Quantization of the Taub cosmological model with extrinsic time

TL;DR

This paper develops a formalism for quantizing the Taub cosmological model using extrinsic time, establishing a conserved inner product and boundary conditions to identify physical solutions in the Wheeler-DeWitt framework.

Contribution

It introduces a novel approach to quantize minisuperspace models with extrinsic time, including a quantum canonical transformation and boundary condition selection.

Findings

Defined a conserved Schrödinger inner product for the model

Established a correspondence between Wheeler-DeWitt solutions and desparametrized wave functions

Identified boundary conditions to select physical states

Abstract

The paper addresses the quantization of minisuperspace cosmological models, with application to the Taub Model. By desparametrizing the model with an extrinsic time, a formalism is developed in order to define a conserved Schr\"{o}dinger inner product in the space of solutions of the Wheeler-De Witt equation. A quantum version of classical canonical transformations is introduced for connecting the solutions of the Wheeler-De Witt equation with the wave functions of the desparametrized system. Once this correspondence is established, boundary conditions on the space of solutions of the Wheeler-De Witt equation are found to select the physical subspace. The question of defining boundary conditions on the space of solutions of the Wheeler-De Witt equation without having reduced the system is examined.