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 Poincaré 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 diffeomorphisminvariant 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 Poincaré charges and their associated long-range fields.

Diffeomorphism invariance and the black hole information paradox

“…We speak of dynamical correlation, and avoid the concept of "entanglement," because the latter refers to a state in a Hilbert space composed of two or more factors. In the setting of quantum gravity, however, a tensor factorization according to spatial localization is not available [22][23][24][25]. Moreover, the particles or field quanta referred to in the previous paragraph are not gauge invariant with respect to diffeomorphisms; and, if they are gravitationally dressed in order to become so, then they are no longer spatially localized.…”

“…Similarly, in the case of the nebulon, we suppose it is because of gravitational dressing required to turn the resonance into a gauge-invariant observable. It has been proved by Donnelly and Giddings [23], using a perturbation expansion in Newton's constant, 9 that for any operator with nonzero Poincaré charge such gravitational dressing always extends to infinity. 10 Put slightly differently, this dressing theorem-which has also been extended to asymptotically AdS gravity [24]-implies that the division of kinematic Hilbert space into a near-boundary tensor factor and bulk tensor factor does not pass to the level of the algebra of gauge invariant observables.…”

The holographic dual of the entanglement wedge symplectic form

“…If one wanted to consider the situation without this gauge-fixing, one would have to introduce degrees of freedom which track the location of Σ. This would lead to a formalism reminiscent of the 'extended phase space' of[33][34][35]. However, an important difference is as follows.…”

IR dynamics from UV divergences: UV/IR mixing, NCFT, and the hierarchy problem

“…(3.8) 5 We note here that the failure of a 'correspondence principle' between commutative and noncommutative theories as θ µν → 0 is clearly intrinsically linked to the appearance of UV/IR mixing. This failure doesn't violate Kontsevich's proof of the existence of deformation quantization for any symplectic manifold [81], as that is confined solely to 'formal' deformation quantization -that is, the production of a formal power series expansion of the algebra of observables in terms of the deformation parameter.…”

“…If we take the UV limit Λ → ∞ first we find an IR divergence 1 p•p , so the noncommutativity has transmogrified the UV divergence into an IR one. 5 Turning to the question of renormalizability, one may naïvely ask if we can absorb all UV divergences into a finite number of counterterms. Under this criterion, it is clear that this procedure works in the noncommutative theory at least when the commutative version is renormalizable.…”