Causal Dynamic Triangulation Question

[cuallito]

Hi, in CDT, when one generates a random spacetime path to use in a calculation, what is the distribution of the angles of the triangles? Is it just the flat distribution between 0 and 90 degrees, some sort of gaussian thing, or something else?
I've asked on Stack Exchange and Quora, and no one answered on there. Hopefully someone on here will know. Thanks! :)

 

[marcus]

Hi, you could clarify your question. What do you mean by a "random spacetime path"?

The most distinctive thing about CDT is the application of Monte Carlo method where you "generate random spacetime geometries" to use in a calculation.

Using computer you generate thousands of random spacetimes and then you take the average---you use them in a Monte Carlo calculation. But I would not call them paths

MAYBE you call a spacetime a "path"? It is a kind of path through a world of possible spatial geometries. If that is how you are talking then I understand your question. It is in fact an excellent question.

 

[marcus]

The best way to understand CDT is to study the 3D case first, get to understand how you generate random 3D geometries, and then extend that to 4D.
A 3D spacetime is a 2+1 D thing. Spatially 2D plus one time dimension. Renate Loll has tutorials online where she goes thru the 2+1D case carefully first, and then extends to 3+1D.

A 3D spacetime geometry is built up causal layer by causal layer by putting down TETRAHEDRONS. They are all the same shape and size, chosen ahead of time, or at most they are all of TWO shapes, chosen in advance. So there is no problem choosing ANGLES all the angles of the building blocks are decided at the outset. The tets are REGULAR except they are allowed to be stretched/compressed in the time direction by a percentage chosen in advance.

The reason this slight (stretch or compress) deformation gives rise to TWO shape categories is that a tet can be put down in the layer in one of two ways
(1) flat side up or down
(2) one edge down like the keel of a boat and another edge up like the ridgepole of a roof.

This deformation by allowing a percentage deformation in the time direction is a fine point. You can imagine all the tets to be regular and the same size. All their edges the same length. And they get randomly jumbled together and build up a 3D spacetime layer by causal layer.

So there is no problem choosing angles, all the angles are multiples of the same set of regular tet angles.

Then if you want to take a RANDOM WALK thru a given spacetime geometry it is all made of regular CELLS and at each stage you go in one side of a cell and you have 4 choices where to go. You can come back out the same way you went in, or you have 3 other triangular sides you can go out.

That is why your question about PATH did not make sense to me in terms of a wandering random walk path. There is no "gaussian probability" needed to choose the angles along a random walk path. You are in a cell complex where all the angles are determined in advance, and they are the same for every spacetime the computer generates. They are basically the angles inherent in a regular tetrahedron (except for that distortion in the time direction which makes for two categories---two classes of "almost regular" tetrahedrons.)

 

[marcus]

BTW in case (you Cuallito or) anyone is especially interested in CDT there is a Jan Ambjorn et al paper "Recent Results in CDT" on the
third quarter most important paper (MIP) poll
https://www.physicsforums.com/threads/poll-third-quarter-2015-mip-most-important-qg-papers.835822/
It draws some connections with the "asymptotic safety" approach to quantum gravity and the dimensional reduction at extreme small scale which both AsymSafe qg and CDT both seem to share.
Measuring the dimension does, in fact, involve considering a random walk and return probabilities. The higher the dimension of the space he is walking in, the less likely that the walker will ever wander back to his starting place.
This is intriguing. One can have a geometry that has fractional dimension . Or one that has a full 4D at ordinary large scale and something like 3/2 dimensionality at very small scale. (Jan Ambjorn et al find, but this also can happen in AsymSafe QG)

One could look up "spectral dimension". Ambjorn et al is the wrong place to start because assumes too much prior research. Is not a beginning introduction. But it is nice to know there is current research going on in this. It is a third quarter 2015 paper.

