Potential energy in quantum gravity

TL;DR

This work seeks a gauge-invariant definition of gravitational potential energy in quantum gravity by expressing the static two-body potential as a vacuum expectation value of a scalar Wilson-line–like functional. It develops both a flat-background (B-G) formulation, which reproduces the Newton potential $V(r) = - G m_1 m_2 / r$, and a background-independent (T-G) formulation, defined via a dynamical Euclidean path integral and invariant separation, enabling nonperturbative evaluations. The study connects these approaches to ADM energy concepts and reports lattice results showing Yukawa-like screening in strong gravity near a critical point, suggesting nonperturbative modifications to the effective Newton constant. Overall, the paper provides a framework for probing quantum-gravity potentials beyond perturbation theory and clarifies how continuum physics may emerge from lattice formulations through a carefully defined correspondence principle.

Abstract

We give a general expression for the static potential energy of the gravitational interaction of two massive particles, in terms of an invariant vacuum expectation value of the quantized gravitational field. This formula holds for functional integral formulations of euclidean quantum gravity, regularized to avoid conformal instability. It could be regarded as the analogue of the Wilson loop of gauge theories and allows in principle, through numerical lattice simulations or other approximation techniques, non perturbative evaluations of the potential or of the effective coupling constant.