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Loop Quantum Gravity a la Aharonov-Bohm

Eugenio Bianchi

19 pages, 1 figure

(Submitted on 24 Jul 2009)

"The state space of Loop Quantum Gravity admits a decomposition into orthogonal subspaces associated to diffeomorphism equivalence classes of graphs. In this paper I discuss the possibility of obtaining this state space from the quantization of a topological field theory with many degrees of freedom. The starting point is a theory of locally-flat connections on a manifold which is non simply-connected because of the presence of a network of defects.The quantization procedure relies on standard field theoretical methods.The functional integral defining the scalar product is shown to reduce to a finite dimensional integral over moduli space. A non-trivial measure given by the Faddeev-Popov determinant is derived. The resulting state space is surprisingly close to the one of ordinary Loop Quantum Gravity. Spin networks arise again and provide the tool for describing gauge- and diffeomorphism-invariant functionals of the connection. The role played by defects and loops in this approach is analogous to the one played by solenoids and Wilson loops in the Aharonov-Bohm effect."

I have highlighted an important point. Loop quantizing geometry does not necessarily rely on a non-standard method of quantization.

This paper is deep perceptive and clearly written. I thought Bianchi was just a PhD student at Luminy. No, I see here he is a postdoc.

http://www.cpt.univ-mrs.fr/~quantumgravity/

It was actually a pleasure to read because it has mathematical ideas. Two years ago, as I recall, Francesca and "fh" (both used to post here frequently) indicated to us to look out for Bianchi and in this case they probably gave us good advice.

This way of developing the kinematical LQG appears to be especially compatible with the Spinfoam treatment of dynamics. This is discussed at the top of page 11, in two bullets following this sentence: "...From this point of view, they are closer to the use that is made of spin networks as boundary states within the context of spin foams...."

The situation with LQG is getting so interesting it is almost amusing. The Corfu school in September is likely to be a blast. Just this week there is the Beijing Loops prep school and next week Beijing Loops 2009. I wonder what new results will come out. Will, for example, Bianchi give a talk. We don't know the titles of the talks being given at Loops 2009. Rovelli will give the public lecture, we know the plenary speakers, but not much more.

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# Noteworthy paper by Bianchi

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