In this talk, we will discuss a specific toy model – the polarized

three-torus Gowdy model – which is basically a one-dimensional field

theory on a circle with gauge group U(1). To this model we couple Gaussian

dust in the context of the so called relational formalism, giving us a

physical phase space that is isomorphic to the kinematical one and in

particular a physical Hamiltonian that is equivalent to the geometrical

Hamiltonian constraint. To quantize this theory, we apply the techniques

of loop quantum gravity and algebraic quantum gravity, giving us a

Schrödinger equation for which we then discuss some formal solutions.