Quantum kinematics of spacetime. II. A model quantum cosmology with real clocks

Nonrelativistic model quantum cosmologies are studied in which the basic time variable is the position of a clock indicator and the time parameter of the Schr\"odinger equation is an unobservable label. Familiar Schr\"odinger-Heisenberg quantum mechanics emerges if the clock is ideal---arbitrarily accurate for arbitrarily long times. More realistically, however, the usual formulation emerges only as an approximation appropriate to states of this model universe in which part of the system functions approximately as an ideal clock. It is suggested that the quantum kinematics of spacetime theories such as general relativity may be analogous to those of this model. In particular it is suggested that our familiar notion of time in quantum mechanics is not an inevitable property of a general quantum framework but an approximate feature of specific initial conditions.