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

One of the key concepts in loop quantum gravity is the quantization of spacetime geometry, with discrete observables such as the quantum area and volume. The quantum state of the gravitational field is encoded in so-called spin networks, and the conventional quantum-mechanical dynamics is substituted by a description in terms of constrained quantum states, in which several constraints define the physical subspace of the Hilbert space. One of these constraints, commonly called the Hamiltonian constraint, remains an elusive object in loop quantum gravity because its action on spin networks 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 this 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 nonapproximated dynamics. Our work sheds light on several 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 the geometric observables. To achieve that, we fill the theoretical gap left in the derivations of the action of the Hamiltonian constraint on spin networks: we provide the first complete derivation of such action for the case of 4-valent spin networks, while updating the corresponding derivation for 3-valent spin networks. Our derivations also include the action of the volume operator. By proposing a new approach to encode spin networks 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 spin networks with graph-changing dynamics perturbatively and verify that the expectation values for the volume 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.

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