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arXiv:quant-ph/9805062 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Decoherent histories and hydrodynamic equations

J. J. Halliwell

Published Oct 9, 1998• 89 citations
arXivHTMLPDF

Abstract

For a system consisting of a large collection of particles, variables that will generally become effectively classical are the local densities (number, momentum, energy). That is, in the context of the decoherent histories approach to quantum theory, it is expected that histories of these variables will be approximately decoherent and that their probabilities will be strongly peaked about hydrodynamic equations. This possibility is explored for the case of the diffusion of the number density of a dilute concentration of foreign particles in a fluid. This system has the appealing feature that the microscopic dynamics of each individual foreign particle is readily obtained and the approach to local equilibrium may be seen explicitly. It is shown that, for certain physically reasonable initial states, the probabilities for the histories of the number density are strongly peaked about evolution according to the diffusion equation. Decoherence of these histories is also shown for a class of initial states which includes nontrivial superpositions of number density. Histories of phase space densities are also discussed. The case of the histories of number, momentum, and energy density for more general systems, such as a dilute gas, is also discussed in outline. When the initial state is a local equilibrium state, it is shown that the histories are trivally decoherent and that the probabilities for histories are peaked about hydrodynamic equations. An argument for the decoherence of more general initial states is given.

DOI
10.1103/physrevd.58.105015

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For a system consisting of a large collection of particles, variables that will generally become effectively classical are the local densities (number, momentum, energy). That is, in the context of the decoherent histories approach to quantum theory, it is expected that histories of these variables will be approximately decoherent and that their probabilities will be strongly peaked about hydrodynamic equations. This possibility is explored for the case of the diffusion of the number density of a dilute concentration of foreign particles in a fluid. This system has the appealing feature that the microscopic dynamics of each individual foreign particle is readily obtained and the approach to local equilibrium may be seen explicitly. It is shown that, for certain physically reasonable initial states, the probabilities for the histories of the number density are strongly peaked about evolution according to the diffusion equation. Decoherence of these histories is also shown for a class of initial states which includes nontrivial superpositions of number density. Histories of phase space densities are also discussed. The case of the histories of number, momentum, and energy density for more general systems, such as a dilute gas, is also discussed in outline. When the initial state is a local equilibrium state, it is shown that the histories are trivally decoherent and that the probabilities for histories are peaked about hydrodynamic equations. An argument for the decoherence of more general initial states is given.
In the context of the decoherent histories approach to quantum theory, it is shown that a class of macroscopic configurations consisting of histories of local densities (number, momentum, energy) exhibits negligible interference. This follows from the close connection of the local densities with the corresponding exactly conserved (and so exactly decoherent) quantities, and also from the observation that the eigenstates of local densities (averaged over a sufficiently large volume) remain approximate eigenstates under time evolution. The result is relevant to the derivation of hydrodynamic equations using the decoherent histories approach.
Abstract Macroscopic systems are described most completely by local densities (particle number, momentum, and energy) yet the superposition states of such physical variables, indicated by the Everett interpretation, are not observed. In order to explain this, it is argued that histories of local number, momentum, and energy density are approximately decoherent when coarse-grained over sufficiently large volumes. Decoherence arises directly from the proximity of these variables to exactly conserved quantities (which are exactly decoherent), and not from environmentally induced decoherence. This chapter discusses the approach to local equilibrium and the subsequent emergence of hydrodynamic equations for the local densities. The results are general but the chapter focuses on a chain of oscillators as a specific example in which explicit calculations may be carried out. It discusses the relationships between environmentally-induced and conservation-induced decoherence, and presents a unified view of these two mechanisms.
We investigate the decoherence of histories of local densities for linear oscillators models. It is shown that histories of local number, momentum and energy density are approximately decoherent, when coarse grained over sufficiently large volumes. Decoherence arises directly from the proximity of these variables to exactly conserved quantities (which are exactly decoherent), and not from environmentally induced decoherence. We discuss the approach to local equilibrium and the subsequent emergence of hydrodynamic equations for the local densities.
In the context of the decoherent histories approach to the quantum mechanics of closed systems, Gell-Mann and Hartle have argued that the variables typically characterizing the quasiclassical domain of a large complex system are the integrals over small volumes of locally conserved densities---hydrodynamic variables. The aim of this paper is to exhibit some simple models in which approximate decoherence arises as a result of local conservation. We derive a formula which shows the explicit connection between local conservation and approximate decoherence. We then consider a class of models consisting of a large number of weakly interacting components, in which the projections onto local densities may be decomposed into projections onto one of two alternatives of the individual components. The main example we consider is a one-dimensional chain of locally coupled spins, and the projections are onto the total spin in a subsection of the chain. We compute the decoherence functional for histories of local densities, in the limit when the number of components is very large. We find that decoherence requires two things: the smearing volumes must be sufficiently large to ensure approximate conservation, and the local densities must be partitioned into sufficiently large ranges to ensure protection against quantum fluctuations.
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.
Using nonrelativistic many body quantum field theory, a master equation is derived for the reduced density matrix of a dilute gas of massive particles undergoing scattering interactions with an environment of light particles. The dynamical variable that naturally decoheres (the pointer basis) is essentially the local number density of the dilute gas. Earlier master equations for this sort of system (such as that derived by Joos and Zeh) are recovered on restricting to the one-particle sector for the distinguished system. The derivation shows explicitly that the scattering environment stores information about the system by ``measuring'' the number density. This therefore provides an important example of the general connection between decoherence and records indicated by the decoherent histories approach to quantum theory. It also brings the master equation for this system into a form emphasizing the role of local densities, which is relevant to current work in deriving hydrodynamic equations from quantum theory.