Diffeomorphism-invariant observables and their nonlocal algebra

TL;DR

The paper constructs diffeomorphism-invariant observables for gravity at leading order in the gravitational coupling, mirroring Dirac dressings in QED with gravitational Wilson lines, Coulomb-like dressings, and worldline dressings that bind a particle to its gravitational field. It then analyzes the algebra of these observables, uncovering nonlocal commutators that depart from local quantum field theory and persist even in the weak-field regime, thereby shedding light on the underlying structure of quantum gravity and potential locality bounds. The work connects gravitational dressing to infrared issues in scattering and to the asymptotic, global structure of spacetime, and discusses potential extensions to AdS, dS, and q-observables. Together, these results suggest that gravity inherently modifies locality at leading order and that the diffeomorphism-invariant observable algebra carries essential information about the quantum nature of spacetime.

Abstract

Gauge-invariant observables for quantum gravity are described, with explicit constructions given primarily to leading order in Newton's constant, analogous to and extending constructions first given by Dirac in quantum electrodynamics. These can be thought of as operators that create a particle, together with its inseparable gravitational field, and reduce to usual field operators of quantum field theory in the weak-gravity limit; they include both Wilson-line operators, and those creating a Coulombic field configuration. We also describe operators creating the field of a particle in motion; as in the electromagnetic case, these are expected to help address infrared problems. An important characteristic of the quantum theory of gravity is the algebra of its observables. We show that the commutators of the simple observables of this paper are nonlocal, with nonlocality becoming significant in strong field regions, as predicted previously on general grounds.