Exploring (2+1)-D Quantum Gravity with Causal Dynamical Triangulations

[marcus]

Another first.

http://arxiv.org/abs/0704.3214
(2+1)-Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
D. Benedetti, R. Loll, F. Zamponi
38 pages, 13 figures

"We perform a non-perturbative sum over geometries in a (2+1)-dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometries. This simplifies the combinatorial problem of counting geometries just enough to enable us to calculate the transfer matrix between boundary states labelled by the area of the spatial universe, as well as the corresponding quantum Hamiltonian of the continuum theory. This is the first time in dimension larger than two that a Hamiltonian has been derived from such a model by mainly analytical means, and opens the way for a better understanding of scaling and renormalization issues."

 

[marcus]

If I remember right, Dario Benedetti got his Masters in Rome in Statistical Physics, and had the good idea to go to Utrecht for his PhD.

He joined Renate Loll's group IIRC in Fall 2005.

Loll and Ambjorn had just had this run of amazing papers where they got Simplex gravity to work not only IN four dimensions but they made the four-dimensionality emerge at macro-scale from microscopic quantum geometric dynamics (where at microscopic Planck scale the dimensionality might not even be four, or even an integer, it might be fractional, if even defined)
the emergent dimensionality becomes a quantum observable, and it is not something predictable with absolute certainty or specified in advance.

We discussed several of the 2005 Loll papers here at PF.

So Benedetti went to Utrecht to work on that kind of simplicial QG (called "Causal Dynamical Triangulations") and his PhD thesis is scheduled for formal acceptance June 2007
D. Benedetti: Quantum gravity from simplices: analytical investigations on causal dynamical triangulations, Doctoral Thesis, Utrecht University, June 2007.

more to discuss, will get back to this as time permits.

 

[marcus]

Whenever I look over a paper by Loll, or Thiemann, or Martin Bojowald, in the back of my mind I am thinking of Hans Kastrup at Aachen.

His PhD thesis advisor was Heisenberg, whose thesis advisor was Sommerfeld, and so on in an unbroken line back to Gauss.
Several similar lines uprooted and transplanted to North America in the course of the last century but this is a native north-european line.

I think that part of the game is not what papers you write but what papers your STUDENTS write----especially the ones you train and motivate.
I see Hans Kastrup as not especially well-known in terms of papers he has written, but I see him as an outstanding north-european QG trainer.
Or as the sensei of that particular dojo, if that means anything.

I don't know the relevant details but I think he influenced all three of those people. And you can see in all three of them a certain savage determination to get the physics to work against extreme odds.
Quantum geometrical QG is a hellish hard problem or, in the understatement Loll uses in the present paper, "challenging".

When Kastrup himself goes into history, it will be interesting to see who the region's top trainers are that replace him.
================

As for Bojowald, largely on his own he created Loop Cosmology, discovered that it has the correct low-energy limit (duplicates Gen Rel anywhere that Gen Rel doesn't fail) and can pass thru the big bang without breaking down. His latest paper described a solvable model of LQC. Now the effort is to extend that to more general cases relaxing restrictions and capturing more of the full LQG.

As for Thiemann he has hammered away with relentless determination for some four years, developing the Master Constraint and AQG (algebraic QG) version. I am still rubbing my eyes and wondering if it can possibly work as well as he claims. But he was the one designated to give the LQG lecture series at this winter's QGQG school and he is the one giving the invited talk on LQG at the Morelia Loops '07 conference. That means quite a few of the insiders think he is on to something proper---that his new way of doing LQG is worth a try-out. Essentially by himself he revamped LQG and this just appeared in the past two years, mostly in 2006. It is a risky business and might blow up on him----took considerable guts to do it.

In both cases they seem to attack problems frontally with a clear idea of the physical goal-----to resolve the cosmological singularity in one case, to construct a satisfactory LQG dynamics in the other. they don't seem to stop and wait much for some new abstract mathematics to be invented or go off very often looking for some untried new methods. Rather they attack the problem in the old iron-beater smithy way, "hammer and tongs", with whatever tools are handy. I exaggerate and distort here---this is a caricature. But I am trying to suggest a distinctive style which these three people have in common.

