. Loll has stressed this and has written explicitly that the theory has no minimal length.

At each stage of approximation to the path integral, characterized by a certain block size, they use the Regge form of the E-H action.

It would seem to me that as you let the block size go to zero what happens is that you get closer and closer to the E-H action, at any fixed scale or energy you want to consider.

So it would seem to me that the Loll method does NOT deviate from E-H at any level of energy. Can this be right? You say something different. What I am saying is, pick some energy or scale where you think Loll might be deviating. Now pick the size of the triangles to be smaller than that by 10 orders of magnitude, and check to see if there is any significant difference from the E-H action. I don't see how there could be, intuitively.

At least to the extent that the conventional E-H could be defined on the rough geometries which Loll's method uses---the jagged piecewise-linear sample chosen as a regularization of the path integral at each scale.

I may be overlooking something here, haven't thought about this. I'd be interested to know what you think, as you may have examined this more carefully.
===================

About Modesto's paper, I'll be interested to see how it is accepted. I have only given it a light once-over. It's too early for me to try to guess if he has succeeded in demonstrating what he claims. I see that he is now at Perimeter. Maybe he will give a talk about this and we will be able to watch the PI online video, see what questions he gets asked...

 

[IMP]

Is this in any way related to the Holographic Principle? Where the inside 2D surface of a sphere contains all the bits to describe a 4D volume?

 

[marcus]

Is this in any way related to the Holographic Principle? Where the inside 2D surface of a sphere contains all the bits to describe a 4D volume?

I believe not. the best way to get an idea of what we are talking about would probably be to look at Loll's Scientific American article. I have a link to it in my sig.

You know how fractals can have fractional dimension---like D = 1.6 or D = 2.314, instead of whole number like D = 2 and D = 3.

Dimensionality is something that you can measure empirically by observing how fast volumes increase as a function of radius, or how rapidly diffusion occurs (intuitively how fast a drop of ink disappears, or a f**t dissipates.)

In quantum geometry, you can't take the dimensionality for granted. At any given location is a quantum observable. It might differ from place to place and might even depend on the scale at which you meansure.

It is possible for a continuum to have dimensionality which varies continuously with scale, so that it is a nice familiar 3D at macro scales----a nice smooth uniform even 3D---but down near Planck scale it might begin to get chaotic and kinky or foamy---so that as you approach micro scale it gradually declines from D = 3 down to D = 2.6 and D = 1.7 and so on.

My guess is that this is turning out to be a kind of generic feature of realistic quantum geometry---that will come up in any quantum geometry that:

1. acknowledges Heisenberg Uncertainty of lengths areas angles etc at very small scale
2. doesn't prescribe some smooth metric geometry for itself at the outset.

That is, we can pretty much forget about theories that require a set up with smooth spaces of some definite dimensionality which is the same everywhere and at all scales. They just aren't realistic, not rough or uncertain enough to model quantum spacetime.
That's my hunch, as I say. Jury's still out.

... LQG starts with the assumption that the classical Hamiltonian in new variables is exact, then quantizes that...

Whenever someone like Rovelli or Speziale does dynamics these days it always seems to be spinfoam. So I wonder about how accurate this picture of LQG actually is. A lot of effort has been going into showing the low energy limit---but when I looked at the papers I don't remember seeing any Hamiltonian. Speciale recently posted an invited review article on deriving n-point functions. I'll have a look.

My impression is that they used the canonical Hamiltonian formalism especially in the 1990s as a kind of heuristic. But since 2000 the approach has mainly been sum-over-histories path-integral-type-----variations on the spinfoam theme. And the spinfoam vertex amplitude issue is still not settled. The last big paper on that was by Laurent Freidel.

The LQG program is not exactly aimed at quantizing GR, as far as I can see. The target is better described as getting a quantum geometry that has GR as largescale limit. My two cents worth.

 

[JustinLevy]

A question about these three approaches (CDT, Asymptotic Safety, LQG):

It is often believed that if matter with intrinsic spin is to be included in GR, you cannot use a (pseudo)Riemannian manifold. Starting with the Einstein-Hilbert action, and allowing affine torsion you get Einstein-Cartan theory.

My question is, since many of the studies of these theories currently are focussing on the case without matter (or it is not agreed yet on how to add matter), what exactly do they allow to vary in the Einstein-Hilbert action?

If they are only allowing the metric to vary, and therefore excluding affine torsion, it seems like they are excluding what was realized even in 1922 must be included even in the classical GR for mathematical consistency with what we know about matter (even though the effect would probably be negligible in astronomical tests of GR ... although there are recent claims that some effects have been seen: Binary pulsar PSR J0737-3039A/B may be a test of relativistic spin-orbit coupling (Breton, et al, 2008) ).

 

[marcus]

A question about these three approaches (CDT, Asymptotic Safety, LQG):
... you cannot use a (pseudo)Riemannian manifold. ... what exactly do they allow to vary in the Einstein-Hilbert action?

If they are only allowing the metric to vary,...

