Scalar Material Reference Systems and Loop Quantum Gravity

In the past, the possibility to employ (scalar) material reference systems in order to describe classical and quantum gravity directly in terms of gauge invariant (Dirac) observables has been emphasised frequently. This idea has been picked up more recently in Loop Quantum Gravity (LQG) with the aim to perform a reduced phase space quantisation of the theory thus possibly avoiding problems with the (Dirac) operator constraint quantisation method for constrained system. In this work, we review the models that have been studied on the classical and/or the quantum level and parametrise the space of theories so far considered. We then describe the quantum theory of a model that, to the best of our knowledge, so far has only been considered classically. This model could arguably called the optimal one in this class of models considered as it displays the simplest possible true Hamiltonian while at the same time reducing all constraints of General Relativity.