As for Loll, it would look as if she has gotten stalled repeatedly with her extremely difficult SIMPLICIAL approach but each time she recovered forward momentum. The basic idea seems solid---in 1961 Tulio Regge discovered that you could formulate Gen Rel WITHOUT COORDINATES, just by letting triangles swarm together. And not only triangles but the higher dimensional analogs of them: tetrahedra and four-simplexes.
the whole Gen Rel could be formulated with this crawling-all-over-itself ANT-HEAP aggregate of four-simplexes-----Loll calls them "LEGO-BLOCKS" after the popular children's construction game.
It has been known since the 1961 paper of Regge that this works. So it seems like an obvious approach to pursue to Quantum Gravity.

However people started failing at this already in the early 1990s. Loll was not the first. I think she got in maybe 1996-1998. The difficulty is there is no efficient mathematics to get a handle on the milions of ways the "Lego-blocks" can combine and stick together. It is a combinatorial problem of the worst sort.

So again and again they looked hopelessly stalled, and one wouldn't hear anything. Then a year or two later somebody (often Loll and some co-workers) would come out with fresh progress. Presumably somebody had been quietly beating their head against the wall and finally the wall cracked a little. Well this is just one way to tell the story and I actually should not be trying to describe it because I have no direct contact with these people, or only the rarest most tangential. I am almost completely unaware of the relevant details. But that is how it looks to me.

Anyway Loll and Benedetti (and Benedetti's old friend from Rome Francesco Zamponi) have gotten a Hamiltonian out this insanely combinatorial buildingblock model of quantum geometry. Those building blocks are the CLAY of gravity. If you want to get your hands dirty and touch gravity and feel it squirm, then get your hands on that kind of RUBIKSCUBE mass of simplexes. And it is the worst possible thing to try to deal with analytically, because the most atomized and irregular and and symmetryless. And they worked on it for two years and got a Hamiltonian. Heh heh. Benedetti probably had to work pretty hard. :-)

 

[marcus]

Here is Ambjorn's abstract of the talk he is to give a Loops '07:

4d quantum gravity as a sum over histories
Jan Ambjorn

"In this plenary talk I will review the attempts to formulate 4d quantum gravity as a sum over histories in such a way that computer simulations can be performed. I will report on computer simulations of a quantum universe with a positive cosmological constant as well as a quantum universe where test matter is included."

Loll group had a series of important papers in 2004 and 2005 reporting results of computer simulations. It looked very promising and one assumed they would continue to run bigger and more realistic computer simulations and report further findings. But to my surprise they published almost nothing in 2006 until the middle of November (except a review paper and some shorter notes).

So I was wondering, did they continue with the numerical work? Did they encounter intractable obstacles? What is the Utrecht group doing? We are still somewhat in suspense about that, although Ambjorn's talk could shed some light.

And then in November 2006 there appeared a major paper by Loll and Benedetti
http://arxiv.org/abs/gr-qc/0611075
Quantum Gravity and Matter: Counting Graphs on Causal Dynamical Triangulations
D. Benedetti, R. Loll
40 pages, 15 figures, 13 tables
(Submitted on 14 Nov 2006)

"An outstanding challenge for models of non-perturbative quantum gravity is the consistent formulation and quantitative evaluation of physical phenomena in a regime where geometry and matter are strongly coupled. After developing appropriate technical tools, one is interested in measuring and classifying how the quantum fluctuations of geometry alter the behaviour of matter, compared with that on a fixed background geometry.
In the simplified context of two dimensions, we show how a method invented to analyze the critical behaviour of spin systems on flat lattices can be adapted to the fluctuating ensemble of curved spacetimes underlying the Causal Dynamical Triangulations (CDT) approach to quantum gravity. We develop a systematic counting of embedded graphs to evaluate the thermodynamic functions of the gravity-matter models in a high- and low-temperature expansion. For the case of the Ising model, we compute the series expansions for the magnetic susceptibility on CDT lattices and their duals up to orders 6 and 12,.."

