Time in quantum theory, the Wheeler-DeWitt equation and the Born-Oppenheimer approximation

We compare two different approaches to the treatment of the Wheeler-DeWitt (WDW) equation and the introduction of time in quantum cosmology. One approach is based on the gauge-fixing procedure in theories with first-class constraints and the construction of the corresponding Hilbert space of quantum states. The other approach uses the Born-Oppenheimer (BO) method, based on the existence of two energy scales in the model under consideration. We apply both to a very simple cosmological model, including a massless scalar field filling a flat Friedmann universe, and observe that they give similar predictions. We also discuss the problem of time in nonrelativistic quantum mechanics and some questions concerning the correspondence between classical and quantum theories.