Separable Hilbert space in Loop Quantum Gravity
Winston Fairbairn, Carlo Rovelli
TL;DR
The paper addresses the nonseparability of the kinematical Hilbert space in Loop Quantum Gravity caused by continuous moduli labeling diff-knots at high-valence nodes. It proposes a minimal extension to the space of classical fields—allowing almost smooth fields—and promotes the gauge group to the extended diffeomorphism $Diff^*$, demonstrating that knot classes become countable and $\mathcal{H}_{\rm diff}$ becomes separable. The authors show that this extension preserves the definitions and spectra of geometric operators (area and volume) and the Hamiltonian when using appropriate regularizations, leaving physics unchanged. They argue that the continuous moduli are likely spurious and that the theory naturally supports a separable, combinatorial description of quantum geometry at the Planck scale.
Abstract
We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a consequence of the fact that the knot-space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the fine-tuning of the mathematical setting. We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable.