Questions on Causal Dynamical Triangulations

[skippy1729]

The 4D triangulations are made up of a sequence of 3D triangulations or slices which are "glued" together with {4,1}, {1,4}, {3,2} and {2,3} 4-simplices. The {A,B} notation denotes A vertices on the time=1 slice and B vertices on the time=t+1 slice. The 3D triangulations are constrained to have the same topology (usually S3) and are made up of equilateral spatial tetrahedrons. So a tetrahedron at time=t forms the base of a {4,1} simplex with one point at time=t+1 and likewise a tetrahedron at time=t+1 forms the base of a {1,4} simplex with one point at time=t. The "gaps" are filled with the {3,2} and {2,3} simplices.

My first question is: Intuitively or algorithmically, how are these "gaps" filled with the {3,2} and {2,3} simplices? I would be happy with an explanation or a reference where it is clearly explained. My 4D geometric intuition is severely challenged.

Second question: There are three or so research centers where CDT is studied. All of their "experiments" are based on numerical simulations. Is any of the simulation source code "open source" or is it all proprietary.

Thanks, Skippy

 

[marcus]

Second question first

...Second question: There are three or so research centers where CDT is studied. All of their "experiments" are based on numerical simulations. Is any of the simulation source code "open source" or is it all proprietary.
...

My understanding is that the Utrecht code, by Loll's group, is freely available. I spoke with Steve Carlip (UC Davis) about their CDT project and what he implied was they COULD HAVE used the already existing Utrecht code but they chose to write their own. Somehow it is more like "repeating the experiment" in the tradition of empirical science, if you write your own code. Then if it happened that the Utrecht group had made a programming error and their beautiful results were all the result of a mistake (! unlikely but conceivable) then hopefully you would not make the mistake and you would get different results. Experiments should be repeated from scratch. And these are in a certain way "numerical" experiments.

So Carlip's UC Davis grad student had to write the whole CDT monte carlo code. And he did. And it runs. I saw some graphic presentation of output.

I haven't personally verified that anybody can get Loll's code. She says it can run on a a work station. I believe however that it is open source and available without charge. If you were to email her she might want to know if you are connected with an institution, using it for research, for a PhD thesis, or whatever.

I don't know if there are any requirements or restrictions. You'd have to write Loll email and say "what requirements? what restrictions? if any." and she would probably answer a simple question like that. Why not?

Carlip said that Joe Henson is running the Utrecht CDT code at Perimeter in Canada. At least I think this is what he said. My understanding is Henson did not write his own, like Carlip's grad student did.

Jerzy Jurkiewicz in Warsaw is another member of Loll's group. He and his PhD students probably run Utrecht code, because Jurkiewicz helped develop it. There must be more than 3 places now if you include Warsaw.

Joe Henson would be another person you could write to. He is postdoc at Perimeter.
Used to be postdoc at Utrecht. Maybe he is less busy than Loll and would respond quicker.
==============

The 4D triangulations are made up of a sequence of 3D triangulations or slices which are "glued" together with {4,1}, {1,4}, {3,2} and {2,3} 4-simplices. The {A,B} notation denotes A vertices on the time=t slice and B vertices on the time=t+1 slice. The 3D triangulations are constrained to have the same topology (usually S3) and are made up of equilateral spatial tetrahedrons. So a tetrahedron at time=t forms the base of a {4,1} simplex with one point at time=t+1 and likewise a tetrahedron at time=t+1 forms the base of a {1,4} simplex with one point at time=t. The "gaps" are filled with the {3,2} and {2,3} simplices.

You have it exactly right. I remember learning this from a 2001 paper that has a lot of diagrams and works up from the 2 and 3D case.

My first question is: Intuitively or algorithmically, how are these "gaps" filled with the {3,2} and {2,3} simplices? I would be happy with an explanation or a reference where it is clearly explained. My 4D geometric intuition is severely challenged.

