Relational, Boundary, and Asymptotic Observables

Gravity admits gauge-invariant observables, but a field evaluated at a bare coordinate point is generally not one of them. Operational quantities instead refer the field to dynamical landmarks, anchor it to a boundary, or extract asymptotic scattering data. Each construction has its own algebra, causal domain, and precision.

Required background. Observable and Regime Matrix for Quantum Gravity distinguishes the regimes in which these objects are defined.

Helpful background. Relational and Gauge-Invariant Gravitational Observables develops their low-energy construction. S-Matrix and T-Matrix Normalization fixes asymptotic-state conventions.

Why a coordinate insertion is not enough

Under an infinitesimal diffeomorphism generated by $\xi^\mu$,

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

Holding the coordinate label $x$ fixed therefore does not produce an invariant. One remedy is to define coordinates $X^A[g,\Psi]$ from dynamical fields and evaluate

$$\Phi(X_0)=\phi(x)\big|_{X^A[g,\Psi](x)=X_0^A}.$$

If both the field and the relational coordinates transform, the coordinate shift cancels order by order. The price is nonlocality and dependence on the reference system.

Three constructions

Relational bulk observables

A worldline, geodesic, clock field, or curvature feature can locate an insertion. For example, one may shoot a geodesic a fixed proper distance from a specified boundary point and evaluate a scalar at its endpoint. Caustics, multiple geodesics, and fluctuations of the reference system bound the domain of this definition; see Marolf 2015 for a review of perturbative constructions.

Timelike-boundary observables

In asymptotically AdS settings, boundary conditions and boundary limits supply a fixed anchor. For a scalar of dimension $\Delta$ in Fefferman–Graham coordinate $z$,

$${\cal O}(x)=\lim_{z\to0}z^{-\Delta}\phi(z,x)$$

schematically defines the extrapolate dictionary after renormalization and normalization are fixed. Its relation to perturbatively reconstructed AdS fields is developed by Hamilton et al. 2006. A boundary-anchored bulk dressing remains nonlocal: its gravitational field reaches the boundary, and different dressings can differ by radiative data.

Asymptotic observables

In a spacetime with suitable in and out regions, wavepackets and infrared-dressed asymptotic states can define scattering amplitudes. Their gauge invariance comes from asymptotic charges and dressing, not from a local bulk insertion. Global AdS boundary correlators and flat-space S-matrix elements are related only through an additional controlled limit.

Comparing one scalar three ways

Consider a scalar excitation.

Quantity	Reference structure	Principal limitation
Geodesically dressed scalar	Boundary point, direction, and proper length	Caustics and perturbative gravitational dressing
Boundary extrapolate	AdS conformal boundary and renormalized falloff	It is boundary data, not an independent local interior operator
Scattering wavepacket	Asymptotic frame, dressed in/out states	Requires an appropriate asymptotic region and infrared prescription

These observables can encode related physics without being interchangeable operators.

Diffeomorphism test

Choose $\xi^\mu$ with compact support in the interior and vanishing boundary value. The undressed $\phi(x)$ changes. A boundary extrapolate remains fixed because the transformation dies at the anchor. A correctly constructed relational observable also remains fixed because its endpoint shifts with the geometry. If the geodesic prescription has multiple endpoints, however, the observable requires an additional branch choice; gauge invariance does not remove that physical ambiguity.

This check establishes perturbative diffeomorphism invariance, not exact bulk locality. Explicit AdS reconstruction and dressing belong to Chapter 12, while rigorous observable algebras belong to Volume XVI.

Evidence cutoff. Interpretive and literature-status statements use a cutoff of 25 July 2026.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Donnelly, William, and Steven B. Giddings. 2016. “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems,” Physical Review D 94, 104038.
• Giddings, Steven B., Donald Marolf, and James B. Hartle. 2006. “Observables in Effective Gravity,” Physical Review D 74, 064018.
• Hamilton, Alex, Daniel Kabat, Gilad Lifschytz, and David A. Lowe. 2006. “Local Bulk Operators in AdS/CFT: A Boundary View of Horizons and Locality,” Physical Review D 73, 086003.
• Marolf, Donald. 2015. “Observables in (Perturbative) Quantum Gravity: A Brief Review,” Classical and Quantum Gravity 32, 245003.