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http://arxiv.org/abs/1109.1290
Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint[/B
Authors: Emanuele Alesci, Thomas Thiemann, Antonia Zipfel
(Submitted on 6 Sep 2011)
Abstract: It is often emphasized that spin-foam models could realize a projection on the physical Hilbert space of canonical Loop Quantum Gravity (LQG). As a first test we analyze the one-vertex expansion of a simple Euclidean spin-foam. We find that for fixed Barbero-Immirzi parameter \gamma=1 the one vertex-amplitude in the KKL prescription annihilates the Euclidean Hamiltonian constraint of LQG. Since for \gamma=1 the Lorentzian part of the Hamiltonian constraint does not contribute this gives rise to new solutions of the Euclidean theory. Furthermore, we find that the new states only depend on the diagonal matrix elements of the volume. This seems to be a generic property when applying the spin-foam projector.
I didn't study the whole paper but looked first at the conclusions: you will find that it's still a a long and stony path ...
Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint[/B
Authors: Emanuele Alesci, Thomas Thiemann, Antonia Zipfel
(Submitted on 6 Sep 2011)
Abstract: It is often emphasized that spin-foam models could realize a projection on the physical Hilbert space of canonical Loop Quantum Gravity (LQG). As a first test we analyze the one-vertex expansion of a simple Euclidean spin-foam. We find that for fixed Barbero-Immirzi parameter \gamma=1 the one vertex-amplitude in the KKL prescription annihilates the Euclidean Hamiltonian constraint of LQG. Since for \gamma=1 the Lorentzian part of the Hamiltonian constraint does not contribute this gives rise to new solutions of the Euclidean theory. Furthermore, we find that the new states only depend on the diagonal matrix elements of the volume. This seems to be a generic property when applying the spin-foam projector.
I didn't study the whole paper but looked first at the conclusions: you will find that it's still a a long and stony path ...