Relating spin-foam to canonical loop quantum gravity by graphical calculus

TL;DR

This paper uses graphical calculus to connect covariant spin-foam models with canonical loop quantum gravity, deriving the EPRL partition function and analyzing the Hamiltonian constraint's action on spin networks, showing their consistency.

Contribution

It generalizes graphical calculus to relate covariant and canonical loop quantum gravity, deriving the EPRL spin-foam partition function and demonstrating the Hamiltonian constraint's weak satisfaction.

Findings

EPRL model provides a rigging map for certain physical states

Quantum dynamics are consistent between covariant and canonical formulations

Hamiltonian constraint is weakly satisfied for Immirzi parameter β=1

Abstract

The graphical calculus method is generalized to study the relation between covariant and canonical dynamics of loop quantum gravity. On one hand, a graphical derivation of the partition function of the generalized Euclidean Engle-Pereira-Rovelli-Livine (EPRL) spin-foam model is presented. On the other hand, the action of a Euclidean Hamiltonian constraint operator on certain spin network states is calculated by graphical method. It turns out that the EPRL model can provide a rigging map such that the Hamiltonian constraint operator is weakly satisfied on certain physical states for the Immirzi parameter $\beta=1$. In this sense, the quantum dynamics between the covariant and canonical formulations are consistent to each other.