Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint

TL;DR

This paper demonstrates that a specific Euclidean spin-foam model can produce new solutions to the Euclidean Hamiltonian constraint in Loop Quantum Gravity, especially at Barbero-Immirzi parameter b3=1, revealing a connection between covariant and canonical approaches.

Contribution

It shows that the one-vertex expansion of a Euclidean spin-foam model yields new solutions to the Euclidean Hamiltonian constraint in LQG at b3=1, highlighting a link between covariant and canonical formalisms.

Findings

The one-vertex amplitude annihilates the Euclidean Hamiltonian constraint at b3=1.

New solutions depend only on diagonal volume matrix elements.

The spin-foam projector may generally produce states with simplified volume dependence.

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.