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Here is some background on Leonardo Modesto's new paper:
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."
marcus said: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."
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