Time evolution in quantum cosmology

Abstract

A commonly adopted relational account of time evolution in generally covariant systems, and more specifically in quantum cosmology, is argued to be unsatisfactory, insofar as it describes evolution relative to observed readings of a clock that does not exist as a bona fide observable object. A modified strategy is proposed, in which evolution relative to the proper time that elapses along the worldline of a specific observer can be described through the introduction of a ``test clock,'' regarded as internal to, and hence unobservable by, that observer. This strategy is worked out in detail in the case of a homogeneous cosmology, in the context of both a conventional Schr\"odinger quantization scheme, and a ``polymer'' quantization scheme of the kind inspired by loop quantum gravity. Particular attention is given to limitations placed on the observability of time evolution by the requirement that a test clock should contribute only a negligible energy to the Hamiltonian constraint. It is found that suitable compromises are available, in which the clock energy is reasonably small, while Dirac observables are reasonably sharply defined.