Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity

TL;DR

This paper develops a reduced phase space quantisation of General Relativity using Loop Quantum Gravity methods, simplifying the algebra of observables and providing a physical Hilbert space where geometrical spectra are physically meaningful.

Contribution

It introduces a novel combination of Brown-Kuchar deparametrisation and Rovelli's relational formalism to explicitly construct and quantise the reduced phase space in LQG.

Findings

The algebra of gauge-invariant observables is simplified and quantised.

The physical Hilbert space aligns with the non-reduced LQG kinematical space.

The spectra of geometrical operators acquire physical significance.

Abstract

We perform a canonical, reduced phase space quantisation of General Relativity by Loop Quantum Gravity (LQG) methods. The explicit construction of the reduced phase space is made possible by the combination of 1. the Brown -- Kuchar mechanism in the presence of pressure free dust fields which allows to deparametrise the theory and 2. Rovelli's relational formalism in the extended version developed by Dittrich to construct the algebra of gauge invariant observables. Since the resulting algebra of observables is very simple, one can quantise it using the methods of LQG. Basically, the kinematical Hilbert space of non reduced LQG now becomes a physical Hilbert space and the kinematical results of LQG such as discreteness of spectra of geometrical operators now have physical meaning. The constraints have disappeared, however, the dynamics of the observables is driven by a physical Hamiltonian which is related to the Hamiltonian of the standard model (without dust) and which we quantise in this paper.