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arXiv:quant-ph/0701186 · Journal of Mathematical Physics

Phase space representations and perturbation theory for continuous-time histories

Aidan Burch

Published Jul 1, 2007• 1 citation
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Abstract

We consider two technical developments of the formalism of continuous-time histories. First, we provide an explicit description of histories of the simple harmonic oscillator on the classical history phase space, comparing and contrasting the Q, P, and Wigner representations; we conclude that a representation based on coherent states is the most appropriate. Second, we demonstrate a generic method for implementing a perturbative approach for interacting theories in the history formalism, using the quartic anharmonic oscillator. We make use of the identification of the closed-time-path generating functional with the decoherence functional to develop a perturbative expansion for the latter up to second order in the coupling constant. We consider both configuration space and phase space histories.

DOI
10.1063/1.2752009

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We consider two technical developments of the formalism of continuous-time histories. First, we provide an explicit description of histories of the simple harmonic oscillator on the classical history phase space, comparing and contrasting the Q, P, and Wigner representations; we conclude that a representation based on coherent states is the most appropriate. Second, we demonstrate a generic method for implementing a perturbative approach for interacting theories in the history formalism, using the quartic anharmonic oscillator. We make use of the identification of the closed-time-path generating functional with the decoherence functional to develop a perturbative expansion for the latter up to second order in the coupling constant. We consider both configuration space and phase space histories.
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.
In this paper we elaborate on the structure of the continuous-time histories description of quantum theory, which stems from the consistent histories scheme. In particular, we examine the construction of history Hilbert space, the identification of history observables and the form of the decoherence functional (the object that contains the probability information). It is shown how the latter is equivalent to the closed-time-path (CTP) generating functional. We also study the phase space structure of the theory first through the construction of general representations of the history group (the analogue of the Weyl group) and the implementation of a histories Wigner-Weyl transform. The latter enables us to write quantum theory solely in terms of phase space quantities. These results enable the implementation of an algorithm for identifying the classical (stochastic) limit of a general quantum system.
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.
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 conclude that the action operator is the generator of time transformations, and is associated with the two types of time-evolution of the standard quantum theory: the wave-packet reduction and the unitary time-evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories. Finally, we show the relation of the action operator to the decoherence functional. 1
We define the action operator for the consistent histories formalism, as the quantum analog of the classical action functional, for the simple harmonic oscillator case. We conclude that the action operator is the generator of time transformations, and is associated with the two types of time evolution of the standard quantum theory: the wave-packet reduction and the unitary time evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories. Finally, we show the relation of the action operator to the decoherence functional.
We consider the classical limit of quantum mechanics from the viewpoint of perturbation theory. The main focus is time dependent perturbation theory, in particular, the time evolution of a harmonic oscillator coherent state in an anharmonic potential. We explore in detail a perturbation method introduced by Bhaumik and Dutta-Roy [J. Math. Phys. 16, 1131 (1975)] and resolve several complications that arise when this method is extended to second order. A classical limit for coherent states used by the above authors is then applied to the quantum perturbation expansions and, to second order, the classical Poincaré–Lindstedt series is retrieved. We conclude with an investigation of the connection between the classical limits of time dependent and time independent perturbation theories, respectively.