Gravitationally Dressed Observables

A scalar field evaluated at a coordinate point is local but not invariant under a diffeomorphism. Perturbative gravity repairs it by attaching the excitation to an asymptotic or relational reference through the metric. The resulting operator is gauge invariant to the calculated order, yet its gravitational field extends beyond the nominal region and generically gives a nonlocal commutator. Dressing solves the gauge problem; it does not restore exact local factorization.

Required background. Quantum Information for Black-Hole Physics supplies operator algebras and subsystem language; What the Equivalence Principles Do and Do Not Imply supplies local frames, invariant measurement, and tidal limitations.

Helpful background. Semiclassical Gravity supplies the weak-backreaction regime; Quantum Fields in Curved Spacetime supplies field observables on a prescribed background; and Black-Hole Information supplies the radiation-algebra problem.

A coordinate-local field is not gauge invariant

Work in four-dimensional perturbation theory about Minkowski spacetime,

$$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}, \qquad \kappa=\sqrt{32\pi G},$$

and retain terms through first order in $\kappa$. Under an infinitesimal diffeomorphism generated by a displacement $\xi^\mu=O(\kappa)$ that vanishes at the asymptotic anchor, choose signs so that

$$\delta_\xi\phi(x)=-\xi^\mu(x)\partial_\mu\phi(x).$$

In this convention, $\delta_\xi h_{\mu\nu}=-\kappa^{-1}(\partial_\mu\xi_\nu+ \partial_\nu\xi_\mu)+O(\kappa)$. Keeping the displacement at the same perturbative order as $V$ makes the cancellation below consistent; some sources instead write the displacement as $\kappa\xi$ and rescale both transformation laws together.

The coordinate label $x$ has shifted while the argument was held fixed, so $\phi(x)$ is not a physical observable. Introduce a metric-dependent vector $V^\mu(x)=O(\kappa)$ whose transformation is $\delta_\xi V^\mu=\xi^\mu+O(\kappa^2)$. Then define

$$\Phi_V(x)=\phi(x)+V^\mu(x)\partial_\mu\phi(x)+O(\kappa^2).$$

Its leading variation cancels:

$$\delta_\xi\Phi_V =-\xi^\mu\partial_\mu\phi +\xi^\mu\partial_\mu\phi+O(\kappa^2) =O(\kappa^2).$$

This is the central construction in Donnelly and Giddings 2016, PRD 94, § II, Eqs. (2)–(3). The cancellation is an operator statement at the retained order; it does not say that all higher orders or nonperturbative diffeomorphisms have been controlled.

A line dressing supplies an asymptotic anchor

Choose a spatial direction $z$. One component of a gravitational Wilson-line dressing is

$$V_W^z(x)=\frac{\kappa}{2} \int_0^\infty ds\,h_{zz}(x+s\widehat z),$$

with accompanying transverse components required for the complete transformation law. Indeed, its longitudinal variation is $\delta_\xi V_W^z=-\int_0^\infty ds\,\partial_z\xi_z(x+s\widehat z) =\xi^z(x)$ when $\xi$ vanishes at infinity. The line reaches an asymptotic anchor and creates the scalar excitation together with a thin gravitational field. Averaging over directions gives a Coulomb-like dressing with the long-range field spread over angles. Both are diffeomorphism-invariant dressings to leading order, but they prepare different gravitational configurations Donnelly and Giddings 2016, PRD 93, § III, Eqs. (39)–(53).

If $V$ and $V'$ obey the same inhomogeneous transformation law, their difference obeys

$$\delta_\xi(V^\mu-V'^\mu)=O(\kappa^2).$$

Thus changing the dressing is not merely changing gauge. The difference is a gauge-invariant radiative or Coulombic gravitational field. An operational protocol must say which boundary anchor or relational reference it prepares; the label “the particle at $x$” is incomplete.