As I recall the way you gradually grow your intuition is you first study the 3D case. Here is the 2001 paper:
http://arxiv.org/abs/hep-th/0105267

For starters forget 4D and forget curved. Focus your mind on 3D and uncurved.
So space is flat 2D. Imagine it paved with equilateral triangles.
Imagine two flat 2D copies of space each paved with equilateral triangles. Now you have to make the tetrahedron sandwich that joins them.

The tets are (3,1) and (1,3) and (2,2) -----when you go up to 4D that (2,2) case will spit into two cases .

Imagine them. (3,1) is like a pyramid point up, sitting on its base. (1,3) is a point down pyramid.

A (2,2) tetrahedron is like resting on a horizontal bottom edge that runs north-south, like the keel of a boat.
And it has a horizontal top edge that runs east-west, like the ridge of a conventional peaked roof of a house.

Say you have two (3,1) tets sitting side by side, their two edges meeting along a N-S line (on the time=t ground)
then that makes a perfect gap or crevice where a 2,2 tet can fit between them.
Its keel runs N-S and its roof ridge runs E-W, joining their two vertices.

Loll allows the tets to not be equilateral. To have their horizontal edges be shorter than their "timelike" rising edges.
When you are learning, FORGET this extra complication. Consider everything simple and flat and equilateral. Make it easy for your mind.

I have even cut equilateral tets out of modeling clay. One time I also cut small tets out of carrot or potato in the kitchen.
Just to fit together and play with. Why not? You can eat the carrot later and it is very healthy.

Then when you are very comfortable with the flat 3D case, it is time to advance to curved 3D, and then to 4D. You can ask again later if you find it helpful to discuss this kind of thing. the main thing is to be patient and gradual with your grey matter and its intuition. Then the whole business actually can proceed quickly, paradoxically because you are not rushing.

 

[skippy1729]

Thanks, Marcus. I have read http://arxiv.org/abs/hep-th/0105267 several times; it seems to be the most complete presentation of the "nuts and bolts" of the method. Once I get the algorithm set in my brain I want to try and see if it can be run on some parallel hardware which I already have. Having the source code of something that works will make debugging easier.

Skippy

 

[RUTA]

marcus,

I read the arXiv paper at your link, I think your explanation is better :-)

Since my approach is so much like CDT and quantum Regge calculus (QRC), perhaps it's safe for me to mention it here and ask a question. If not, I guess this post will just be deleted in which case I hope you will respond via a private message.

The main difference between my approach and those of QRC and CDT is that I don't have empty spacetime, so I want my 4D building blocks to directly connect sources (in parlance of QFT). [This is motivated by choosing constitutive, rather than causal, non-locality to account for violations of Bell's inequality, see http://arxiv.org/abs/0908.4348 if you're interested.] Now that I've a picture of how CDT uses simplices to foliate spacetime (thanks to you), I don't see a basis for worldlines. As I understand RC the energy-momentum content resides on links, but I don't see worldlines composed of simplicial links, it's too "rough" given the triangular nature of simplices, cf., hypercubes. I see the stress-energy tensor (SET) describing momentum flux through the faces of variously oriented 2D surfaces (standard GR). I know the boundary of a boundary principle (BBP) is associated with the structure dual to the Regge thatch [Miller, W.A.: The Geometrodynamic Content of the Regge Equations as Illuminated by the Boundary of a Boundary Principle. Foundations of Physics 16, 143-169 (1986)] and the BBP is responsible for divergence-free SET (local conservation of 4 momentum). Sorry to prattle on, here's my question, how do you see the Regge thatch encoding a divergence-free SET? From your answer to that question I can figure out where worldlines reside. Thanks, marcus.

 

[marcus]

...
The main difference between my approach and those of QRC and CDT is that I don't have empty spacetime, so I want my 4D building blocks to directly connect sources (in parlance of QFT). [This is motivated by choosing constitutive, rather than causal, non-locality to account for violations of Bell's inequality, see http://arxiv.org/abs/0908.4348 if you're interested.] Now that I've a picture of how CDT uses simplices to foliate spacetime (thanks to you), I don't see a basis for worldlines. As I understand RC the energy-momentum content resides on links, but I don't see worldlines composed of simplicial links,...

