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arXiv:gr-qc/9303036 · International Journal of Modern Physics D

DECOHERENCE FUNCTIONAL AND INHOMOGENEITIES IN THE EARLY UNIVERSE

R. LAFLAMME, A. MATACZ

Published Jun 1, 1993• 32 citations
arXivHTMLPDF

Abstract

We investigate the quantum to classical transition of small inhomogeneous fluctuations in the early Universe using the decoherence functional of Gell-Mann and Hartle. We study two types of coarse graining; one due to coarse graining the value of the scalar field and the other due to summing over an environment. We compare the results with a previous study using an environment and the off-diagonal rule proposed by Zurek. We show that the two methods give different results.

DOI
10.1142/s0218271893000155

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We investigate the quantum to classical transition of small inhomogeneous fluctuations in the early Universe using the decoherence functional of Gell-Mann and Hartle. We study two types of coarse graining; one due to coarse graining the value of the scalar field and the other due to summing over an environment. We compare the results with a previous study using an environment and the off-diagonal rule proposed by Zurek. We show that the two methods give different results.
A specific proposal is discussed for how to identify decohering paths in a wavefunction of the universe. The emphasis is on determining the correlations among subsystems and then considering how these correlations evolve. The proposal is similar to earlier ideas of Schroedinger and of Zeh, but in other ways it is closer to the decoherence functional of Griffiths, Omnes, and Gell-Mann and Hartle. There are interesting differences with each of these which are discussed. Once a given coarse-graining is chosen, the candidate paths are fixed in this scheme, and a single well defined number measures the degree of decoherence for each path. The normal probability sum rules are exactly obeyed (instantaneously) by these paths regardless of the level of decoherence. Also briefly discussed is how one might quantify some other aspects of classicality. The important role that concrete calculations play in testing this and other proposals is stressed.
We investigate the retrieval of classical behavior from the wave function of the Universe. It has been previously argued that to do this two conditions are necessary: decoherence (or lack of quantum interference) and the existence of classical correlations. Using the Wigner function we study the mutual compatibility of these requirements. Assuming the correlation interpretation of Geroch and Hartle, we show that some form of coarse graining is essential for the theory to predict classical behavior from WKB wave functions. We also show that there must be a compromise between the degree of decoherence and the sharpness of classical correlations. A minisuperspace model is used as an example to illustrate these points.
We study the issue of decoherence and dissipation in the wave function of the Universe for a Bianchi type-I universe with classical and quantum matter. We obtain a coarse-grained description by tracing over the matter degrees of freedom. Provided that for small universes the wave function of the universe is concentrated on a neighborhood of the isotropic configuration, then the coarse-grained density matrix of the universe will show an even more marked peak around isotropy for large universes. In this sense we can say that, while decoherence makes the reduced density matrix of the universe diagonal, dissipation causes the universe to be isotropic with a high probability for large radii.
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.
Decoherence and dissipation in quantum systems has been studied extensively in the context of Quantum Brownian Motion. Effective decoherence in coarse grained quantum systems has been a central issue in recent efforts by Zurek and by Hartle and Gell-Mann to address the Quantum Measurement Problem. Although these models can yield very general classical phenomenology, they are incapable of reproducing relevant characteristics expected of a local environment on a quantum system, such as the characteristic dependence of decoherence on environment spatial correlations. I discuss the characteristics of Quantum Brownian Motion in a local environment by examining aspects of first principle calculations and by the construction of phenomenological models. Effective quantum Langevin equations and master equations are presented in a variety of representations. Comparisons are made with standard results such as the Caldeira-Leggett master equation.