Physical Diffeomorphisms in Loop Quantum Gravity

TL;DR

This paper constructs a separable Hilbert space for Loop Quantum Gravity by analyzing the action of stratified diffeomorphisms, leading to a basis labeled by knot classes and spin quantum numbers, with implications for quantum constraints.

Contribution

It introduces a physical principle for constructing the diffeomorphism-invariant Hilbert space, identifying stratified diffeomorphisms as key, and establishes a spin-knot basis labeled by knot classes and spins.

Findings

Hilbert space constructed with a spin-knot basis.

Orbits of graphs are labeled by knot classes.

Diffeomorphism invariance compatible with flux operators.

Abstract

We investigate the action of diffeomorphisms in the context of Hamiltonian Gravity. By considering how the diffeomorphism-invariant Hilbert space of Loop Quantum Gravity should be constructed, we formulate a physical principle by demanding, that the gauge-invariant Hilbert space is a completion of gauge- (i.e. diffeomorphism-)orbits of the classical (configuration) variables, explaining which extensions of the group of diffeomorphisms must be implemented in the quantum theory. It turns out, that these are at least a subgroup of the stratified analytic diffeomorphisms. Factoring these stratified diffeomorphisms out, we obtain that the orbits of graphs under this group are just labelled by their knot classes, which in turn form a countable set. Thus, using a physical argument, we construct a separable Hilbert space for diffeomorphism invariant Loop Quantum Gravity, that has a spin-knot basis, which is labelled by a countable set consisting of the combination of knot-classes and spin quantum numbers. It is important to notice, that this set of diffeomorphism leaves the set of piecewise analytic edges invariant, which ensures, that one can construct flux-operators and the associated Weyl-operators. A note on the implications for the treatment of the Gauss- and the Hamilton-constraint of Loop Quantum Gravity concludes our discussion.