I also do not see how matter will be included in the CDT approach. As far as I know, they have only included "dark energy"---since they have the positive cosmological constant. No type of usual matter, at least in their regular 4D version. I feel confident they will do it, but I didn't see it yet.

I do not see what would correspond to worldlines. But I am not an expert in CDT, just an interested observer. I also am interested in QRC but am so far less familiar with the literature.

In 4D CDT the curvature is all concentrated on the 2D faces*. From my non-expert intuitive perspective, I would (perhaps merely superstition on my part) somehow expect that when they arrive at including matter the matter worldlines will also not be worldlines---they could also be chains of timelike 2D faces. Pardon the rank speculation.

stress-energy tensor (SET) ... Regge thatch [Miller, W.A.: The Geometrodynamic Content of the Regge Equations as Illuminated by the Boundary of a Boundary Principle. Foundations of Physics 16, 143-169 (1986)]... ... how do you see the Regge thatch encoding a divergence-free SET? ... figure out where worldlines reside...

I hesitated to post, actually. Because I cannot answer your question and I don't like "covering" your post by one of my own. I wish someone else more familiar with the Regge ideas would see your question and respond to it.

Feel free to start a "question about Regge Calculus" thread, which will make your question more visible, if you wish. There is no guarantee of getting an answer, but it does no harm to try.

However you also indicated that you are wondering if it is appropriate to inject questions about Regge Calculus into this thread. I am not a moderator but I personally see no reason to be overscrupulous about this. It is all one bag of ideas and I say more power to you for extending the discussion in what I think is a coherent way. Only your question is very hard.

BTW here is a random observation. As a sideline observer it seems to me that all the simplicial approaches are now struggling with the problem of how to include matter and also the spinfoam approach faces that problem.
So you will see in John Baez TWF 281 some reference to papers by Baez-Perez and by Perez-Fairbairn and maybe also by Freidel-Baratin.
In the struggle to find the way thru the jungle to the hidden city of matter these people, or some of them, have come to where they consider stringlike matter. But not the vibrating strings of stringtheory with its extra dimensions. It is all in 4D. They have merely come to consider worldsheets in 4D instead of worldlines in 4D.

I notice that spinfoam uses 2-cell complexes---even though it is not necessarily to be regarded as a simplicial approach. I wildly speculate that when the spinfoam people succeed in including matter it may have something to do with chains of 2-cells.

Well I may regret this later in the day when in a more sober frame of mind---just waking up and having morning coffee.

*In N-dimension simplicial work, the N-2 simplices are called "hinges", or sometimes "bones". I prefer the former term, "hinges". The curvature always resides on the hinges. Thus in 2D CDT the curvature is on points, and in 3D CDT the curvature resides on line segments. This is familiar to us in the 2D case where we know that the curvature resides at points because it is the deficit angle. In equilateral 2D CDT one counts the triangles surrounding the point. If it is 6 then she is flat at that point. If there are less than 6 then she has positive curvature and if there are more than 6 surrounding the point then she has negative. So measuring curvature is reduced to counting.
(This is not for you, RUTA, since you read the 2001 paper. But someone else might read here and want to know.)

 

[RUTA]

marcus,

Thanks very much for your response. I'm coming at this from the "other direction," i.e., instead of quantizing Regge calculus (RC) I've a quantum graphical formalism that I'm trying to link to GR. Obviously (to me anyway) that connection should be most easily forged via RC, since RC is the unique discrete version of GR, GR is the unique continuum version of RC, and RC is a graphical form of GR. Incredibly, I earned my PhD in GR cosmology, published and referee therein yet only this summer found RC, so I have limited experience with it. Thus, it was very helpful to know that QRC and CDT are working on a similar problem, i.e., how to incorporate matter. [I don't have a choice, my graphs represent the co-construction of space, time and matter (discrete QFT sources), so it's not like I already have a spacetime and I'm trying to "add" sources.]

