The Background Field Approximation in (quantum) cosmology

TL;DR

This paper investigates how the Hamilton-Jacobi action in quantum cosmology relates to the Schrödinger equation in a fixed background, clarifying the conditions under which the background field approximation is valid.

Contribution

It provides a detailed analysis of the background field approximation in quantum cosmology, connecting classical Hamilton-Jacobi solutions to quantum WKB states.

Findings

The change in gravitational action matches the matter energy in the background geometry to first order.

Higher order corrections depend on the geometry's susceptibility.

Classical properties inform the validity of the background approximation in quantum cosmology.

Abstract

We analyze the Hamilton-Jacobi action of gravity and matter in the limit where gravity is treated at the background field approximation. The motivation is to clarify when and how the solutions of the Wheeler-DeWitt equation lead to the Schr\"odinger equation in a given background. To this end, we determine when and how the total action, solution of the constraint equations of General Relativity, leads to the HJ action for matter in a given background. This is achieved by comparing two neighboring solutions differing slightly in their matter energy content. To first order in the change of the 3-geometries, the change of the gravitational action equals the integral of the matter energy evaluated in the background geometry. Higher order terms are governed by the ``susceptibility'' of the geometry. These classical properties also apply to quantum cosmology since the conditions which legitimize the use of WKB gravitational waves are concomitant with those governing the validity of the background field approximation.