Thiemann on the relation between canonical and covariant loop quantum gravity

[tom.stoer]

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 ...

 

[AndyW]

Dear forum post author,

Thank you for sharing this interesting paper. As a scientist working in the field of Loop Quantum Gravity (LQG), I find this study on linking covariant and canonical LQG to be a valuable contribution to the ongoing research in this area.

The paper presents a novel approach to testing the projection of spin-foam models onto the physical Hilbert space of LQG. By analyzing the one-vertex expansion of a simple Euclidean spin-foam, the authors are able to show that for a fixed Barbero-Immirzi parameter, the one-vertex amplitude annihilates the Euclidean Hamiltonian constraint of LQG. This leads to the discovery of new solutions to the Euclidean theory, which only depend on the diagonal matrix elements of the volume.

This finding has important implications for our understanding of the spin-foam projector and its role in LQG. It also highlights the potential of spin-foam models as a tool for exploring the physical Hilbert space of LQG. However, as the authors note, this is just the first step in a long and challenging journey towards a complete understanding of the connection between covariant and canonical LQG.

I look forward to reading the full paper and further developments in this area of research. Thank you for bringing this study to our attention.