Nice LQG paper by Modesto about dimension down with scale

[marcus]

This confirms a beautiful result of Modesto's earlier paper, that LQG seems to coincide with two other very different types of quantum geometry/gravity on the business of dimensionality declining continuously with scale. Fractal-like microstructure of space time. As scale decreases the observed dimensionality goes smoothly down from 4D to 2D.

Analogous to a fractal sponge which looks like a cube to a large wavelength probe but has a lot of holes, that can be revealed and probed with higher energy, and that make it have less dimensionality when seen under magnification. Like soapsuds is really 2D not 3D.

Analogous to a crumpled ball of wire. It looks like a 3D ball but if you examine closely you see a 1D wire.

The surprising thing is that both Loll Triangulations QG and Reuter AsymptSafe QG which are extremely different from each other, both got this result. And then, to top it off, Modesto showed that you get this kind of behavior with LQG, which is different from the other two.

Anyway, MTd2 found and posted the abstract for Modesto's new paper:
http://arxiv.org/abs/0905.1665
Fractal Quantum Space-Time
Leonardo Modesto
(Submitted on 11 May 2009)

"In this paper we calculated the spectral dimension of loop quantum gravity (LQG) using the scaling property of the area operator spectrum on spin-network states and using the scaling property of the volume and length operators on Gaussian states. We obtained that the spectral dimension of the spatial section runs from 1.5 to 3, and under particular assumptions from 2 to 3 across a 1.5 phase when the energy of a probe scalar field decreases from high to low energy in a fictitious time T. We calculated also the spectral dimension of space-time using the scaling of the area spectrum operator calculated on spin-foam models. The main result is that the effective dimension is 2 at the Planck scale and 4 at low energy. This result is consistent with two other approaches to non perturbative quantum gravity: "causal dynamical triangulation" and "asymptotically safe quantum gravity". We studied the scaling properties of all the possible curvature invariants and we have shown that the singularity problem seems to be solved in the covariant formulation of quantum gravity in terms of spin-foam models. For a particular form of the scaling (or for a particular area operator spectrum) all the curvature invariants are regular also in the Trans-Planckian regime."

BTW Modesto includes a picture of that fractal sponge that I was talking about.

You also get that in Loll's SciAm article. That is very good for a nontechnical article. I have the link in my sig. It gives a lot of intuition about ways in which spacetime microgeometry might be different, essentially for Heisenberg or quantum reasons, from the more familiar macro-scale geometry. Something that seems to happen consistently in Loll's computer simulations of universes.

It seems bizarre that this same behavior is coming up in the context of several different approaches to quantum geometry/gravity. Happened in the Horava-Lifgarbagez case too. And Benedetti recenty drew a connection with non-commutative geometry.

 

[Finbar]

I'd heard of this in the CDT and asymtotic theories. Now in LQG too! Ok this might be a bit of a way out idea... if we think of strings they trace 2d world sheets "in spacetime" right? But really there should be no spacetime in which these strings are moving instead the stings should define spacetime(assuming BI here). So could this 4 to 2d behaviour in other theories QG could be understood as them making conntact with string theory?

 

[marcus]

... really there should be no spacetime in which these strings are moving instead the stings should define spacetime(assuming BI here). So could this 4 to 2d behaviour in other theories QG could be understood as them making conntact with string theory?

I seem to recall having heard that idea coming up now and then other places. I haven't heard anything conclusive along those lines. (I think by BI you mean Background Independence, a putative version of string that is constructed without any prior geometric background for the strings to live in). Someone else besides me could better answer your question. Perhaps Modesto's result is suggestive of a contact between the two theoretical frameworks. Can't say.

 

[atyy]

I'd heard of this in the CDT and asymtotic theories. Now in LQG too! Ok this might be a bit of a way out idea... if we think of strings they trace 2d world sheets "in spacetime" right? But really there should be no spacetime in which these strings are moving instead the stings should define spacetime(assuming BI here). So could this 4 to 2d behaviour in other theories QG could be understood as them making conntact with string theory?

There's a comment about this in section 1.4 of http://relativity.livingreviews.org/Articles/lrr-2006-5/

 

[MTd2]

This result is not that surprising, I spoke about that earlier, and former papers indicated such link. But, let me put a physical interpratation for such phenomenon: the thing about fractals comes indeed from the failure from h-cobordism theorem in 4 dimensions, which means that it is impossible to map all kinds of topological equivalent manifolds using finitely many worldsheets, that is, smooth strings. You can do that with topological strings, I bet. But on the smooth side, you just solve that by the use of a casson handle, that is, a tree fractal like structure of strings.

The use of either triangulations or loops simulate the use of fractal like strings, and as one progresses towards infinite complexity, builing a casson handle, one gets closer to find a general resolvent for any general 4 manifolds, which also yiels an averaged (any of these results are based on average measures) 4-smooth well behaved as is used in GR. But this gives is a naturaly correspondence because when one go back from the averaged smooth to the general kind, one must consider all different perturbations, which is the case of assymtotic safety. The former one means that assymptotic safety is identifying all the expansions of that fractal struture to the path integrals obtained from GR.

 

[MTd2]

BTW, the use of 2d surfaces knoted in 4dimensions or above can emulate any group. So, there is a chance that any physics of string theory in whatever dimensonsm can be equivalentely be obtained in 4 dimensions by sufficiently complicated knots. Given the infinite compexity of these tree like structures, it would not be surprising that, for example, F-Theory is realisticaly equal to a Loopy Theory in 4 dimensions.