Sorkin's paper, "Time-evolution problem in Regge calculus," Phys. Rev. D12, #2, 385-396 (1975), shows very simply how to get T_ij from the Regge action -- it's exactly what you'd think, i.e., the variation of the "additional" terms with respect to l²_ij (see p 389). Immediately thereafter he writes, "Since T(ij) must represent 'matter' we can say that, simplectically, 'energy-momentum is concentrated in the legs of the net,' even though curvature is diffused throughout [2D hinges]." However, he doesn't address the divergence-free nature of T(ij). Miller ("The Geometrodynamic Content of the Regge Equations as Illuminated by the Boundary of a Boundary Principle," Foundations of Physics 16, 143-169 (1986)) concludes his paper with the following:

We recall that the "net" moment of rotation vector for any given P_L was introduced in Sec. 7. To obtain the contracted Bianchi we need only sum these vector quantities; however, each moment vector is evaluated at the center of their corresponding edge L_j. The evaluation of this sum involves the parallel transportation of the vectors to a common point, say A. This summation at point A, we believe, is a skeletal analogue of Eq. (1). Once obtained, it may be used in conjunction with the full Regge
equation [Eq. (40)] to obtain a skeletal conservation law. These issues, which we believe are of utmost importance, will be discussed in another place.(19)

He has introduced a structure S* dual to the Regge skeleton S (or "thatch" or "net" or "graph") in exactly the fashion you would expect, i.e., vertices A of S correspond to 4D simplices in S*, links (or edges L_j) in S correspond to 3D polyhedra P_L in S*, 3D tetrahedra in S correspond to links in S*, and 4D simplices in S correspond to vertices in S*. He then characterizes the boundary of a boundary principle (divergence-free T(ij), local conservation of E-p, contracted Bianchi identity) via S*. His Eq. (1) is the divergence-free nature of the Einstein tensor and his Eq. (40) is a statement of Einstein's equations per his characterization using S*. Thus, his concluding sentence is in reference to the fact that he still hasn't characterized the boundary of a boundary principle on the Regge skeleton. Clearly this is relevant to my project and I suspect it is relevant to the problem of introducing matter in QRC and CDT. I have requested his reference 19, but it's a "paper in print" at that time so I don't know how long it will take. I wrote Miller yesterday hoping for a direct (and current) answer. I'll let you know what he says if he responds.

 

[skippy1729]

RE: Worldlines in CDT. Would this work?

The curvature is idealized as being concentrated at the bones (triangles in 4D). Construct the dual cell to the bone (the set of all points closer to the bone than any other bone). Take the curvature in the cell to be its average which would equal the deficit angle at the bone times the area of the bone. Each cell would now be a region of constant curvature. Calculate the geodesics piecewise as they pass from cell to cell. The geodesics would have discontinuous derivatives at cell interfaces. Skippy

 

[RUTA]

RE: Worldlines in CDT. Would this work?

The curvature is idealized as being concentrated at the bones (triangles in 4D). Construct the dual cell to the bone (the set of all points closer to the bone than any other bone).

Where are these points? Are they simply the vertices of the bone? I'm trying to picture your construct. I appreciate you weighing in on the question.

 

[marcus]

... Construct the dual cell to the bone (the set of all points closer to the bone than any other bone). Take the curvature in the cell to be its average which would equal the deficit angle at the bone times the area of the bone. ...

Where are these points? Are they simply the vertices of the bone? I'm trying to picture your construct. I appreciate you weighing in on the question.

Skippy and RUTA, any extra links to online sources, or definitions, that you want to throw in would be welcome. The "dual" idea mentioned by Skippy comes up in the context of CW-complexes---I would rather use the vague terminology "cell complex".
I have an intuitive notion of the dual of a 2D complex which I can exemplify by picturing the plane tiled with blue triangles. Then in each triangle choose a red point (think of it as "center of gravity") and then connect the red "central" points in adjacent triangles by new red lines that cross over the blue edges.

