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arXiv:gr-qc/0011111 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum correlation functions and the classical limit

Charis Anastopoulos

Published May 24, 2001• 15 citations
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Abstract

We study the transition from the full quantum mechanical description of physical systems to an approximate classical stochastic one. Our main tool is the identification of the closed-time-path (CTP) generating functional of Schwinger and Keldysh with the decoherence functional of the consistent histories approach. Given a degree of coarse graining in which interferences are negligible, we can explicitly write a generating functional for the effective stochastic process in terms of the CTP generating functional. This construction gives particularly simple results for Gaussian processes. The formalism is applied to simple quantum systems, quantum Brownian motion, and quantum fields in curved spacetime. Perturbation theory is also explained. We conclude with a discussion on the problem of the back-reaction of quantum fields in spacetime geometry.

DOI
10.1103/physrevd.63.125024

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We study the transition from the full quantum mechanical description of physical systems to an approximate classical stochastic one. Our main tool is the identification of the closed-time-path (CTP) generating functional of Schwinger and Keldysh with the decoherence functional of the consistent histories approach. Given a degree of coarse graining in which interferences are negligible, we can explicitly write a generating functional for the effective stochastic process in terms of the CTP generating functional. This construction gives particularly simple results for Gaussian processes. The formalism is applied to simple quantum systems, quantum Brownian motion, and quantum fields in curved spacetime. Perturbation theory is also explained. We conclude with a discussion on the problem of the back-reaction of quantum fields in spacetime geometry.
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.
Using the Schwinger-Keldysh (closed time path or CTP) and Feynman-Vernon influence functional formalisms we obtain an expression for the influence functional in terms of Bogoliubov coefficients for the case of spinor quantum electrodynamics. Then we derive a CTP effective action in semiclassical approximation and its cumulant expansion. Using it we obtain a equation for the description of the charged particle creation in electric field and of backreaction of charged quantum fields and their fluctuations on time evolution of this electric field. Also an intimate connection between CTP effective action and decoherence functional will allow us to analyze how macroscopic electromagnetic fields are ``measured'' through interaction with charges and thereby rendered classical.
We use a λΦ 4 scalar quantum field theory to illustrate a new approach to the study of quantum to classical transition. In this approach, the decoherence functional is employed to assign probabilities to consistent histories defined in terms of correlations among the fields at separate points, rather than the field itself. We present expressions for the quantum amplitudes associated with such histories, as well as for the decoherence functional between two of them. The dynamics of an individual consistent history may be described by a Langevintype equation, which we derive. Dedicated to Professor Brill on the occasion of his sixtieth birthday, August 1993 Introduction Interpretations of Quantum Mechanics and Paradigms of Statistical Mechanics This paper attempts to bring together two basic concepts, one from the foundations of statistical mechanics and the other from the foundations of quantum mechanics, for the purpose of addressing two basic issues in physics: the quantum to classical transition, and the quantum origin of stochastic dynamics. Both issues draw in the interlaced effects of dissipation, decoherence, noise, and fluctuation. A central concern is the role played by coarse-graining —the naturalness of its choice, the effectiveness of its implementation and the relevance of its consequences.
We study a formulation of quantum mechanics in which the central notion is that of a quantum-mechanical history---a sequence of events at a succession of times. The primary aim is to identify sets of ``decoherent'' (or ``consistent'') histories for the system. These are quantum-mechanical histories suffering negligible interference with each other, and, therefore, to which probabilities may be assigned. These histories may be found for a given system using the so-called decoherence functional. When the decoherence functional is exactly diagonal, probabilities may be assigned to the histories, and all probability sum rules are satisfied exactly. We propose a condition for approximate decoherence, and argue that it implies that most probability sum rules will be satisfied to approximately the same degree. We also derive an inequality bounding the size of the off-diagonal terms of the decoherence functional. We calculate the decoherence functional for some simple one-dimensional systems, with a variety of initial states. For these systems, we explore the extent to which decoherence is produced using two different types of coarse graining. The first type of coarse graining involves imprecise specification of the particle's position. The second involves coupling the particle to a thermal bath of harmonic oscillators and ignoring the details of the bath (the Caldeira-Leggett model). We argue that both types of coarse graining are necessary in general. We explicitly exhibit the degree of decoherence as a function of the temperature of the bath, and of the width to within which the particle's position is specified. We study the diagonal elements of the decoherence functional, representing the probabilities for the possible histories of the system. To the extent that the histories decohere, we show that the probability distributions are peaked about the classical histories of the system, with the distribution of their initial positions and momenta given by a smeared version of the Wigner function. We discuss this result in connection with earlier uses of the Wigner function in this context. We find that there is a certain amount of tension between the demands of decoherence and peaking about classical paths.
The aim of this paper is twofold: to probe the statistical mechanical properties of interacting quantum fields, and to provide a field theoretical justification for a stochastic source term in the Boltzmann equation. We start with the formulation of quantum field theory in terms of the set of Schwinger-Dyson equations for the correlation functions, which we describe by a closed-time-path master $(n=\ensuremath{\infty}\mathrm{PI})$ effective action. When the hierarchy is simply truncated to a certain order, one obtains the usual closed system of correlation functions up to that order, and from the nPIn\mathrm{PI}nPI effective action, a set of time-reversal invariant equations of motion. (This is the Dyson equation, the quantum field theoretical parallel of the collisionless Boltzmann equation.) But when the effect of the higher order correlation functions is included through a causal factorization condition (such as the molecular chaos assumption in Boltzmann's theory) called slaving, the dynamics of the lower order correlations shows dissipative features, as familiar in the usual (dissipative yet noiseless) Boltzmann equation, the field-theoretical version of which being the dissipative Dyson equations. We show that a fluctuation-dissipation relation should exist for such effectively open systems, and use this fact to show that a stochastic term, which explicitly introduces quantum fluctuations in the lower order correlation functions, necessarily accompanies the dissipative term. This leads to a stochastic Dyson equation, which is the quantum field theoretic parallel of the classical Boltzmann-Langevin equation, encompassing both the dissipative and stochastic dynamics of correlation functions.