Loop quantum cosmology of the Bianchi I model: complete quantization

TL;DR

This paper completes the canonical quantization of the vacuum Bianchi I model in loop quantum cosmology, detailing the Hilbert space structure, observables, and superselection sectors, ensuring a well-posed initial value problem.

Contribution

It provides a full quantization framework for the Bianchi I model, including the structure of superselection sectors and the well-posedness of the initial value problem.

Findings

Characterization of the Hilbert space of physical states

Identification of superselection sectors due to polymeric quantization

Proof of well-posedness of the initial value problem

Abstract

We complete the canonical quantization of the vacuum Bianchi I model within the improved dynamics scheme of loop quantum cosmology, characterizing the Hilbert structure of the physical states and providing a complete set of observables acting on them. In order to achieve this task, it has been essential to determine the structure of the separable superselection sectors that arise owing to the polymeric quantization, and to prove that the initial value problem obtained when regarding the Hamiltonian constraint as an evolution equation, interpreting the volume as the evolution parameter, is well-posed.