Time dependent Green functions in Quantum Cosmology

The aim of this article is twofold. First we examine from a new angle the question of the 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. 1 present address: Laboratoire de Mathématique et Physique Théorique, Faculté des Sciences, Université de Tours, 37200 Tours, France Unité propre de recherche du C.N.R.S. associée à l’ENS et à l’Université de Paris Sud.