Whoever wants to check connections between the two approaches has, i guess, to look for analogy between these three papers:
Daniele Oriti
The Feynman propagator for spin foam quantum gravity
http://arxiv.org/abs/gr-qc/0410134
J. Ambjorn, J. Jurkiewicz, R. Loll
Emergence of a 4D World from Causal Quantum Gravity
http://arxiv.org/hep-th/0404156
Semiclassical Universe from First Principles
J. Ambjorn, J. Jurkiewicz, R. Loll
15 pages, 4 figures
http://arxiv.org/abs/hep-th/0411152
Abstract:
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over space-time geometries in nonperturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor."
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It looks to me like AJL are stabilizing their nomenclature
they are calling what they do "causal dynamical triangulations"
over the years research in this direction has been called various things: simplicial gravity, simplicial quantum geometry, quantum gravity---sometimes Lorentzian instead of causal---as in
"Lorentzian dynamical triangulations".
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the idea is simple
they gamble with the geometry of the universe
or more precisely with the whole history of the geometry of the universe
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imagine you have 200,000 small tetrahedra---little 3sided pyramids---jumbled into a box
and there are coils so you can suddenly turn on a magnetic field in the box and make the tets want to stick together
so you say "Tets, cohere!" and you throw the switch and they all jump up out of their heap and fit together (triangular side to triangular side, vertex touching vertex) and this gives you a random geometry
and as an experiment, you do that over and over again and photograph the result each time-----so eventually you get an idea of the probabilities of various configurations and an idea of average properties etc.
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they have an idea of causal ordering, so they only allow a certain kind of assembly.
so their game is not quite as free as what i described. but it is a game and falls in the category of "Monte Carlo methods"
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Notice that a 3sided pyramid has FOUR triangular faces because you have to count the bottom too. so it is called a Tet (to indicate the four-ness)
A triangle is what you use to triangulate a 2D surface
A tetrahedron is what you use to
triangulate a 3D region
What is it called that you use to
triangulate a 4D spacetime
what is the analog of a tetrahedron in 4D? It could be called a "pent" because it has
FIVE faces (each of which is a tetrahedron) and pent indicates five-ness.
Technically it is called a 4-simplex. and a 4-simplex has 5 faces, each of which is a 3D solid, namely a tetrahedron.
So if you want to play the AJL game in 4D then you have to have a heap of "pents" in a box. you have to have a heap of 4-simplexes.
A computer finds it very easy to imagine a 4-simplex. But I do not. therefore I imagine them as little tetrahedra----a bit like hard cream-colored plastic dice.
AJL use large numbers of them: 100 thousand or 200 thousand, the most this time seems to have been 360 thousand
When you say "Pents! Cohere!" they all jump together and form a random 4D shape.
Then the computer can project that for you into lower dimension so you can have the experience of viewing it. Or it can calculate statistics of certain measures and features, and keep a record. then you do it over and over again and get an idea of the probabilities.
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and AJL have some extra rules in how they can fit together
having to do with causality.
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So the question is HOW CLOSE IS Oriti spin foam model to this
AJL Causal Dynamations "pent" random geometry game?
Can you get similar results?
AJL report a kind of Epiphany when a semiclassical Lagrangian for the scalefactor of the universe appeared to them out of the midst of Monte Carlo randomness. they didnt put it in.
A familiar Lagrangian came to dinner uninvited. This is quite interesting.
I didnt yet hear of anything like this happening with Spin Foam.
So it seems like an urgent question to put to Daniele Oriti. Can he assimilate or link to anything in the AJL program?
Heck, he got the whole game started! 