About LQG:
LQG does not use a pseudoRiemannian manifold at any point in the construction.
There is no metric. So naturally they are not "allowing the metric to vary."
A quantum state of geometry is a labeled graph---spin labels on the edges.
Matter fields to be included by adding labels---Rovelli discussed this in his invited talk at Strings 2008.

About CDT:
CDT does not even use a differential manifold much less a pseudo-Riemannian. There is of course no metric anywhere in the picture. The geometry of the universe is represented by a triangulation by equal size blocks. The Einstein-Hilbert action is implemented combinatorially, by counting (the different orders of triangular) blocks. You might want to take a look at the Loll SciAm article in my sig---and refs therein. It turns out that counting blocks tells you collective curvature, because it tells stuff analogous to how many equilateral triangles are meeting around a point, on average. The E-H action turns out to have an elegant combinatorial form.

The basic space that is triangulated is about the simplest thing imaginable, a topological space like S³ x R¹. No coordinate charts, no smoothness, no metric, not even a predetermined (e.g. Hausdorff) dimensionality...

About AS:
Asymptotic Safety QG is unusual in that it does involve a metric. In fact there is a whole sequence of metrics depending on scale or energy. I'm sorry to say I don't know of any obstacle to including matter with spin in Reuter's picture. Matter is included in several of Reuter's papers, and as I recall, matter terms are added to the action. I don't specifically remember spin-ful matter terms, but there could have been. I'll keep an eye out and see if we can clarify this point regarding Asymptotic Safety QG.

 

[JustinLevy]

About LQG:
LQG does not use a pseudoRiemannian manifold at any point in the construction.
There is no metric. So naturally they are not "allowing the metric to vary."
A quantum state of geometry is a labeled graph---spin labels on the edges.
Matter fields to be included by adding labels---Rovelli discussed this in his invited talk at Strings 2008.

I don't understand at what point the Einstien-Hilbert action is used, but I thought they used GR written with Ashtekar variables ... which still requires a metric does it not? http://en.wikipedia.org/wiki/Ashtekar_variables

About CDT:
CDT does not even use a differential manifold much less a pseudo-Riemannian. There is of course no metric anywhere in the picture. The geometry of the universe is represented by a triangulation by equal size blocks. The Einstein-Hilbert action is implemented combinatorially, by counting (the different orders of triangular) blocks. You might want to take a look at the Loll SciAm article in my sig---and refs therein. It turns out that counting blocks tells you collective curvature, because it tells stuff analogous to how many equilateral triangles are meeting around a point, on average. The E-H action turns out to have an elegant combinatorial form.

It may be discrete, but the geometry is still there. The emergence of an 'effective' differential manifold is superfluous to my question.

It seems to me that since their definition of the geometry is purely combinatorial (and nothing they use depends on the order of counting), then it REQUIRES the discrete version of the Ricci curvature tensor to be symmetric and therefore there is no affine torsion.

It seems to me they are effectively only varying the metric in the Einstein-Hilbert action, and restricting themselves to configurations with no affine torsion which we know must be allowed if matter with spin is to be included later.

About AS:
Asymptotic Safety QG is unusual in that it does involve a metric. In fact there is a whole sequence of metrics depending on scale or energy. I'm sorry to say I don't know of any obstacle to including matter with spin in Reuter's picture. Matter is included in several of Reuter's papers, and as I recall, matter terms are added to the action. I don't specifically remember spin-ful matter terms, but there could have been. I'll keep an eye out and see if we can clarify this point regarding Asymptotic Safety QG.

Remember, Einstein-Cartan comes from the same action (Einstein-Hilbert). It just depends on what you allow to vary (just the metric -> Pseuo-Reimannian manifold), or more (to allow affine torsion -> Riemann–Cartan geometry).

Without affine torsion you could only allow scalar particles in the theory (no electrons, quarks, etc.).

 

[atyy]

There are several versions of AS quantum gravity but the most common is probably Reuter's Einstein-Hilbert truncation where he takes the E-H action (with cosmo constant Lambda) and let's both G and Lambda run with scale.

So to me this looks rather much like E-H action but with different G and Lambda used at high energy. I suppose the real question is how different does it have to get before you get a qualitative difference in micro behavior.

Please let me know if I am overlooking some significant detail here.

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

With Loll CDT it may be important to notice that the triangular building blocks have a uniform size and that the size goes to zero in the limit. Loll has stressed this and has written explicitly that the theory has no minimal length.

At each stage of approximation to the path integral, characterized by a certain block size, they use the Regge form of the E-H action.

It would seem to me that as you let the block size go to zero what happens is that you get closer and closer to the E-H action, at any fixed scale or energy you want to consider.

So it would seem to me that the Loll method does NOT deviate from E-H at any level of energy. Can this be right? You say something different. What I am saying is, pick some energy or scale where you think Loll might be deviating. Now pick the size of the triangles to be smaller than that by 10 orders of magnitude, and check to see if there is any significant difference from the E-H action. I don't see how there could be, intuitively.

At least to the extent that the conventional E-H could be defined on the rough geometries which Loll's method uses---the jagged piecewise-linear sample chosen as a regularization of the path integral at each scale.

