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. An operational quantity must instead say how the event is found: by dynamical clocks and rods, by an anchor at a timelike boundary, or by radiative data and detectors in an asymptotic region. These three choices evade coordinate gauge dependence in different ways. They are naturally organized in different algebras or subalgebras unless an additional dictionary identifies them, and the relevant support—reference system or dressing, boundary point or region, or asymptotic wavepacket—has a different notion of locality, causal domain, and precision in each case.

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.

Gauge invariance fixes the reference, not the coordinate label

Work perturbatively about a background $\bar g_{\mu\nu}$,

$$g_{\mu\nu}=\bar g_{\mu\nu}+\kappa h_{\mu\nu},$$

where $\kappa$ counts powers of the gravitational coupling. For a small diffeomorphism generated by $\kappa\xi^\mu$, a scalar at a fixed coordinate label transforms as

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

Thus $\phi(x)$ is a gauge-dependent intermediary, not an observable. A perturbatively dressed scalar instead has the form

$$\begin{aligned} \Phi_V(x) &\equiv \phi\bigl(x+V[h](x)\bigr)\\ &=\phi(x)+V^\mu[h](x)\partial_\mu\phi(x)+O(\kappa^2), \end{aligned}$$

with $V^\mu=O(\kappa)$. If the dressing satisfies

$$\delta_\xi V^\mu(x)=\kappa\xi^\mu(x)+O(\kappa^2),$$

then the two variations cancel:

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

This is a perturbative construction, not a proof of exact locality. The compensating term depends on the metric away from the nominal insertion, and different solutions for $V^\mu$ can differ by physical, gauge-invariant gravitational radiation. Explicit line and Coulomb dressings are constructed in Donnelly and Giddings 2016, §§III.A–III.B.3, printed pp. 9–13, and their nonlocal commutators are derived in §IV.B, printed pp. 18–22.

The transformations above are small gauge transformations: they approach the identity sufficiently rapidly at every relevant boundary. A diffeomorphism with an allowed nonzero boundary value can instead be an asymptotic symmetry generated by a charge. Its action on an observable is physical and need not vanish.

Relational bulk fields require a valid clock or anchor

Classically—and semiclassically in states where the reference fields are sharply peaked—one coordinate-free construction in $D$ bulk dimensions uses $D$ dynamical scalar reference fields $X^A[g,\Psi]$, with $A=0,\ldots,D-1$. In a region where

$$\det\!\left(\partial_\mu X^A\right)\ne0,$$

the equations $X^A(x_X)=X_0^A$ select a local branch $x_X(X_0)$, and

$$\Phi_X(X_0)\equiv\phi\bigl(x_X(X_0)\bigr)$$

is relational: both the scalar and the event selected by the reference fields move under a diffeomorphism. A coordinate gauge such as $X^A=x^A$ is one representation of this classical or semiclassical observable; the gauge-fixed quantity $\phi(x)$ without the defining reference system is not. In the quantum theory, one uses a smeared or integrated composite operator concentrated near $X^A=X_0^A$ rather than assuming that an exact operator-valued inverse $x_X(X_0)$ exists. Its localization is state-dependent, its composite products require renormalization, and the construction fails or becomes multivalued when the Jacobian vanishes, two branches share the same clock readings, or reference-field fluctuations exceed the intended resolution. Integrated clock-and-rod constructions and these localization limits are developed in Giddings, Marolf, and Hartle 2006, §4.1, printed pp. 11–16.

A second construction fires an inward geodesic from a specified boundary point and direction, then selects its endpoint by a fixed Fefferman–Graham or other radial label. Because the AdS conformal boundary lies at infinite physical proper distance, an ordinary finite proper length from that boundary is not valid defining data; one may instead use a cutoff surface together with a regulated or renormalized distance. The construction is well defined only on a chosen branch: before caustics, with a unique geodesic, and while fluctuations of the boundary frame and geometry remain controlled. The geodesic and the gravitational field that reaches the anchor are part of the observable’s support. Boundary-relational examples are reviewed in Marolf 2015, §2, printed pp. 2–4, while explicit leading-order AdS line dressings appear in Giddings and Kinsella 2018, §II.B.1–II.B.4, printed pp. 5–8.

Neither prescription restores an exactly local bulk algebra. Two dressed fields whose coordinate endpoints are spacelike separated can have a nonzero, dressing-dependent commutator because their gravitational fields overlap. Gauge invariance fixes the redundancy; it does not select a unique dressing or erase the gravitational Gauss law.

