Extrinsic Time in Quantum Cosmology

TL;DR

This paper introduces a method to identify an extrinsic time in isotropic and homogeneous quantum cosmological models, enabling their quantization to be analogous to an ideal clock system, resulting in a unitary theory.

Contribution

It presents a novel approach to define extrinsic time in cosmological models and quantizes them similarly to an ideal clock, ensuring unitarity and consistency.

Findings

Successfully matches cosmological models with an ideal clock

Constructs a Hilbert space with a time-dependent inner product

Achieves a unitary quantum theory equivalent to phase space reduction

Abstract

An extrinsic time is identified in most isotropic and homogeneous cosmological models by matching them with the ideal clock - a parametrized system whose only "degree of freedom" is time -. Once this matching is established, the cosmological models are quantized in the same way as the ideal clock. The space of solutions of the Wheeler-DeWitt equation is turned out into a Hilbert space by inserting a time dependent operator in the inner product, yielding a unitary theory equivalent to the phase space reduced theory.