Atyy,
Suppose we look at two of the three papers in your OP and consider the original question as simply as possible "why do people like polytopes?" We can put the question more specifically, in the case of these two papers, as "
Why do these 6 people like polyhedra?"It may give us some clues as to where Freidel Speziale Livine etc will be going with the new "Principle of Relative Locality". Most of these 6 people are repeat co-authors who must certainly share ideas. Both about polyhedra and about "relative locality" (a new initiative as yet only partially grasped.)
atyy said:
...
http://arxiv.org/abs/0905.3627
Holomorphic Factorization for a Quantum Tetrahedron
Laurent Freidel, Kirill Krasnov, Etera R. Livine
http://arxiv.org/abs/1009.3402
Polyhedra in loop quantum gravity
Eugenio Bianchi, Pietro Dona', Simone Speziale
...
...
I guess the idea of a "space of shapes" is basic. Spinnetwork nodes are intertwiners--the space of intertwiners is a hilbert space of (quantum) shapes. Roughly speaking. A quantum geometry should be built by gluing together quantum (uncertain, indefinite) shapes. This is not rigorous, just trying to sense why the idea of quantumpolyhedra is so basic. Let's see what Freidel Krasnov Livine (FKL) say:
========quote FKL=========
Our discussion has so far been quite mathematical, so we would now like to switch to a more heuristic description and explain the significance of our results for the field of quantum gravity. As we have already mentioned, the n = 4 intertwiner that we have characterized in this paper in most details plays a very important role in both the loop quantum gravity and the spin foam approaches. These intertwiners have so far been characterized using the real basis... In particular, the main building blocks of the spin foam models – the (15j)-symbols and their analogs – arise as simple pairings of 5 of such intertwiners (for some choice of the channels ij). The main result of this paper is a holomorphic description of the space of intertwiners, and, in particular, an explicit basis in H
j1,...,j4 given by the holomorphic intertwiner... While the basis ..., being discrete, may be convenient for some purposes, the underlying geometric interpretation in
it is quite hidden. Indeed, recalling the interpretation of the intertwiners from H
j1,...,j4 as giving the states of a quantum tetrahedron, the states...describe a tetrahedron whose shape is maximally uncertain. In contrast, the intertwine..., being holomorphic, are coherent states in that they manage to contain the complete information about the shape of the tetrahedron coded into the real and imaginary parts of the cross-ratio coordinate Z. We give an explicit description of this in the main text.
Thus, with the holomorphic intertwiner... at our disposal, we can now characterize the “quantum geometry” much more completely than it was possible before. Indeed, we can now build the spin networks – states of quantum geometry – using the holomorphic intertwiners. The nodes of these spin networks then receive a well-defined geometric interpretation as corresponding to particular tetrahedral shapes. Similarly, the spin foam model simplex amplitudes can now be built using the coherent intertwiners, and then the basic object becomes not the (15j)-symbol of previous studies, but the (10j)-(5Z)-symbol with a well-defined geometrical interpretation. Where this will lead the subjects of loop quantum gravity and spin foams remains to be seen, but the very availability of this new technology opens way to many new developments and, we hope, will give a new impetus to the field that is already very active after the introduction of the new spin foam models in [11–15].
The organization of this paper is as follows. In section II we describe how the phase space that we would like to quantize arises as a result of the symplectic reduction of a simpler phase space... semi-classical limit of large spins and show that it takes the form precisely as is expected from the point of view of geometric quantization. ...
===endquote===
