Clocks and Trajectories in Quantum Cosmology

Przemysław Małkiewicz,Patrick Peter,Sandro Dias Pinto Vitenti

doi:10.3390/universe8020071

Abstract

We consider a simple cosmological model consisting of an empty Bianchi I Universe, whose Hamiltonian we deparametrise to provide a natural clock variable. The model thus effectively describes an isotropic universe with an induced clock given by the shear. By quantising this model, we obtain various different possible bouncing trajectories (semiquantum expectation values on coherent states or obtained by the de Broglie–Bohm formulation) and explicit their clock dependence, specifically emphasising the question of symmetry across the bounce.

Highlights

The problem of time in quantum cosmology [1,2] is well-known and, as of unsolved
It rests on the fact that general relativity (GR) is a totally constrained theory, and its canonically quantised counterpart can be reduced to the Wheeler–DeWitt (WDW) equation HΨ = 0, which is a timeless Schrödinger equation
We have reviewed the question of clock transformation and trajectories in quantum cosmology by means of a simple deparametrised and quantised Bianchi I model

Summary

Introduction

The problem of time in quantum cosmology [1,2] is well-known and, as of unsolved. [3], where in particular it was shown that there exist two categories of possible choices, namely the so-called fast- and slow-time gauges In the former case, the singularity is somehow not removed upon quantisation, in the sense that the wavefunction asymptotically shrinks towards a δ−function around the vanishing scale factor ( a singularity) after an infinite amount of time. We shall restrict out attention in what follows to the slow-time gauge only and assume the arbitrary function to take the simple form T = V−1, leading the relevant variable q to be identified with the volume V.

Affine Quantisation

Clock Transformations

Trajectories

Conclusions