Minisuperspaces: Observables and Quantization

TL;DR

This paper simplifies the phase space of homogeneous cosmologies via canonical transformations to explicitly find Dirac observables, enabling a complete quantization and analysis of quantum singularities, with a comparison of different quantization approaches.

Contribution

It introduces a method to explicitly construct Dirac observables in homogeneous cosmologies, facilitating a complete quantization and comparison of different quantum formulations.

Findings

Initial singularities persist in the quantum theory.

Explicit Dirac observables can be obtained after canonical transformation.

Quantum theories based on different approaches are shown to be equivalent.

Abstract

A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e.\ phase space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through ``deparametrization'' and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory {\it inspite of the fact that the evolution is implemented by a 1-parameter family of unitary transformations}. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to quantum theory following an independent avenue. The two quantum theories --based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables-- are compared and shown to be equivalent.