Why spin foams just use tetrahedron and pentachoron?

[MTd2]

There is a big list of possible ways to tessellate space. But why just those 2 for 3 and 4 dimensions?

http://en.wikipedia.org/wiki/List_of_regular_polytopes

 

[Finbar]

There is a big list of possible ways to tessellate space. But why just those 2 for 3 and 4 dimensions?

http://en.wikipedia.org/wiki/List_of_regular_polytopes

I'm not sure but i would guess that using simplexes to triangulate rather than uses cubes a space is just a choice. You could use cubes but its just "easier" to use simplexes as they are the simplest polytype. Of coarse the question of whether if you used polytopes other than simplexes will effect physical quantities is an open one. The introduction of simplexes should be understood as a regularisation scheme and not a statement that the space-time is made of pentachorons(4-simplexes). Maybe a simpler answer is the Regge calculus is defined in terms of simplexes and spin foams are supposed to have the classical limit of the Regge action for GR.

 

[marcus]

I'm not sure but i would guess that using simplexes to triangulate rather than uses cubes a space is just a choice...

Finbar, I think the short answer to the question (Why spin foams just use tetrahedron and pentachoron?) is that spinfoams doesn't.

As far as i know in spinfoam QG the spinfoam is made of vertices, edges, and faces.
The faces do not have to be any particular type of polygon---any n-side polygon is OK.

Any number of edges can meet at a given vertex. If you were to take the dual you would not expect to get a "triangulation" or "tesselation" comprised of tets or pents or anything specific.

In some research papers they temporarily make restrictions and consider only limited cases and they take the dual etc etc. But that is not a normal or universal practice. In a general spinfoam paper there is nothing like a "tesselation by pentachorons". That sounds more like Regge and CDT.
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MTd2,
The most up to date LQG and SF paper is Rovelli's April 2010.
http://arxiv.org/abs/1004.1780
It has some pictures with vertices, edges, and faces.
See if you can find something about pentachorons in the paper. I can't.

 

[MTd2]

Eq. 59 shows a pentachoron! :)

 

[marcus]

Eq. 59 shows a pentachoron! :)

True enough! Actually it just shows a complete graph on 5 vertices, chosen as an example to illustrate a calculation that is general in character---could be done on some other graph.
But as Rovelli points out the complete graph on 5 vertices is the graph formed by the edges of a 4-simplex (pentachoron). It is a good example, but I don't think it shows a significant connection between today's spinfoam QG and 4-simplices.

 

[MTd2]

Yes, it could be done in any other graph, but each choice will give a different approximation, right? So, maybe graphs with the pentachoron will give the lowest contributions if you approach the scales required by GR. I mean, didn't smolin showed that LGG should obey the holographic principle? So, we should look for the states that maximizes the indistinguibility of the system, that is, vertexes that correspond to tesselations.

 

[marcus]

I think we should stop talking about pentachorons in connection with spinfoam. It confuses people. A spinfoam contains no 4-simplices in its makeup.

It consists of 0-dim, 1-dim, and 2-dim objects---called vertices, edges, faces. The faces are flat, and not in general triangular: they can be n-gons. Thus incompatible with 4-simplices.

As an example of something to analyze I guess you can take a 4-simplex. But that is not the spinfoam. That is a very simple toy spacetime that you take as a problem to analyze.
It is just a "for instance".

In no way does the 4-simplex characterize the theory. Am I wrong? Can you show me an essential connection to 4-simplices in Rovelli's paper?
In that paper the spin foam is quite general. A vertex can have any number of edges.

 

[MTd2]

No, there is no essential connection with 4-simplex. But, as you can read in my previous post, I am talking about a chosen approximation to find GR. Why simplex and not a honeycomb?

 

[tom.stoer]

In a general spin network there are no polyhedra. The spin network (or its dual) does not even tessellate space, as this would restrict the class of graphs - and I cannot see such a restriction in the generic spin network models.

Look at 2-dim. space. You get a tesselation only of you restrict to planar graphs, but this restriction is absent in the spin network picture (unfortunately they always draw planar graphs as examples ...).

 

[genneth]

It important to realize that without a geometry to start with, spin networks are only defined by their topology. In trying to build back up, cubes, etc. aren't very stable topologies. Simplicial networks are.

 

[tom.stoer]

Taking all the new results seriously my conclusion is that Ashtekar variables, simplices, etc. are history. Just as the rigid body is history I am QM when you talk about spin. You cannot derive spin from pure classical reasoning ...

 

[ensabah6]

Taking all the new results seriously my conclusion is that Ashtekar variables, simplices, etc. are history. Just as the rigid body is history I am QM when you talk about spin. You cannot derive spin from pure classical reasoning ...

What has replaced Ashtekar variables?

 

[f-h]

The best reference for spin foams on CW-complexes rather than on triangulations is Oeckls:

http://arxiv.org/abs/hep-th/0110259

and his book especially.

More recently KKL rediscovered this and applied it to the EPRL spin foam models:

http://arxiv.org/abs/0909.0939

Triangulations are technically simpler to work with though and better suited for the definition of TQFTs du to the fact that rigorous and simple results relating the triangulations of a manifold are known.

Also non planar 3-valent spin networks still give a tesselation of some 2d surface (or better, delta-complex), just not the plane. LQG type spin networks (as opposed to diagrammatical calculus spin networks) naturally live in 3d anyways.

 

[MTd2]

In a general spin network there are no polyhedra. The spin network (or its dual) does not even tessellate space, as this would restrict the class of graphs - and I cannot see such a restriction in the generic spin network models.

What is the problem in restricting the class of graphs? I am trying to understan here is what the biggest contribution should be to yield GR at classical levels. here are infinite ways to tessellate a manifold, but simplexes are just a tessellation in 2 dimensions, above a simplex is not a tessellation. A tessellation, and its first order perturbations, would be like analyzing the most important scattering amplitudes all points, looking for the biggest entropy.

 

[tom.stoer]

What is the problem in restricting the class of graphs? I am trying to understan here is what the biggest contribution should be to yield GR at classical levels. here are infinite ways to tessellate a manifold, but simplexes are just a tessellation in 2 dimensions, above a simplex is not a tessellation.

I think that the problem is that it's unnatural. You start with a generic graph with two SU(2) colorings (= a spin network with Su(2) intertwiners at the vertices + SU(2) variables at the links). If this seems to be natural to you, then any further restriction of the underlying graph is unnatural.

I do not know what the main contribution to a classical 4-dim. manifold really is. Perhaps it comes from the restricted graphs but I am not sure about that.

What do you mean by "above a simplex is not a tessellation"? I am not talking about regular simplices. Just the topology of an simplex. Does it still not work?

 

[tom.stoer]

What has replaced Ashtekar variables?

Spin networks.

The kinematical Hilbert space is derived from the Ashtekar variables. But the calculations today are done with spin networks- The Ashtekar variables are an intermediate step.

 

[tom.stoer]

Have a look at Rovelli's new paper and the following thread: https://www.physicsforums.com/showthread.php?t=394757