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arXiv:gr-qc/9503063 · Journal of Mathematical Physics

Continuous histories and the history group in generalized quantum theory

Christopher John Isham, Noah Linden

Published Oct 1, 1995• 130 citations
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Abstract

We treat continuous histories within the histories approach to generalized quantum mechanics. The essential tool is the ‘‘history group:’’ the analog, within the generalized history scheme, of the canonical group of single-time quantum mechanics.
DOI
10.1063/1.531267

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These lecture notes cover the important developments in histories approach to quantum mechanics with overall content and emphasis somewhat different from other reviews and books on the subject.The idea of Houtappel, Van Dam and Wigner of employing objects based on primitive concepts of physical theories is discussed in some detail and the fact that histories are such objects is emphasized. Application of histories formalism to the problem of understanding the quasiclassical domain is treated in some detail. Other topics discussed include generalized histories-based quantum mechanics and its application to the quantum mechanics of space-time,generalization of the notion of time sequences employing partial semigroups,quasitemporal structures, history projection operator (HPO) formalism, the algebraic scheme of Isham and Linden, an axiomatic scheme for quasitemporal histories-based theories and symmetries and conservation laws in histories-based theories.
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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.