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We consider a reduced phase space quantisation of a model with $T^3$ Gowdysymmetry in which gravity has been coupled to Gaussian dust. We complete thequantisation programme in reduced loop quantum gravity (LQG) as well asalgebraic quantum gravity (AQG) and derive a Schr\"odinger-like equation with aphysical Hamiltonian operator encoding the dynamics. Due to the classicalsymmetries of the physical Hamiltonian, the operators are quantised in agraph-preserving way in both cases -- a difference to former models availablein the literature. As a first step towards applications of the model in AQG, weconsider an ansatz that we use to first construct zero volume states asspecific solutions of the Schr\"odiger-like equation. We then also find stateswith a vanishing action of the Euclidean part of the physical Hamiltonian andinvestigate the degeneracies these states experience via the action of theLorentzian part of the physical Hamiltonian. The results presented here can betaken as a starting point for deriving effective models as well as analysingthe dynamics numerically in future work.

A reduced phase space quantisation of a model in Algebraic Quantum Gravity with polarised $T^3$ Gowdy symmetry