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

We consider a reduced phase space quantisation of a model with $T^3$ Gowdy symmetry in which gravity has been coupled to Gaussian dust. We complete the quantisation programme in reduced loop quantum gravity (LQG) as well as algebraic quantum gravity (AQG) and derive a Schr\"odinger-like equation with a physical Hamiltonian operator encoding the dynamics. Due to the classical symmetries of the physical Hamiltonian, the operators are quantised in a graph-preserving way in both cases -- a difference to former models available in the literature. As a first step towards applications of the model in AQG, we consider an ansatz that we use to first construct zero volume states as specific solutions of the Schr\"odiger-like equation. We then also find states with a vanishing action of the Euclidean part of the physical Hamiltonian and investigate the degeneracies these states experience via the action of the Lorentzian part of the physical Hamiltonian. The results presented here can be taken as a starting point for deriving effective models as well as analysing the dynamics numerically in future work.