Observables, gravitational dressing, and obstructions to locality and subsystems

Abstract

Quantum field theory - our basic framework for describing all non-gravitational physics - conflicts with general relativity: the latter precludes the standard definition of the former's essential principle of locality, in terms of commuting local observables. We examine this conflict more carefully, by investigating implications of gauge (diffeomorphism) invariance for observables in gravity. We prove a dressing theorem, showing that any operator with nonzero Poincare charges, and in particular any compactly-supported operator, in flat-spacetime quantum field theory must be gravitationally dressed once coupled to gravity, i.e. it must depend on the metric at arbitrarily long distances, and we put lower bounds on this nonlocal dependence. This departure from standard locality occurs in the most severe way possible: in perturbation theory about flat spacetime, at leading order in Newton's constant. The physical observables in a gravitational theory therefore do not organize themselves into local commuting subalgebras: the principle of locality must apparently be reformulated or abandoned, and in fact we lack a clear definition of the coarser and more basic notion of a quantum subsystem of the Universe. We discuss relational approaches to locality based on diffeomorphism-invariant nonlocal operators, and reinforce arguments that any such locality is state-dependent and approximate. We also find limitations to the utility of bilocal diffeomorphism-invariant operators that are considered in cosmological contexts. An appendix provides a concise review of the canonical covariant formalism for gravity, instrumental in the discussion of Poincare charges and their associated long-range fields.