Applications of a new proposal for solving the ‘‘problem of time’’ to some simple quantum cosmological models

We apply a recent proposal for defining states and observables in quantum gravity to some simple models. First, we consider the toy model of a Klein-Gordon particle in an external potential in Minkowski spacetime and compare our proposal to the theory obtained by deparametrizing with respect to a choice of time slicing prior to quantization. We show explicitly that the dynamics defined by the deparametrization approach depends upon the choice of time slicing. On the other hand, our proposal automatically yields a well defined dynamics, manifestly independent of the choice of time slicing at intermediate times, but there is a ‘‘memory effect’’: After the potential is turned off, the dynamics no longer returns to the standard, free particle dynamics. Next, we apply our proposal to the closed Robertson-Walker quantum cosmology with a homogeneous massless scalar field. We choose our time variable to the size of the Universe, so the only dynamical variable is the scalar field. It is shown that the resulting theory has the expected semiclassical behavior up to the classical turning point from expansion to contraction; i.e., given a classical solution which expands for much longer than the Planck time, there is a quantum state whose dynamical evolution closely approximates this classical solution during the expanding phase. However, when the ‘‘time’’ takes a value larger than the classical maximum, the scalar field becomes ‘‘frozen’’ at the value it had when it entered the classically forbidden region. The Taub model, with and without a homogeneous scalar field, also is studied, and similar behavior is found. In an Appendix, we derive the form of the Wheeler-DeWitt equation on minisuperspace for the Bianchi models by performing a proper quantum reduction of the momentum constraints; this equation differs from the usual form of the Wheeler-DeWitt equation, which is obtained by solving the momentum constraints classically, prior to quantization.