Life in an energy eigenstate: Decoherent histories analysis of a model timeless universe

Inspired by quantum cosmology, in which the wave function of the universe is annihilated by the total Hamiltonian, we consider the internal dynamics of a simple particle system in an energy eigenstate. Such a system does not possess a uniquely defined time parameter, and all physical questions about it must be posed without reference to time. We consider in particular the following question: what is the probability that the system’s trajectory passes through a set of regions of configuration space without reference to time? We first consider the classical case, where the answer has a variety of forms in terms of a phase-space probability distribution function. We then consider the quantum case, and we analyze this question using the decoherent histories approach to quantum theory, adapted to questions which do not involve time. When the histories are decoherent, the probabilities approximately coincide with the classical case, with the phase-space probability distribution replaced by the Wigner function of the quantum state. For some initial states, decoherence requires an environment, and we compute the required influence functional and examine some of its properties. Special attention is given to the inner product used in the construction (the induced or Rieffel inner product), the construction of class operators describing the histories, and the extent to which reparametrization invariance is respected. Our results indicate that simple systems without an explicit time parameter may be quantized using the decoherent histories approach, with the expected classical limit extracted. The results support, for simple models, the usual heuristic proposals for the probability distribution function associated with a semiclassical wave function satisfying the Wheeler-DeWitt equation.