Why spacelike dressed operators need not commute

For two spacelike-separated bare scalar fields, microcausality gives $[\phi(x),\phi(y)]=0$. Expanding two dressed operators instead gives terms of the form

$$[\Phi_V(x),\Phi_W(y)] =[V^\mu(x),W^\nu(y)] \partial_\mu\phi(x)\partial_\nu\phi(y)+\cdots.$$

Each dressing is $O(\kappa)$, so the first generic correction is $O(\kappa^2)=O(G)$. The metric integrals can overlap or share asymptotic support even when $x$ and $y$ are spacelike separated. Explicit line and Coulomb examples have nonlocal commutators at this order Donnelly and Giddings 2016, PRD 93, § IV.2, Eqs. (116)–(132). A particular commutator can vanish because of geometry, state, or dressing choice; there is no universal net of exactly local commuting algebras for charged excitations.

The constraint argument is broader. Energy, momentum, and angular momentum are measurable through asymptotic gravitational fields. A compactly supported operator that changes a Poincaré charge must therefore act on those fields at arbitrarily large distance. The dressing theorem makes this precise at leading order Donnelly and Giddings 2016, PRD 94, § III.1, theorem and Eqs. (15)–(16). Gravity cannot screen positive energy in the same way that opposite electric charges can screen net charge.

Checks and discriminating limits

Weak-gravity limit. As $G\to0$ with the matter configuration fixed, $\kappa\to0$, $V^\mu\to0$, and $\Phi_V\to\phi$. The dressing commutator is suppressed by $G$, recovering local field theory within its regime.

Dressing difference. Apply the same diffeomorphism to a line and a Coulomb dressing. Their inhomogeneous variations cancel in the difference, confirming that the changed gravitational field is physical rather than a failed gauge cancellation.

Charge test. If an operator changes asymptotic energy or momentum, a strictly compact dressing would make every asymptotic surface charge commute with it, contradicting the known transformation. A Poincaré singlet can evade that particular obstruction, but constructing it commonly requires integration over spacetime and is already nonlocal.

Dimensionally, $\kappa h$ is dimensionless in the metric expansion and $V^\mu$ has length dimension, so $V^\mu\partial_\mu\phi$ has the same dimension as $\phi$. This check catches a frequent error in which the factor of $\kappa$ is counted twice.

Consequences for black-hole subsystems

An exterior radiation algebra cannot be defined solely as “fields with $r\gt r_h$.” It needs a dressing prescription, boundary conditions, perturbative order, and treatment of the gravitational constraints. The resulting algebra may share centers or asymptotic data with an interior algebra. Entropy and recovery claims must therefore name the algebra or regulated factor being used rather than silently assuming exact tensor factorization.

This result does not show that arbitrary information travels acausally, nor does it derive islands, holography, or an interior decoder. The nonlocal commutator is the constrained gravitational field of a physical excitation. Its magnitude, measurability, and causal use require a state and protocol. The derivation is limited to weak perturbations about a fixed asymptotic background; strong curvature, finite boundaries, and nonperturbative quantum gravity require a new construction.

Exercises

Suppose two candidate dressings satisfy $\delta_\xi V^\mu=\delta_\xi V'^\mu=\xi^\mu$ at leading order. Show that both dressed scalar operators are gauge invariant and explain why their difference can nevertheless be experimentally meaningful.

Solution

For either dressing, the variation of the bare scalar is $-\xi^\mu\partial_\mu\phi$, while the variation of the dressing term contributes $+\xi^\mu\partial_\mu\phi$; hence each sum is invariant through the retained order. The vector $V^\mu-V'^\mu$ has no inhomogeneous variation, so the operator difference is itself gauge invariant at that order. It represents a different gravitational field—such as line-like versus Coulombic flux—and can change commutators or detector response even though neither construction is a gauge error.

References

• Donnelly, William, and Steven B. Giddings. 2016a. “Diffeomorphism-Invariant Observables and Their Nonlocal Algebra.” Physical Review D 93, 024030. DOI. arXiv:1507.07921.
• Donnelly, William, and Steven B. Giddings. 2016b. “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems.” Physical Review D 94, 104038. DOI. arXiv:1607.01025.