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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Wigner function and decoherence in quantum cosmology

Salman Habib, Raymond Laflamme

Published Dec 15, 1990• 83 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.42.4056

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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.
The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.
We study the classical limit of quantum mechanics as applied to quantum cosmology. Conventional wisdom regards the peaking of the Wigner function of the Universe around a classical trajectory as being a quantum prediction of that trajectory. We show that, with quantum interference correctly taken into account, the hoped for "classical correlation" does not exist. There are, therefore, serious difficulties with the notion that a pure quantum state has a classical limit relevant to the description of our world. Some form of quantum decoherence appears necessary for a strict classical limit to exist. This alternative is briefly discussed.
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Two basic properties defining classical behavior are "decoherence" and "correlations between co-ordinates and momenta." We study how the correlations that define the semiclassical decohering histories of the relevant cosmological variables are affected by the interaction with an environment formed by unobserved ("irrelevant") degrees of freedom. For some quantum cosmological models we analyze under what conditions the semiclassical coarse-grained histories obey the so-called semiclassical Einstein's equations (i.e., ${G}_{\ensuremath{\mu}\ensuremath{\nu}}=\ensuremath{\kappa}〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$). These equations are shown to be valid only as a description of adiabatic regions of histories for which the interference effects have been suppressed. We also discuss the problem related to the existence of divergences in the decoherence factor of various quantum cosmological models.
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.