On Propagation in Loop Quantum Gravity

A rigorous implementation of the Wheeler–Dewitt equations was derived in context Loop Quantum Gravity (LQG) and coined Spin Dynamics (QSD). The Hamiltonian constraint QSD criticised as being too local to prevent “propagation” canonical LQG. That criticism based on an algorithm developed for generating solutions Wheeler–DeWitt equations. fine details that could not be worked out because makes crucial use volume operator, which cannot diagonalised analytically. In this paper, we consider U(1)3 model Euclidean vacuum LQG consists replacing structure group SU(2) by otherwise keeps all properties theory intact. This enables analytical calculations ingto out. By considering one simplest possible non-trivial classes very small graphs, show (1) infinite number ingexist are (2) generically normalisable with respect inner product space spatially diffeomorphism invariant distributions (3) display propagation. Due closeness LQG, it is extremely likely three hold also case even more so physical Lorentzian These arguments can principle made water tight using modern numerical (e.g., ML or QC) methods combined techniques paper reserve future work.