I wasn't sure what to make of this. Perhaps we can interpret it better in light of the paper of Loll Benedetti Zamponi which just appeared and which seems to head out in still another completely new direction----in a restricted special case to derive a Hamiltonian governing the volume of the (spatially 2D toy) universe.
Is the approach using "product" triangulations legitimate? (Loll has an argument that it is at least in 3D) Will they be able to extend this to 4D?

 

[jal]

Here is another paper.
http://arxiv.org/PS_cache/hep-th/pdf/0505/0505004v2.pdf
Foliations and 2+1 Causal Dynamical Triangulation Models
Tomasz Konopka
10 Feb 2006

If you do not want to read the paper then here is a short explanation.

http://astro.uwaterloo.ca/~tkonopka/phys/p_foliations.htm

Put another way, I play around with the definition of the building blocks and substitution rules and see whether or not the conclusions that can be drawn from the causal dynamical triangulation models change.

I would presume that not all foliation structure are allowed. Volume would only be preserved within a hex configuration to a cubic configuration.
jal

 

[Chronos]

Getting a workable Hamiltonian beyond 2d is an extraordinary accomplishment. I am curious where this will go. Stunned silence appears to be the immediate reaction.

 

[jal]

Chronos
Getting a workable Hamiltonian beyond 2d is an extraordinary accomplishment. I am curious where this will go. Stunned silence appears to be the immediate reaction.

I’m not so sure that the silence is intentional. Look at the links, names, group, (enrage), and Related Networks from Renate Loll home page. http://www.phys.uu.nl/~loll/Web/title/title.html

Not all of the eggs are in one basket.

jal
‘Deterministic systems’, ‘Limiting Curvature Construction’, ‘Quantum Geometry’, and ‘QMLS’.
Who will be the “math kid” that can combine all of the approaches?

Since I’m not on their mailing list, I will presume that they are searching for a 3d model that they can “crunch” with the numbers.
They will need to move the time slices to arrive at some probable models that they can take to dynamic simulations.
My thought are that they will move their time slice to half way between the hex. lattice and the cubic lattice. As a result, they would get a sine curve/wave which will also resolve time and the “arrow of time”. It would arise from the dynamics.
Doe anyone have any thoughts on moving the time slice? Other approaches?
Maybe some of the “math kids” will enlighten us or maybe a fly on the wall?
jal

 

[Chronos]

I think I follow the concept, but 1/2 the distance rule strikes me as too linear. Surely it must be more like 1/pi, 2gc/ha, or some derivable of some combination of fundamental constants [background independence].

 

[jal]

My thought are that they will move their time slice to half way between the hex. lattice and the cubic lattice.

Perhaps a different phrasing would be better.
Look at
http://arxiv.org/PS_cache/gr-qc/pdf/0610/0610140v1.pdf
Multiple-event probability in general-relativistic quantum mechanics
Frank Hellmann, Mauricio Mondrago, Alejandro Pere, Carlo Rovelli
27 Oct 2006

This is based on the observation that a multiple-event probability, such as P ⇒ ′ ′′ can always be reinterpreted as a single-event probability, once the dynamics and the quantum nature of the apparatus making the measurements are taken into account. If we do so, the time order gets naturally coded into the dynamics of the system. This strategy provides a general way for dealing with multiple-event probabilities in general relativistic quantum mechanics.

Here is the follow up paper.
http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.0006v1.pdf
Multiple-event probability in general-relativistic quantum mechanics: a discrete model
Mauricio Mondragon, Alejandro Perez, Carlo Rovelli
30 April 2007

Abstract
We introduce a simple quantum mechanical model in which time and space are discrete and periodic.

(a two-dimensional extended configuration space)
I understand that they are positioning the time slice between the two observation.
jal