Time and Evolution in Quantum and Classical Cosmology

Alexander Yu Kamenshchik,Jeinny Nallely Pérez Rodríguez,Tereza Vardanyan

doi:10.3390/universe7070219

Abstract

We analyze the issue of dynamical evolution and time in quantum cosmology. We emphasize the problem of choice of phase space variables that can play the role of a time parameter in such a way that for expectation values of quantum operators the classical evolution is reproduced. We show that it is neither necessary nor sufficient for the Poisson bracket between the time variable and the super-Hamiltonian to be equal to unity in all of the phase space. We also discuss the question of switching between different internal times as well as the Montevideo interpretation of quantum theory.

Highlights

The problem of “disappearance of time” in quantum gravity and cosmology is well known and has a rather long history
The structure of this paper is the following: we present for completeness some basic formulae of the Arnowitt-Deser-Misner formalism and the Dirac quantization of gravity, arriving to the general form of the Wheeler-DeWitt equation
In this review various ideas concerning the problem of time in quantum gravity and cosmology were studied by using a simple model of a flat Friedmann universe filled with a massless minimally coupled scalar field

Summary

Introduction

The problem of “disappearance of time” in quantum gravity and cosmology is well known and has a rather long history (see, e.g., References [1,2,3] and references therein). We shall identify the parameter t, defining the hypersurfaces of the 3+1 foliation, with time This vector field is tangent to the integral curves, which are the curves along which the three spatial coordinates are constant. (we shall use a different convention for the signature of the spacetime to simplify the comparison with the papers [17,21].) This universe is filled with a massless spatially homogeneous scalar field φ(t) minimally coupled to gravity For this minisuperspace model, the Lagrangian can be written as. Let us try to use the following one: ξ(α, pα, t) This gauge condition coincides with the classical solution of the Friedmann equation, giving the dependence of the scale factor a on the cosmic time t. To obtain these results, we have used the super-Hamiltonian constraint

Introducing Time without Using the Super-Hamiltonian Constraint

Switching Internal Times and the Montevideo Interpretation of Quantum Theory

Discussion