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arXiv:gr-qc/9302013 · Cambridge University Press eBooks

Decoherence of Correlation Histories

Esteban Calzetta, B. L. Hu

Published Nov 16, 1956• 35 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1017/cbo9780511524653.007

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•Jan 1, 2013•0 citations
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.
The statistical mechanical properties of interacting quantum fields in terms of the dynamics of the correlation functions are investigated. We show how the Dyson - Schwinger equations may be derived from a formal action functional, the n-particle irreducible (nPI,n→∞nPI, n \to \inftynPI,n→∞) or the `master' effective action. It is related to the decoherence functional between histories defined in terms of correlations. Upon truncation of the Dyson - Schwinger hierarchy at a certain order, the master effective action becomes complex, its imaginary part arising from the higher order correlation functions, the fluctuations of which we define as the correlation noises of that order. Decoherence of correlation histories via these noises gives rise to classical stochastic histories %driven by the flucutations of these higher correlation functions. Ordinary quantum field theory corresponds to taking the lowest order functions, usually the mean field and the 2-point functions. As such, our reasoning shows that it is an effective theory which can be intrinsically dissipative. The relation of loop expansion and correlation order as well as the introduction of an arrow of time from the choice of boundary conditions are expounded with regard to the origin of dissipation in quantum fields. Relation with critical phenomena, quantum transport, molecular hydrodynamics and potential applications to quantum gravity, early universe processes and black hole physics are mentioned.
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.
This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. It is in part a continuation of an earlier paper on the uncertainty principle for quantum-mechanical histories. We consider histories characterized by position samplings at n moments of time. This includes the case of position sampling continuous in time in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, F=mx\ifmmode\ddot\else\textasciidieresis\fi{}+V'(x), at n-2 times. We find that it is peaked around F=0; thus classical correlations are exhibited. We show that the width of the peak \ensuremath{\Delta}F is largely independent of the initial state. We show that the uncertainty principle takes the form 2${\mathrm{\ensuremath{\sigma}}}^{2}$(\ensuremath{\Delta}F)2{)}^{2})2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/t2{\mathit{t}}^{2}t2, where \ensuremath{\sigma} is the width of the position samplings, and t is the time scale between projections. We determine the modifications to this result when the system is coupled to a thermal environment, thus obtaining a measure of the comparative sizes of quantum and thermal fluctuations. We show that the thermal fluctuations become comparable to the quantum fluctuations under the same conditions that decoherence effects come into play, in agreement with earlier work. We also study an alternative measure of classical correlations, namely, the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred.We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. Under appropriate conditions, which we describe, they admit an interpretation as a statistical ensemble of classical solutions, with the probability of each individual classical solution given by a smeared Wigner function of its initial data. We study the decoherence properties of histories characterized by the value of the field equations, F, at a sequence of times. We argue that they will be decoherent if their initial data are decoherent.
In this paper we are interested in studying coarse graining in field theories using the language of quantum open systems. Motivated by the ideas of Hu and Calzetta on correlation histories we employ the Zwanzig projection technique to obtain evolution equations for relevant observables in self-interacting scalar field theories. Our coarse-graining operation consists in concentrating solely on the evolution of the correlation functions of degree less than n,n,n, a treatment which corresponds to the familiar truncation of the BBKGY hierarchy at the nnnth level. We derive the equations governing the evolution of mean-field and two-point functions thus identifying the terms corresponding to dissipation and noise. We discuss possible applications of our formalism, the emergence of classical behavior, and the connection to the decoherent histories framework.
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.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.