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arXiv:quant-ph/9711031 · Journal of Mathematical Physics

Continuous time and consistent histories

C. J. Isham, N. Linden, K. Savvidou, S. Schreckenberg

Published Apr 1, 1998• 66 citations
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Abstract

We discuss the use of histories labelled by a continuous time in the approach to consistent-histories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [C. J. Isham and N. Linden, J. Math. Phys. 36, 5392–5408 (1995)] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.

DOI
10.1063/1.532265

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We discuss the use of histories labelled by a continuous time in the approach to consistent-histories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [C. J. Isham and N. Linden, J. Math. Phys. 36, 5392–5408 (1995)] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.
We discuss the use of histories labelled by a continuous time in the approach to consistent-histories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us \\cite{IL95} where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.
We discuss the case of histories labelled by a continuous time parameter in the {\em History Projection Operator} consistent-histories quantum theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about time averages of the energy. We define the action operator for the consistent histories formalism, as the quantum analogue of the classical action functional, for the simple harmonic oscillator case. We show that the action operator is the generator of two types of time transformations that may be related to the two laws of time-evolution of the standard quantum theory: the `state-vector reduction' and the unitary time-evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories; we demonstrate how the requirement of the temporal logic structure of the theory is sufficient for the definition of classical histories. Furthermore, we show the relation of the action operator to the decoherence functional which describes the dynamics of the system. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.
An introduction is given to an algebraic formulation and generalisation of the consistent histories approach to quantum theory. The main technical tool in this theory is an orthoalgebra of history propositions that serves as a generalised temporal analogue of the lattice of propositions of standard quantum logic. Particular emphasis is placed on those cases in which the history propositions can be represented by projection operators in a Hilbert space, and on the associated concept of a `history group'.
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.
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.