Reduced phase space approach to Kasner universe and the problem of time in quantum theory

TL;DR

This paper applies reduced phase space quantization to the Kasner universe, constructing a physical Hilbert space and exploring the problem of time, revealing inequivalent quantum evolutions depending on clock choices.

Contribution

It introduces a method to quantize the Kasner universe using reduced phase space, explicitly constructing Dirac observables and analyzing the problem of time in quantum gravity.

Findings

Physical Hilbert space constructed with self-adjoint Dirac observables

Different clock choices lead to inequivalent quantum theories

Hubble operator spectrum varies with clock variables

Abstract

We apply the reduced phase space quantization to the Kasner universe. We construct the kinematical phase space, find solutions to the Hamilton equations of motion, identify Dirac observables and arrive at physical solutions in terms of Dirac observables and an internal clock. We obtain the physical Hilbert space, which is the carrier space of the self-adjoint representation of the Dirac observables. Then we discuss the problem of time. We demonstrate that the inclusion of evolution in a gravitational system, at classical level as well as at quantum level, leads respectively to canonically and unitarily inequivalent theories. The example of Hubble operator in two different clock variables and with two distinct spectra is given.