The criteria for a solution of the field equations to be a classical limit of a quantum cosmology

TL;DR

This paper establishes criteria for when solutions to Einstein's field equations can be considered classical limits of quantum cosmology, emphasizing the role of path integrals and saddlepoint approximations in the quantum-to-classical transition.

Contribution

It introduces specific conditions involving the classical action and saddlepoints that determine the classical limit of quantum cosmological solutions.

Findings

Classical solutions must dominate the path integral.

Stationarity of the action with respect to gravitational degrees is essential.

A single saddlepoint should significantly contribute to each integral.

Abstract

If the gravitational field is quantized, then a solution of Einstein's field equations is a valid cosmological model only if it corresponds to a classical limit of a quantum cosmology. To determine which solutions are valid requires looking at quantum cosmology in a particular way. Because we infer the geometry by measurements on matter, we can represent the amplitude for any measurement in terms of the amplitude for the matter fields, allowing us to integrate out the gravitational degrees of freedom. Combining that result with a path-integral representation for quantum cosmology leads to an integration over 4-geometries. Even when a semiclassical approximation for the propagator is valid, the amplitude for any measurement includes an integral over the gravitational degrees of freedom. The conditions for a solution of the field equations to be a classical limit of a quantum cosmology are: (1) The effect of the classical action dominates the integration, (2) the action is stationary with respect to variation of the gravitational degrees of freedom, and (3) only one saddlepoint contributes significantly to each integration.