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Feynmans of geometry: computable trans. amps. by K+Le+P algorithm

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marcus
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Mar31-12, 01:45 PM
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PF Gold
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Kisielowski Lewandowski Puchta have made a significant advance in calculating the transition amplitudes between quantum states of geometry.

They have found a systematic algorithm that enumerates the (generalized) spinfoam-like histories by which one state evolves into another.

And in this case the amplitude of each history is conveniently accessible. Each history that one enumerates more or less "comes with" its probability amplitude.

This looks to me like a systematic combinatorial algorithm that can be PROGRAMMED, which is one reason it's potentially important.

In the September 2011 seminar discussion Laurent Freidel objected that the algorithm does not offer any advantage in the case where one restricts to spinfoams which are dual to triangulations of a 4d manifold. (The vertices at most 5-valent etc.)

But Frank Hellmann (AEI) countered by observing that one HAS to consider more general spinfoams because they, for example, arise naturally in the simplest applications to cosmology. He and co-workers have been studying the "hourglass" spinfoam picture of the cosmological bounce and have found they have to go to cases which are not based on triangulations He reported that they had found the Warsaw algorithm a big help.

Our two sources on this are the paper http://arxiv.org/abs/1107.5185 and the online recorded seminar talk by Puchta.
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marcus
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Mar31-12, 02:09 PM
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I think a lot depends on whether this Feynman diagram approach to geometry quantum dynamics turns out to work (and provides an efficient computer algorithm), so I will copy the links and source summary to make looking it over convenient:
Here's Puchta's recorded seminar slides and audio:
http://relativity.phys.lsu.edu/ilqgs/puchta092011.pdf
http://relativity.phys.lsu.edu/ilqgs/puchta092011.wav
The Feynman Diagrammatics for the Spinfoam Models
It's especially good because it has a lot of question, comment, and discussion by Ashtekar, Freidel, Hellmann, even at one or two points Lewandowski, and of course Puchta.
Listening to the talk makes the paper quite a lot easier to understand. At several points where Ashtekar senses the audience isn't getting it, he asks Puchta to back up 2 or 3 slides and take it again slowly. This is very new stuff.

The new algorithm involves new unfamiliar graph manipulations. They are good because systematic and programmable (combinatorial) steps not requiring geometric intuition. But the geometric motivation is not always immediately clear. Personally my hunch is that this is an indication of depth, non-triviality.

I tend to think of the paper by the initials of the authors Kisielowski Lewandowski Puchta. That sounds nice because it reminds me of the Russian word kleb for bread. Think of dynamic spacetime geometry as the "bread" of reality .

Here is the abstract for the "KLeP" paper
http://arxiv.org/abs/1107.5185
Feynman diagrammatic approach to spin foams
Marcin Kisielowski, Jerzy Lewandowski, Jacek Puchta
(Submitted on 26 Jul 2011)
"The Spin Foams for People Without the 3d/4d Imagination" could be an alternative title of our work. We derive spin foams from operator spin network diagrams} we introduce. Our diagrams are the spin network analogy of the Feynman diagrams. Their framework is compatible with the framework of Loop Quantum Gravity. For every operator spin network diagram we construct a corresponding operator spin foam. Admitting all the spin networks of LQG and all possible diagrams leads to a clearly defined large class of operator spin foams. In this way our framework provides a proposal for a class of 2-cell complexes that should be used in the spin foam theories of LQG. Within this class, our diagrams are just equivalent to the spin foams. The advantage, however, in the diagram framework is, that it is self contained, all the amplitudes can be calculated directly from the diagrams without explicit visualization of the corresponding spin foams. The spin network diagram operators and amplitudes are consistently defined on their own. Each diagram encodes all the combinatorial information. We illustrate applications of our diagrams: we introduce a diagram definition of Rovelli's surface amplitudes as well as of the canonical transition amplitudes. Importantly, our operator spin network diagrams are defined in a sufficiently general way to accommodate all the versions of the EPRL or the FK model, as well as other possible models. The diagrams are also compatible with the structure of the LQG Hamiltonian operators, which is an additional advantage. Finally, a scheme for a complete definition of a spin foam theory by declaring a set of interaction vertices emerges from the examples presented at the end of the paper.
36 pages, 23 figures


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