On the notion of potential in quantum gravity

TL;DR

This paper investigates the gauge dependence of quantum corrections to the gravitational potential in quantum gravity, showing that classical terms are gauge-independent while quantum corrections depend on gauge choices.

Contribution

It provides a detailed analysis of gauge dependence in quantum gravitational potentials using the $S$-matrix approach and Slavnov identities at one-loop level.

Findings

Classical $1/r^2$ terms are gauge-independent.

Quantum $1/r^3$ corrections are gauge-dependent.

Explicit calculation of gauge dependence using Slavnov identities.

Abstract

The problem of consistent definition of the quantum corrected gravitational field is considered in the framework of the $S$-matrix method. Gauge dependence of the one-particle-reducible part of the two-scalar-particle scattering amplitude, with the help of which the potential is usually defined, is investigated at the one-loop approximation. The $1/r^2$-terms in the potential, which are of zero order in the Planck constant $\hbar,$ are shown to be independent of the gauge parameter weighting the gauge condition in the action. However, the $1/r^3$-terms, proportional to $\hbar,$ describing the first proper quantum correction, are proved to be gauge-dependent. With the help of the Slavnov identities, their dependence on the weighting parameter is calculated explicitly. The reason the gauge dependence originates from is briefly discussed.