Continuous-time histories: Observables, probabilities, phase space structure and the classical limit

The continuous-time histories program stems from the consistent histories approach to quantum theory and aims to provide a fully covariant formalism for quantum mechanics. In this paper we examine some structural points of the formalism. We demonstrate a general construction of history Hilbert spaces and identify a large class of time-averaged observables. We pay particular attention to the construction of the decoherence functional (the object that encodes probability information) in the continuous-time limit and its relation to the temporal structure of the theory. Phase space observables are introduced, through the study of general representations of the history group, which is the analog of the canonical group in the formalism. We can also define a closed-time-path (CTP) generating functional for each observable, which encodes the information of its correlation functions. The phase space version of the CTP generating functional leads to the implementation of Wigner-Weyl transforms, that gives a description of quantum theory solely in terms of phase space histories. These results allow the identification of an algorithm for going to the classical (stochastic) limit for a generic quantum system. (C) 2001 American Institute of Physics.