Of course in CDT the random walk used to calculate the spectral dimension is very simple because the spacetime is all divided up into almost identical cells so at each step the walker's choice is finite and definite. There is no dilemma of choosing "angles" to turn===all is cut and dried.

 

[cuallito]

Yes Marcus, that is what I was trying to ask. By "path" I meant each of the spacetimes generated in the MC calculation.

Question: deforming a tetrahedron changes the angles, does it not? (Did you mean that the tet's are all uniformly scaled in the time direction?)

In reply to your third post: Sorry I really confused you! I was asking how each of the spacetimes in the sum/integral is generated, not really about taking a random walk along one of them! Hopefully we understand each other now!

(I called it a random walk since each spacetime is a random evolution from an initial state to a final state)

 

[cuallito]

To further clarify my question: What is the distribution of the angles of the triangles in these diagrams?
(Reference: the paper available here: http://www.sciencedirect.com/science/article/pii/S0370269313004711)

Spacetime with preferred foliation:

Without preferred foliation:

 

[marcus]

Ah ha! There is a catch. concentrate on the top picture---with a foliation.
You should imagine that as merely a schematic. Think of all the triangles as identical equilateral triangles. Then you will see that putting them together in various ways generates CURVATURE.

That is the whole idea of CDT---which uses identical building blocks---or at most two classes of identical blocks where one is allowed to stretch all the members of a class by the same amount in the time direction. The amount is predetermined.

So in this 2D spacetime they would be identical ISOSCELES triangles stretched or compressed in the time direction. This is decided in advance. So all the angles are predetermined.

The curvature is determined by how many blocks come together at point. In the equilateral case, if fewer than 6 come together at a point, then that point has positive curvature (a mountain) If more than 6 come together at that point, it has negative curvature (warped looking).

 

[cuallito]

Thanks!
So it's the number of triangles that meet at each vertex.
You can probably guess my next question :D
So what equation determines the distribution of the number of triangles at each vertex?

 

[marcus]

There is a Monte Carlo recipe for assembling the blocks into a random spacetime. So you generate thousands and measure what you want and average it over the whole batch. You probably don't have any simple equation giving probabilities of most of the things you might ask about. You discover probabilities by the Monte Carlo method, numerically, experimentally, so to speak.

I don't recall details of the recipe for generating random CDT spacetimes. But I can try to get you links. I read about this method about 10 years ago, in papers like this:
http://arxiv.org/abs/0711.0273
The Emergence of Spacetime, or, Quantum Gravity on Your Desktop
R. Loll
(Submitted on 2 Nov 2007)
Is there an approach to quantum gravity which is conceptually simple, relies on very few fundamental physical principles and ingredients, emphasizes geometric (as opposed to algebraic) properties, comes with a definite numerical approximation scheme, and produces robust results, which go beyond showing mere internal consistency of the formalism? The answer is a resounding yes: it is the attempt to construct a nonperturbative theory of quantum gravity, valid on all scales, with the technique of so-called Causal Dynamical Triangulations. Despite its conceptual simplicity, the results obtained up to now are far from trivial. Most remarkable at this stage is perhaps the fully dynamical emergence of a classical background (and solution to the Einstein equations) from a nonperturbative sum over geometries, without putting in any preferred geometric background at the outset. In addition, there is concrete evidence for the presence of a fractal spacetime foam on Planckian distance scales. The availability of a computational framework provides built-in reality checks of the approach, whose importance can hardly be overestimated.
22 pages, 11 figures

and earlier papers like these:
http://arxiv.org/abs/hep-th/0604212
Quantum Gravity, or The Art of Building Spacetime
J. Ambjorn, J. Jurkiewicz, R. Loll
(Submitted on 28 Apr 2006)
The method of four-dimensional Causal Dynamical Triangulations provides a background-independent definition of the sum over geometries in quantum gravity, in the presence of a positive cosmological constant. We present the evidence accumulated to date that a macroscopic four-dimensional world can emerge from this theory dynamically. Using computer simulations we observe in the Euclidean sector a universe whose scale factor exhibits the same dynamics as that of the simplest mini-superspace models in quantum cosmology, with the distinction that in the case of causal dynamical triangulations the effective action for the scale factor is not put in by hand but obtained by integrating out in the quantum theory the full set of dynamical degrees of freedom except for the scale factor itself.
22 pages, 6 figures. Contribution to the book "Approaches to Quantum Gravity", ed. D. Oriti, Cambridge University Press

