Euclidean Correlation Functions in Quantum Gravity

TL;DR

This paper develops and applies a relational (master-coordinate) formalism to construct diffeomorphism-invariant Euclidean two-point functions in the low-energy effective theory of gravity. By incorporating coordinate corrections and carefully handling the conformal mode, the authors compute curvature- and volume-operator correlators through one-loop order, obtaining gauge-invariant results that depend only on Newton's constant $G$ and the source-sink separation. The curvature correlator yields a $rac{3G^2}{π^2 x_E^8}$-type long-range tail at one-loop, while the volume correlator similarly exhibits universal nonlocal behavior with both $rac{1}{x_E^2}$ and $rac{1}{x_E^4}$ fall-offs, consistent with reflection positivity. These results provide robust predictions for comparison with nonperturbative lattice gravity and demonstrate the essential role of relational observables in extracting gauge-invariant content from quantum gravity in a low-energy EFT setting.

Abstract

We calculate Euclidean correlation functions through next-to-leading order in the low energy effective theory of gravity. We focus on correlation functions of curvature and volume operators, calculating these functions through one-loop order. We show that quantum fluctuations of the background spacetime must be taken into account in order to obtain gauge invariant expressions, and we point out a subtlety associated with the analytic continuation of the conformal mode. Our final expressions for the correlation functions involve only Newton's constant and the source-sink separation, and they are a universal prediction of the low energy effective theory. Thus, they serve as a useful point of comparison for nonperturbative lattice formulations of gravity.