# Fractal LQG spacetime and renormalization of the Immirzi parameter

1. Jul 1, 2010

### tom.stoer

Hi,

this is not based on detailed work but just an idea which arised comparing causal dynamical triangulations and loop quantum gravity.

In CDT it seems reasonable to treat spacetime as a fractal. That means there is no limit or minimum length in the triangulations, but the triangulations are "nested" like a fractal. This corresponds to coarse graining like the block-spin transformations in the renormalization group approach for the Ising model.

Wouldn't it be possible to apply a similar method in LQG? That would mean nesting of spin networks. In a sense a vertex would be an effective description of a volume which (via coarse graining) resolves into finer volumina with new vertices. In one paper studying LQG black holes I found a similar idea. The isolated horizon in LQG translates into the idea that one can replace all vertices (intertwiners) forming the spacetime inside the horizon by one single huge (!) interwiner representing the whole black hole.

If one applies this idea to LQG as a whole one must answer the question what happens with a minimum length. This could be achieved via renormalization of the Immirzi parameter. So coarse graining of spin networks goes hand in hand with a running Immirzi parameter. Btw.: the length, area and volume operators are not necessarily Dirac observables, therefore their quantized spectrum does not automatically carry over to physical observables.

Does this idea make sense?

2. Jul 1, 2010

### marcus

That reminds me of this paper: http://arxiv.org/abs/0905.4916

3. Jul 1, 2010

### tom.stoer

Thanks for the reference; this is the paper I am talking about.

What do you think? Am I crazy or could this be a viable picture of LQG spacetime?

4. Jul 1, 2010

### marcus

I can't say anything useful, Tom. The idea of progressively refining spin networks while letting some parameter run is attractive. It would be related to LQG, but something different.
I'm too sleepy to think about it right now. It would be nice if someone like f-h would comment. I will turn in now and get back to it in the morning.

5. Jul 1, 2010

### francesca

literature

Fractal Quantum Space-Time
Leonardo Modesto
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.

Fractal Dimension in 3d Spin-Foams
Francesco Caravelli, Leonardo Modesto
In this paper we perform the calculation of the spectral dimension of the space-time in 3d quantum gravity using the dynamics of the Ponzano-Regge vertex (PR) and its quantum group generalization (Turaev-Viro model (TV)). We realize this considering a very simple decomposition of the 3d space-time and introducing a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the space-time runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decreases from high to low energy. For the TV model the spectral dimension at hight energy increase with the value of the cosmological constant. At low energy the presence of the cosmological constant does not change the spectral dimension.

Fractal Space-Time from Spin-Foams
Elena Magliaro, Claudio Perini, Leonardo Modesto
In this paper we perform the calculation of the spectral dimension of spacetime in 4d quantum gravity using the Barrett-Crane (BC) spinfoam model. We realize this considering a very simple decomposition of the 4d spacetime already used in the graviton propagator calculation and we introduce a boundary state which selects a classical geometry on the boundary. We obtain that the spectral dimension of the spacetime runs from $\approx 2$ to 4, across a $\approx 1.5$ phase, when the energy of a probe scalar field decreases from high $E \lesssim E_P/25$ to low energy. The spectral dimension at the Planck scale $E \approx E_P$ depends on the areas spectrum used in the calculation. For three different spectra $l_P^2 \sqrt{j(j+1)}$, $l_P^2 (2 j+1)$ and $l_P^2 j$ we find respectively dimension $\approx 2.31$, 2.45 and 2.08.

6. Jul 2, 2010

7. Jul 2, 2010

### tom.stoer

thanks for the references ... so I will cancel my flight to Stockholm :-)