http://arxiv.org/abs/hep-th/0509010
The Universe from Scratch
R. Loll, J. Ambjorn, J. Jurkiewicz
(Submitted on 1 Sep 2005)
A fascinating and deep question about nature is what one would see if one could probe space and time at smaller and smaller distances. Already the 19th-century founders of modern geometry contemplated the possibility that a piece of empty space that looks completely smooth and structureless to the naked eye might have an intricate microstructure at a much smaller scale. Our vastly increased understanding of the physical world acquired during the 20th century has made this a certainty. The laws of quantum theory tell us that looking at spacetime at ever smaller scales requires ever larger energies, and, according to Einstein's theory of general relativity, this will alter spacetime itself: it will acquire structure in the form of "curvature". What we still lack is a definitive Theory of Quantum Gravity to give us a detailed and quantitative description of the highly curved and quantum-fluctuating geometry of spacetime at this so-called Planck scale. - This article outlines a particular approach to constructing such a theory, that of Causal Dynamical Triangulations, and its achievements so far in deriving from first principles why spacetime is what it is, from the tiniest realms of the quantum to the large-scale structure of the universe.
31 pages, 5 figures, commissioned review article.

http://arxiv.org/abs/hep-th/0505154
Reconstructing the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
(Submitted on 17 May 2005)
We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations. A closer look at the quantum geometry reveals a number of highly nonclassical aspects, including a dynamical reduction of spacetime to two dimensions on short scales and a fractal structure of slices of constant time.
52 pages, 20 postscript figures, Physical Review D.

 

[cuallito]

Thanks, Marcus!

 

[cuallito]

Hey, question for Marcus or anyone else that would know...
So CDT in the continuum limit becomes Horava-Lifschitz gravity, at least for 1+1 and 2+1 dimensions.
However, once the causality (preferred foliation) requirement is relaxed to make *locally causal* dynamical triangulations (LCDT), do we know the continuum limit?
In R. Loll's recent paper on LCDT in 1+1, she said it looks like it may belong to a new different universality class than CDT.

 

[marcus]

Hey, question for Marcus or anyone else that would know...
So CDT in the continuum limit becomes Horava-Lifschitz gravity, at least for 1+1 and 2+1 dimensions.
However, once the causality (preferred foliation) requirement is relaxed to make *locally causal* dynamical triangulations (LCDT), do we know the continuum limit?
In R. Loll's recent paper on LCDT in 1+1, she said it looks like it may belong to a new different universality class than CDT.

You are way ahead of me now, Cuallito
I have not kept up with Loll's latest work or with CDT in general as much as I should. Hopefully someone else here will be able to reply. For our convenience, since your pointing it out makes me curious, I'll copy the abstract of that paper. Other people may want to take a look too.
http://arxiv.org/abs/1507.04566
Locally Causal Dynamical Triangulations in Two Dimensions
Renate Loll, Ben Ruijl
(Submitted on 16 Jul 2015)
We analyze the universal properties of a new two-dimensional quantum gravity model defined in terms of Locally Causal Dynamical Triangulations (LCDT). Measuring the Hausdorff and spectral dimensions of the dynamical geometrical ensemble, we find numerical evidence that the continuum limit of the model lies in a new universality class of two-dimensional quantum gravity theories, inequivalent to both Euclidean and Causal Dynamical Triangulations.
Phys. Rev. D 92, 084002 (2015)
http://inspirehep.net/record/1383120?ln=en