Boundary limits are boundary observables, not reconstructed bulk points

In asymptotically $AdS_{d+1}$, let $z$ be a Fefferman–Graham defining coordinate and consider a scalar in standard quantization with

$$m^2L^2=\Delta(\Delta-d).$$

Near the conformal boundary, the generic source/response structure is

$$\varphi(z,x) =z^{d-\Delta}\bigl[\varphi_{(0)}(x)+\cdots\bigr] +z^\Delta\bigl[A(x)+\cdots\bigr].$$

Additional logarithmic terms occur at resonant dimensions. The leading coefficient $\varphi_{(0)}$ is the source, whereas $A$ contains state-dependent response data. Indeed,

$$z^{-\Delta}\varphi(z,x) =z^{d-2\Delta}\varphi_{(0)}(x)+A(x)+\cdots.$$

For $\Delta>d/2$, the naive rescaled limit therefore diverges when the source is nonzero. It selects the normalizable coefficient only in the source-free case or after the source terms have been subtracted. If the bulk action and source normalization are collected in $\mathcal N_\varphi$, the simple standard-quantization relation is

$$\langle\mathcal O(x)\rangle_{\mathrm{ren}} =\mathcal N_\varphi(2\Delta-d)A(x) +\text{local or contact terms},$$

where $\mathcal N_\varphi$ includes the sign convention and overall normalization of the bulk action and source coupling. Finite local counterterms can change contact terms, and alternate quantization or mixed boundary conditions change which coefficient is held fixed. The source/response split, counterterms, and one-point function are derived in de Haro, Skenderis, and Solodukhin 2001, §5.1, eqs. (5.2), (5.5), and (5.8)–(5.11), printed pp. 14–16.

Once the boundary theory, state, normalization, and renormalization scheme are specified, $\mathcal O(x)$ belongs to the boundary observable algebra and obeys its boundary causal structure. It is not, by itself, an independent local operator at an interior point. Constructing an interior field from boundary operators requires a smearing domain, a state or code subspace, and an approximation; those questions belong to Extrapolate Dictionaries versus Interior Reconstruction.

Asymptotic data require an infrared and symmetry prescription

For a massless scalar in an asymptotically flat spacetime, an incoming wavepacket may be formed by smearing radiative modes,

$$a_{\mathrm{in}}^\dagger[f] =\int d\mu(p)\,f(p)\,a_{\mathrm{in}}^\dagger(p), \qquad \int d\mu(p)\,\lvert f(p)\rvert^2=1.$$

Here $d\mu(p)=d^{D-1}\mathbf p/[(2\pi)^{D-1}2\lvert\mathbf p\rvert]$ is the positive-energy invariant one-particle measure in the linked normalization convention. The incoming packet is defined at $\mathscr I^-$; an outgoing packet is defined analogously at $\mathscr I^+$. The packet profile, asymptotic frame, detector resolution, and chosen component of null infinity are part of the definition. In gravity one must also specify the soft and memory sector and the infrared prescription. Ordinary Fock-space matrix elements are not a complete answer in four-dimensional gravity because generic scattering changes the long-range field.

Ware, Saotome, and Akhoury construct Faddeev–Kulish-type asymptotic dynamics and coherent states in perturbative quantum gravity in Ware, Saotome, and Akhoury 2013, §§4–5, printed pp. 10–18, and demonstrate all-orders cancellation of infrared divergences for gravitational potential scattering in §6.2, printed pp. 23–26. That result does not establish a universal nonlinear Hilbert-space completion. Prabhu, Satishchandran, and Wald show that the analogous charge-eigenstate construction is obstructed in nonlinear gravity and propose an algebraic scattering framework instead Prabhu, Satishchandran, and Wald 2022, §6.3 and Theorem 1, printed pp. 72–74, and §8, printed pp. 78–80, PDF.

One concrete development of that algebraic direction defines a superscattering map between asymptotic algebraic states and, after assuming generalized asymptotic completeness and using improper BMS-particle states, obtains infrared-finite amplitudes and a soft theorem Prabhu and Satishchandran 2024, §§4.1–4.2 and Theorem 3, printed pp. 30–41, PDF. These assumptions are part of the result; it is not a conventional unitary operator on one preferred Fock space. A complementary 2026 preprint keeps the finite-time dependence of Faddeev–Kulish dressings and memory detectors in perturbative gravity, reproducing linear memory and higher-order Christodoulou contributions through inclusive in-in calculations Oertel 2026, §§3–4, arXiv preprint. This is a perturbative finite-time construction, not a nonperturbative nonlinear Hilbert-space completion. The durable conclusion is therefore narrower: specified asymptotic algebras, inclusive observables, and perturbative dressed amplitudes can be meaningful in their declared domains, but “the gravitational S-matrix” is incomplete unless its infrared and memory data are stated.

Asymptotic symmetries are not extra gauge redundancies to divide out. In the active-transformation convention $\delta_\eta\mathcal A=i[Q_\eta,\mathcal A]$, an allowed transformation with Bondi–Metzner–Sachs (BMS) or boundary charge $Q_\eta$ acts as

$$\delta_\eta\mathcal A=i[Q_\eta,\mathcal A]$$

on an asymptotic observable $\mathcal A$. This physical action must be distinguished from invariance under a small diffeomorphism that is the identity in a neighborhood of the asymptotic boundary.

One scalar, three noninterchangeable observables

The phrase “the same scalar excitation” can refer to related states in different descriptions, but it does not make the following quantities the same operator. A duality dictionary, reconstruction map, or flat-space limit is an additional claim that must be checked separately.

