QFT in Curved Spacetime from Quantum Gravity: proper WKB decomposition of the gravitational component

TL;DR

This paper derives the low-energy quantum field theory in curved spacetime from a full quantum gravity framework using a Born-Oppenheimer approach, clarifying the role of graviton averaging and Wheeler-DeWitt constraints.

Contribution

It demonstrates how QFT in curved spacetime emerges from quantum gravity through a WKB expansion and averaging over gravitons, providing a rigorous justification for the Wheeler-DeWitt equation's application.

Findings

QFT in curved spacetime arises after averaging over graviton degrees of freedom.

The Wheeler-DeWitt constraint is justified as a condition on the graviton wave functional.

The approach clarifies the low-energy limit of quantum gravity in a cosmological setting.

Abstract

Starting from a re-analysis of previous work, we construct the proper low energy quantum field theory (QFT) limit of a full quantum gravity theory in the Born-Oppenheimer approach. We separate the gravitational sector into a classical background, given by a vacuum diagonal Bianchi I cosmology, and its quantum perturbations represented by the two graviton degrees of freedom; we further include quantum matter in the form of a test scalar field. We then implement a Born-Oppenheimer separation, where the gravitons and matter play the role of "slow" and "fast" quantum components respectively, and perform a Wentzel-Kramers-Brillouin (WKB) expansion in a Planckian parameter. The functional Schr\"odinger evolution for matter is recovered after averaging over quantum gravitational effects, provided that a condition is imposed on the gravitons' wave functional. Such a condition fixes the graviton dynamics and is equivalent to the purely gravitational Wheeler-DeWitt constraint imposed in previous approaches. The main accomplishment of the present work is to clarify that QFT in curved spacetime can be recovered in the low energy limit of quantum gravity only after averaging over the graviton degrees of freedom, in the spirit of effective field theory. Furthermore, it justifies a posteriori the implementation of the gravitational Wheeler-DeWitt equation on the "slow" gravitons' wave functional rather than assuming its validity a priori.