Observables and a Hilbert Space for Bianchi IX

TL;DR

This paper presents a novel quantization approach for the Bianchi IX cosmological model, constructing a physical Hilbert space with gauge-invariant operators that reflect classical recollapse behavior.

Contribution

It introduces a self-adjoint operator-based quantization with a continuous spectrum at zero and constructs a complete set of gauge-invariant operators on the physical space.

Findings

The constraint has a continuous spectrum at zero.

The physical Hilbert space is infinite-dimensional.

Operators indicate classical recollapse behavior.

Abstract

We consider a quantization of the Bianchi IX cosmological model based on taking the constraint to be a self-adjoint operator in an auxiliary Hilbert space. Using a WKB-style self-consistent approximation, the constraint chosen is shown to have only continuous spectrum at zero. Nevertheless, the auxiliary space induces an inner product on the zero-eigenvalue generalized eigenstates such that the resulting physical Hilbert space has countably infinite dimension. In addition, a complete set of gauge-invariant operators on the physical space is constructed by integrating differential forms over the spacetime. The behavior of these operators indicates that this quantization preserves Wald's classical result that the Bianchi IX spacetimes expand to a maximum volume and then recollapse.