Classical and quantum behavior of the generic cosmological solution

TL;DR

This paper extends Misner's quantum analysis of Bianchi IX cosmology to a generic inhomogeneous universe, deriving a quantum dynamics framework and identifying conditions for semiclassical behavior.

Contribution

It generalizes the quantum treatment of cosmological models to inhomogeneous universes by solving constraints and decoupling spatial points, providing a new functional quantum dynamics representation.

Findings

Quantum dynamics factorizes into independent components for each spatial point.

Semiclassical behavior requires mean occupation numbers of order around 100.

The approach extends previous homogeneous models to generic inhomogeneous cosmologies.

Abstract

In the present paper we generalize the original work of C.W. Misner \cite{M69q} about the quantum dynamics of the Bianchi type IX geometry near the cosmological singularity. We extend the analysis to the generic inhomogeneous universe by solving the super-momentum constraint and outlining the dynamical decoupling of spatial points. Firstly, we discuss the classical evolution of the model in terms of the Hamilton-Jacobi approach as applied to the super-momentum and super-Hamiltonian constraints; then we quantize it in the approximation of a square potential well after an ADM reduction of the dynamics with respect to the super-momentum constraint only. Such a reduction relies on a suitable form for the generic three-metric tensor which allows the use of its three functions as the new spatial coordinates. We get a functional representation of the quantum dynamics which is equivalent to the Misner-like one when extended point by point, since the Hilbert space factorizes into $\infty^3$ independent components due to the parametric role that the three-coordinates assume in the asymptotic potential term. Finally, we discuss the conditions for having a semiclassical behavior of the dynamics and we recognize that this already corresponds to having mean occupation numbers of order $\mathcal{O}(10^2)$.