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Inspired by quantum cosmology, in which the wave function of the universe isannihilated by the total Hamiltonian, we consider the internal dynamics of asimple particle system in an energy eigenstate. Such a system does not possessa uniquely defined time parameter and all physical questions about it must beposed without reference to time. We consider in particular the question, whatis the probability that the system's trajectory passes through a set of regionsof configuration space without reference to time? We first consider theclassical case, where the answer has a variety of forms in terms of a phasespace probability distribution function. We then consider the quantum case, andwe analyze this question using the decoherent histories approach to quantumtheory, adapted to questions which do not involve time. When the histories aredecoherent, the probabilities approximately coincide with the classical case,with the phase space probability distribution replaced by the Wigner functionof the quantum state. For some initial states, decoherence requires anenvironment, and we compute the required influence functional and examine someof its properties. Special attention is given to the inner product used in theconstruction (the induced or Rieffel inner product), the construction of classoperators describing the histories, and the extent to which reparametrizationinvariance is respected. Our results indicate that simple systems without anexplicit time parameter may be quantized using the decoherent historiesapproach, and the expected classical limit extracted. The results support, forsimple models, the usual heuristic proposals for the probability distributionfunction associated with a semiclassical wave function satisfying theWheeler-DeWitt equation.Comment: 57 pages, plain Tex, 1 figur

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