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

Understand

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.