I may be overlooking something here, haven't thought about this. I'd be interested to know what you think, as you may have examined this more carefully.
===================

My impression is that in principle, Asymptotic Safety is the possibility that the high energy action may contain all sorts of additional terms in the metric which will cure the unrenormalizability, and reduce to Einstein-Hilbert form at low energy. The problem is they don't know which additional terms, if any, will do such a thing. The truncated forms are just guesses they started with, in case something simple works, but not a fundamental part of the point of view.

For example, the commentary just before Eq. 3.5 where the unknown terms are represented by " ..." in Litim, http://arxiv.org/abs/0810.3675. Also, Eq. 1.3.2 in Percacci, http://arxiv.org/abs/0709.3851.

CDT takes the E-H action in non-redundant variables, then for simulations they impose a cut-off, making it a low-energy action. Their simulations suggests a high energy limit exists, but is it the E-H action? It looks like they make a guess for an action in Eq. 27, which only becomes something from the E-H action on large spatial scales Eq. 31, 33. Ambjorn et al http://arxiv.org/abs/hep-th/0604212.

This is why I wonder if CDT and AS aren't closer to one of the emergent approaches like strings or condensed matter, since both say that new things have to be added? CDT and AS consider new terms but the same number of degrees of freedom. The emergent guys consider that even the number of degrees of freedom changes.

 

[atyy]

Is this in any way related to the Holographic Principle? Where the inside 2D surface of a sphere contains all the bits to describe a 4D volume?

Apprently a possibility: "In the context of the asymptotic safety scenario, on the other hand, the presumed reduction to effectively two-dimensional propagating degrees of freedom is a consequence of the renormalization group dynamics, which in this case acts like a ‘holographic map’. This holographic map is of course not explicitly known, nor is it off-hand likely that it can be described by some effective string theory." Niedermaier and Reuter, http://relativity.livingreviews.org/Articles/lrr-2006-5/index.html

 

[atyy]

A question about these three approaches (CDT, Asymptotic Safety, LQG):

It is often believed that if matter with intrinsic spin is to be included in GR, you cannot use a (pseudo)Riemannian manifold. Starting with the Einstein-Hilbert action, and allowing affine torsion you get Einstein-Cartan theory.

"The typical starting point of Loop Quantum Gravity is a modification of the Einstein-Cartan action. The Einstein-Cartan action differs from the Einstein-Hilbert action in that it allows for non-zero torsion in the presence of matter. ... the Einstein-Cartan and Einstein-Hilbert approaches will likely yield different quantum theories. This has been verified in three-dimensional gravity, where the quantization is fairly well understood. In 3 + 1 gravity, torsion likely does play a fundamental, though poorly understood, role in the standard formulation of Loop Quantum Gravity. Thus, in choosing to begin with the Einstein-Cartan action, we are taking a theoretical leap of faith. However, again from theoretical arguments, this leap is not without ample justification."

In Search of Quantum de Sitter Space: Generalizing the Kodama State
Andrew Randono
http://arxiv.org/abs/0709.2905

 

[JustinLevy]

Thank you for that. It is good to see LQG does allow torsion.
He also confirms that this is important even in vacuum formulations (which is why I am still worried about the CDT approach)
"From a quantum perspective, allowing for torsion versus excluding it by hand changes things considerably. Even if we restrict attention to the vacuum sector, since quantum dynamics depends on contributions from the the entire space, including off-shell contributions where the equations of motion do not hold, the Einstein-Cartan and Einstein-Hilbert approaches will likely yield different quantum theories. This has been verified in three-dimensional gravity, where the quantization is fairly well understood."


While it may just be a semantics issue, I don't understand why he refers to the Einstein-Hilbert and Einstein-Cartan theories as different actions. You get different dynamics because you allow variations that contain torsion in the Cartan theory, but you still use the same action. Even in the paper he admits both use the action
$$S=\frac{1}{2\kappa}\int (R -2\Lambda)\sqrt{-g} d^4 x$$
but that when not using the Einstein-Cartan approach, the spin-torsion coupling "is excluded by hand".

Am I missing something here, or is it merely a semantics thing that since different variations are allowed he refers to them as different actions?

 

[p-brane]

It is good to see LQG does allow torsion

Unfortunately, what it doesn't allow is a correct description of nature.

 

[JustinLevy]

Unfortunately, what it doesn't allow is a correct description of nature.

Aren't they free to add in any matter lagrangian they want? Or is there some constraint, or lacking a part of quantum-gravity, that hinders LQG?

 

[atyy]

A question about these three approaches (CDT, Asymptotic Safety, LQG):

It is often believed that if matter with intrinsic spin is to be included in GR, you cannot use a (pseudo)Riemannian manifold. Starting with the Einstein-Hilbert action, and allowing affine torsion you get Einstein-Cartan theory.

Rovelli formulates GR on p24 in terms of a tetrad field rather than a metric so that fermions can be added, similar to supergravity it seems. He discusses fermions in LQG on p207, which apparently can be added naturally, and surprisingly it is adding a scalar that requires contortion. Maybe he should predict non-existence of the Higgs?
http://www.cpt.univ-mrs.fr/~rovelli/book.pdf