Quantum constraints, Dirac observables and evolution: group averaging versus Schroedinger picture in LQC

A general quantum constraint of the form $C= - \partial_T^2 \otimes B - I\otimes H$ (realized in particular in Loop Quantum Cosmology models) is studied. Group Averaging is applied to define the Hilbert space of solutions and the relational Dirac observables. Two cases are considered. In the first case, the spectrum of the operator $(1/2)\pi^2 B - H$ is assumed to be discrete. The quantum theory defined by the constraint takes the form of a Schroedinger-like quantum mechanics with a generalized Hamiltonian $\sqrt{B^{-1} H}$. In the second case, the spectrum is absolutely continuous and some peculiar asymptotic properties of the eigenfunctions are assumed. The resulting Hilbert space and the dynamics are characterized by a continuous family of the Schroedinger-like quantum theories. However, the relational observables mix different members of the family. Our assumptions are motivated by new Loop Quantum Cosmology models of quantum FRW spacetime. The two cases considered in the paper correspond to the negative and, respectively, positive cosmological constant. Our results should be also applicable in many other general relativistic contexts.