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.