Taming Thiemann's Hamiltonian constraint in canonical loop quantum gravity: reversibility, eigenstates and graph-change analysis

Abstract

The Hamiltonian constraint remains an elusive object in loop quantum gravity because its action on spinnetworks leads to changes in their corresponding graphs. As a result, calculations in loop quantum gravity are often considered unpractical, and neither the eigenstates of the Hamiltonian constraint, which form the physical space of states, nor the concrete effect of its graph-changing character on observables are entirely known. Much worse, there is no reference value to judge whether the commonly adopted graph-preserving approximations lead to results anywhere close to the non-approximated dynamics. Our work sheds light on many of these issues, by devising a new numerical tool that allows us to implement the action of the Hamiltonian constraint without the need for approximations and to calculate expectation values for geometric observables. To achieve that, we fill the theoretical gap left in the derivations of the action of the Hamiltonian constraint on spinnetworks: we provide the first complete derivation of such action for the case of 4-valent spinnetworks, while updating the corresponding derivation for 3-valent spinnetworks. Our derivations also include the action of the volume operator. By proposing a new approach to encode spinnetworks into functions of lists and the derived formulas into functionals, we implement both the Hamiltonian constraint and the volume operator numerically. We are able to transform spinnetworks with graph-changing dynamics perturbatively and verify that volume expectation values have rather different behavior from the approximated, graph-preserving results. Furthermore, using our tool we find a family of potentially relevant solutions of the Hamiltonian constraint. Our work paves the way to a new generation of calculations in loop quantum gravity, in which graph-changing results and their phenomenology can finally be accounted for and understood.