In that way every blue 2-cell is assigned a red 0-cell (the central point)
every blue 1-cell (the side of the triangle) is assigned a red 1-cell that crosses over it
and every blue 0-cell (a vertex of the old triangle) automatically now has a red 2-cell, a red hexagon that surrounds it.

You start with a blue triangle tiling and you get a red honeycomb hexagonal tiling.

Or if you start with a hexagonal, and construct the dual in the analogous way, you get back the triangle tiling.

In dimension D, the idea is
to every D-cell assign a 0-cell, a point
and to every (D-1)-cell assign a 1-cell
to every (D-2)-cell assign a 2-cell
...
...
to every 0-cell you automatically get a D-cell
(Sorry if that sounds like a flashlight battery. )

When you take the dual of a triangulation you don't always get a simplicial complex but you get some kind of cell complex----there may be analogous things to polygons.

I think maybe you two (Skippy, RUTA) are both familiar with this dual cellcomplex idea but in case someone else is reading...
=================

So then Skippy proposes a way to represent a world line in Renate Loll's CDT approach. D = 4.
There is a triangulation using 4-simplices. A "bone" is a triangle, a 2-simplex. The bone or hinge is always a cell of dimension D - 2, two less than the full dimensionality.
Now when you take the dual of that you get a 2-cell, not necessarily a triangle.
The idea is that there is a systematic way to construct timelike geodesics through these 2-cells. And these could serve as worldlines for matter.

=================

Personally I can't gauge how useful this construction might be. RUTA you may want to check it out, but as I understand it you already have a simplicial or cell-complex way to represent geometry+matter that you are working on. If you feel like explaining it in basic elementary terms, please do.

 

[RUTA]

Warner Miller sent me this link: http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.3041v3.pdf. Here's a teaser quote:

The contracted Bianchi identity for RC has clear implications for the coupling of energy-momentum to the lattice as well as to our understanding of diffeomorphism invariance in RC. Furthermore, if we expect the quantization of spacetime to produce an inherently discrete spacetime, then grasping the meaning of the BBP in a discrete theory becomes essential to understanding the quantum theory of gravity. RC serves naturally as an underlying framework since simplicial spacetimes provide one of the most elegant and universal descriptions of discrete spacetime (Regge T and Williams R M 2000 J. Math. Phys 41 p. 3964).

BBP = boundary of a boundary principle
RC = Regge calculus

 

[RUTA]

Miller's paper http://arxiv.org/PS_cache/arxiv/pdf/...807.3041v3.pdf ends with this statement (bold mine):

"In particular, the doubly projected stress-energy along the edges emanating from a vertex must sum to zero, to at least second order in the length scale of the lattice. While not exact, this gives the interpretation of a Kirchhoff-like conservation principle for the geometry, and (with Einstein’s equations) the flow of energy and momentum (Figure 4). As a result, we obtain a set of vertex-based constraints for edge-based expressions that constrain energy-momentum. This exercise indicates that energy-momentum is naturally wired to the simplicial lattice at each vertex and is naturally wired to each edge in its coupling with the simplicial field equations.

For applications of RC to pre-geometric quantum spacetime, one must necessarily formulate an appropriate stress-energy tensor arising from the quantum dynamics. For applications to classical spacetimes a simplicial form of the stress-energy tensor must be constructed from the non-gravitational sources. This work indicates that the stress-energy will most naturally be expressed as a vertex-based tensor, and that its coupling to the RC equations will be through its double projection on the edges of the lattice. We will explore this coupling in future work."

This is very helpful in my approach since I associate the field with individual detector clicks. For a scalar field the source and field are automatically associated with vertices, but the source and field are associated with plaquettes in a tensor theory so having stress-energy associated with "a vertex-based tensor" gives me a natural identification for clicks (spatiotemporally localized events) in my graphical tensor field theory. In fact, for fun, just read section 2 (through middle of p 11) in http://arxiv.org/abs/0908.4348, keeping in mind that this material was written without knowledge of Miller's 2008 paper (there is reference to his 1986 paper).