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TROUBLES WITH QUANTUM ANISOTROPIC COSMOLOGICAL MODELS: LOSS OF UNITARITY

Flávio Gimenes Alvarenga, A. B. Batista, J. C. Fabris, S. V. B. Gonçalves

Published Jan 1, 2004• 31 citations

Abstract

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

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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.

An analogy between non-relativistic quantum mechanics in the Madelung formulation and quantum geometrodynamics in the case of the maximally symmetric space is drawn. The equations equivalent to the continuity equation and the hydrodynamic Euler equation describing the evolution of the velocity introduced for the case of hypothetical fluid flow characterizing the cosmological system are obtained. It is shown that the perfect nature of the fluid is broken by the quantum Bohm potential. The quantum potential is calculated in the semi-classical approximation for different forces acting in the system both in standard quantum mechanics and in the minisuperspace model of quantum geometrodynamics. The explicit dependences of the cosmic scale factor on the conformal time, which account for the quantum additive, are found for the empty space with spatial curvature and for a spatially flat universe with dust and radiation.

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.

Recently it has been claimed that the Wheeler--DeWitt quantization of gravity is unable to avoid cosmological singularities. However, in order to make this assertion, one must specify the underlying interpretation of quantum mechanics which has been adopted. For instance, several nonsingular models were obtained in Wheeler--DeWitt quantum cosmology in the framework of the de Broglie--Bohm quantum theory. Conversely, there are specific situations where the singularity cannot be avoided in the framework of the consistent histories approach to quantum mechanics. In these specific situations, the matter content is described by a scalar field, and the Wheeler--DeWitt equation looks like a Klein--Gordon equation. The aim of this work is to study a possible singular behavior of quantum cosmological models in which the matter content is described by a hydrodynamical perfect fluid, where the resulting wave equation is a genuine Schr\"odinger equation. In this case, it is shown that the conclusions of the consistent histories and the de Broglie--Bohm approaches coincide in the quantum cosmological models where the curvature of the spatial sections is not positive definite, namely, that the cosmological singularities are eliminated. In the case of positive spatial curvature, the family of histories is no longer consistent, and no conclusion can be given in this framework.

This note studies how a universe can generate its own classical records without any external environment or inserted measurement registers. In a solvable anisotropic quantum cosmology model with a Bianchi-I-lite constraint, the two kernel branches correspond to opposite relational arrows of time. The anisotropy degree of freedom evolves differently in the two branches, and this internal motion dynamically writes a record of which relational arrow the universe follows. The paper proves an exact self-decoherence law: the visibility between the two arrow branches is given by the characteristic function of the universe’s own clock-energy distribution. The decoherence rate is determined by the energy variance, and the record–visibility relation satisfies the exact Englert duality. Thus, the same mathematical structure governing information deficits, records, and revival phenomena in the companion framework appears here as a self-generated cosmological decoherence mechanism. The results establish a complete chain: constraint → relational motion → internal records → classicality. Unlike previous exact emergence-of-classicality models in the series, where matter records were introduced as fixed registers, this model generates the records dynamically from the universe’s own degrees of freedom. The universe acts as its own environment. The paper also identifies the boundaries of the result: dispersion returns because of the nonlinear spectrum, and finite-spectrum recurrences appear beyond the declared window. These effects are explicitly analyzed and distinguished from the continuum behavior. The work is limited to a solvable anisotropic minisuperspace model with one anisotropy degree of freedom and a fixed volume-momentum sector. It does not claim a result for full Bianchi cosmology or complete Einstein dynamics. All results are based on pre-registered predictions, computational verification, and reproducibility materials.