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arXiv:1603.01406 · Physica Scripta

Unitary evolution for anisotropic quantum cosmologies: models with variable spatial curvature

Sachin Pandey, Narayan Banerjee

Published Oct 12, 2016• 4 citations
arXivHTMLPDF

Abstract

Abstract Contrary to the general belief, there has recently been quite a few examples of unitary evolution of quantum cosmological models. The present work gives more examples, namely Bianchi type VI and type II. These examples are important as they involve varying spatial curvature unlike the most talked about homogeneous but anisotropic cosmological models like Bianchi I, V and IX. We exhibit either an explicit example of the unitary solutions of the Wheeler–DeWitt equation, or at least show that a self-adjoint extension is possible.

DOI
10.1088/0031-8949/91/11/115001

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Abstract Contrary to the general belief, there has recently been quite a few examples of unitary evolution of quantum cosmological models. The present work gives more examples, namely Bianchi type VI and type II. These examples are important as they involve varying spatial curvature unlike the most talked about homogeneous but anisotropic cosmological models like Bianchi I, V and IX. We exhibit either an explicit example of the unitary solutions of the Wheeler–DeWitt equation, or at least show that a self-adjoint extension is possible.
A brief summary of the application of coherent states in the examination of quantum dynamics of cosmological models is given. We discuss quantization maps, phase space probability distributions and semiclassical phase spaces. The implementation of coherent states based on the affine group resolves the hardest singularities, renders self-adjoint Hamiltonians without boundary conditions and provides a completely consistent semiclassical description of the involved quantum dynamics. We consider three examples: the closed Friedmann model, the anisotropic Bianchi Type I model and the deep quantum domain of the Bianchi Type IX model.
The anisotropic Bianchi I cosmological model coupled with perfect fluid is quantized in the minisuperspace. The perfect fluid is described by using the Schutz formalism which allows to attribute dynamical degrees of freedom to matter. It is shown that the resulting model is non-unitary. This breaks the equivalence between the many-worlds and dBB interpretations of quantum mechanics. PACS number(s): 04.20.Cv., 04.20.Me The aim of this work is to study the anisotropic Bianchi I quantum cosmological model coupled to a perfect fluid described by the Schutz formalism[1, 2]. Hence, we start with the classical Lagrangian given by A = d M 4 x √ ∫ −gR + 2 d ∂M 3 x √ hhabK ab + d 4 x √ −gp (1) M where K ab is the extrinsic curvature, and hab is the induced metric over the three-dimensional spatial hypersurface, which is the boundary ∂M of the four dimensional manifold M; the factor 16πG is made equal to one. The first two terms were first obtained in the classical work of Arnowitt, Deser and Misner[3]; the last term of (1) represents the matter contribution to the total action in the Schutz’s formalism for perfect fluids, p being the pressure, which is linked to the energy density by the equation of state p = αρ. In the Schutz’s formalism[1, 2], the four-velocity is expressed in terms of five potentials ǫ, ζ, β, θ and S: Uν = 1 (ǫ,ν + ζβ,ν + θS,ν) (2) µ where µ is the specific enthalpy. The variable S is the specific entropy, while the potentials ζ and β are connected with rotation and are absent for FRW’s type models. The variables ǫ and θ have no clear physical meaning. The four velocity is subject to the condition U ν Uν = 1. (3) The metric describing a Bianchi I anisotropic model is given by
In the isotropic quantum cosmological perfect fluid model, the initial singularity can be avoided, while the classical behaviour is recovered asymptotically. We verify if initial anisotropies can also be suppressed in a quantum version of a classical anisotropic model where gravity is coupled to a perfect fluid. Employing a Bianchi I cosmological model, we obtain a "Schrödinger-like" equation where the matter variables play de role of time. This equation has a hyperbolic signature. It can be explicitly solved and a wave packet is constructed. The expectation value of the scale factor, evaluated in the spirit of the many-worlds interpretation, reveals an isotropic Universe. On the other hand, the bohmian trajectories indicate the existence of anisotropies. This is an example where the Bohm-de Broglie and the many-worlds interpretations are not equivalent. It is argued that this inequivalence is due to the hyperbolic structure of the "Schrödinger-like" equation.
We consider the problem of extracting physical predictions from the wave function of the universe in quantum cosmological models. We state the features of quantum cosmology an interpretational scheme should confront. We discuss the Everett interpretation, and extensions of it, and their application to quantum cosmology. We review the steps that are normally taken in the process of extracting predictions from solutions to the Wheeler-DeWitt equation for quantum cosmological models. Some difficulties and their possible resolution are discussed. We conclude that the usual wave function-based approach admits at best a rather heuristic interpretation, although it may in the future be justified by appeal to the decoherent histories approach.
A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focusing on the particular example of the classically recollapsing Bianchi type-IX universe. The decoherence functional for such universes is exhibited. We show how the probabilities of decoherent sets of alternative, coarse-grained histories of these model universes can be calculated. We consider in particular the probabilities for classical evolution defined by a suitable coarse graining. For a restricted class of initial conditions and coarse grainings we exhibit the approximate decoherence of alternative histories in which the universe behaves classically and those in which it does not. For these situations we show that the probability is near unity for the universe to recontract classically if it expands classically. We also determine the relative probabilities of quasiclassical trajectories for initial states of WKB form, recovering for such states a precise form of the familiar heuristic $``J\ensuremath{\cdot}d\ensuremath{\Sigma}''$ rule of quantum cosmology, as well as a generalization of this rule to generic initial states.