Separable Hilbert space in Loop Quantum Gravity

TL;DR

This paper demonstrates that by extending the classical field class in loop quantum gravity, the nonseparable kinematical Hilbert space can be made separable, simplifying the mathematical structure without affecting the physical content.

Contribution

It shows that a minor extension of the classical fields' functional class renders the state space of loop quantum gravity separable, addressing a key mathematical issue.

Findings

The standard kinematical Hilbert space is nonseparable due to uncountable knot classes.

Extending the classical fields' functional class removes the continuous moduli.

The resulting Hilbert space becomes countable and separable.

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.