Time dependent Green functions from Wheeler De Witt solutions

TL;DR

This paper constructs Green functions for matter fields in quantum cosmology from Wheeler De Witt solutions, clarifying the emergence of time and the validity of approximations, and setting the stage for analyzing quantum matter transitions.

Contribution

It introduces a new approach to recover time in quantum cosmology using Green functions derived from Wheeler De Witt solutions, and critically examines the standard inverse Planck mass expansion.

Findings

Background geometry is determined by Einstein equations driven by mean matter energy.

The formal inverse Planck mass expansion is shown to be illegitimate.

Green functions facilitate the study of quantum matter transitions in cosmology.

Abstract

The aim of this article is twofold. First we examine from a new angle the question of recovery of time in quantum cosmology. We construct Green functions for matter fields from the solutions of the Wheeler De Witt equation. For simplicity we work in a mini-superspace context. By evaluating these Green functions in a first order development of the energy ``increment'' induced by matrix elements of field operators, we show that the background geometry is the solution of Einstein equations driven by the mean matter energy and that it is this background which determines the time lapses separating the field operators. Then, by studying higher order corrections, we clarify the nature of the small dimensionless parameters which guarantee the validity of the approximations used. In this respect, we show that the formal expansion in the inverse Planck mass which is sometime presented as the ``standard procedure'' is illegitimate. Secondly, by the present analysis of Green functions, we prepare the study of quantum matter transitions in quantum cosmology. In a next article, we show that the time parametrization of transition amplitudes appears for the same reasons that it appeared in this article. This proves that the background is dynamically determined by the transition under examination.