Definitions, support, and causal domains for three scalar observables

Candidate quantity	Anchor and defining data	Algebra and support	Causal or operational domain
Geodesically dressed bulk scalar	Boundary point, inward direction, radial or Fefferman–Graham label, branch, and dressing prescription	Support includes the geodesic and gravitational field to the anchor; commutators depend on the dressing and need not be local	A caustic-free normal neighborhood within a perturbative gravitational-EFT regime
Renormalized AdS boundary response	Conformal boundary, boundary conditions, source, state, operator normalization, and counterterm scheme	A boundary operator or response coefficient in the boundary algebra; no interior support is implied by the limit alone	The boundary spacetime and its causal structure; interior reconstruction requires further data
Asymptotic scattering wavepacket	In/out frame, packet profile, radiative component, soft-memory sector, and detector or dressing prescription	An asymptotic radiative algebra or a specified dressed/inclusive scattering observable	Spacetimes with suitable asymptotic regions and the stated falloff, matching, and completeness assumptions

Control, evidence, and downgrade conditions for the three constructions

Candidate quantity	Control and retained precision	Logical and evidence status	Leading uncertainty	Failure or downgrade
Geodesically dressed bulk scalar	Dimensionless components of κh small in a declared gauge and frame, energy below the EFT cutoff, unique geodesic, and a stated perturbative order	Definition plus explicit leading-order constructions and commutator calculations	Reference fluctuations, composite renormalization, backreaction, and dressing choice	At a caustic the single-branch definition fails; extra branch or routing data define a different observable, while loss of perturbative control leaves only a gauge-fixed proxy
Renormalized AdS boundary response	Controlled near-boundary expansion, fixed quantization and sources, and complete counterterms	Direct boundary observable in a specified boundary theory; its bulk identification is dictionary-dependent	Finite counterterms, contact terms, logarithms, and source-response mixing	The naive rescaled limit fails with an unsubtracted source and never establishes interior locality
Asymptotic scattering wavepacket	Asymptotic falloffs, packet resolution, soft sector, IR prescription, and stated perturbative order	Direct asymptotic-algebra observable or conditional perturbative S-matrix construction	Memory sectors, asymptotic completeness, collinear or soft limits, and nonlinear gravitational flux	Without a valid nonlinear state-space prescription, retain only the specified inclusive, perturbative, or algebraic claim

A diffeomorphism that exposes the anchor

Run the test separately in the two asymptotic settings.

AdS test. Choose $\xi_{\mathrm{AdS}}^\mu$ with compact support in the bulk and separated from the timelike conformal boundary. Choose a point and field configuration in that support for which $\xi_{\mathrm{AdS}}^\mu\partial_\mu\phi\ne0$. The bare insertion then changes by $-\kappa\xi_{\mathrm{AdS}}^\mu\partial_\mu\phi$ and is not gauge-invariant. The relational or boundary-dressed scalar is unchanged through the retained order because its endpoint or dressing shifts by the compensating amount. The renormalized boundary datum is also unchanged because the diffeomorphism is the identity in an entire boundary collar, so its source, response, and boundary frame are fixed.

Asymptotically flat test. Choose a separate $\xi_{\mathrm{flat}}^\mu$ with compact support away from past and future null infinity, and a point and field configuration for which $\xi_{\mathrm{flat}}^\mu\partial_\mu\phi\ne0$. The bare insertion then changes, whereas a correctly dressed scalar remains invariant through the retained order. The asymptotic wavepacket is unchanged because the radiative data, smearing function, and frame at $\mathscr I^\pm$ are fixed.

Now replace $\xi_{\mathrm{AdS}}$ by an allowed nonzero boundary transformation, or $\xi_{\mathrm{flat}}$ by a BMS generator. The corresponding dressed, boundary, and asymptotic quantities can transform under the relevant charge. That is a physical symmetry action, not a failure of gauge invariance.

The strongest claim surviving this test is precise: under the declared small diffeomorphisms, the correctly defined relational scalar through its retained perturbative order, the renormalized boundary datum, and the specified asymptotic observable are invariant; the bare coordinate insertion is not. The test does not prove exact bulk locality, uniqueness of a dressing, invariance under charged asymptotic symmetries, equivalence of the three algebras, or nonperturbative existence. A vanishing reference-field Jacobian or a geodesic caustic removes the relational branch; an unsubtracted source invalidates the naive boundary limit; and an unresolved nonlinear infrared prescription downgrades a universal S-matrix claim to the particular inclusive, perturbative, or algebraic observable actually defined.

Common pitfalls

Gauge invariant does not mean local. The dressing or reference system can extend far outside the coordinate neighborhood, so spacelike-separated coordinate endpoints do not guarantee commuting observables.

A boundary limit is not an interior reconstruction. The renormalized response belongs to the boundary algebra. An interior representation needs a separate reconstruction domain and approximation.

A large diffeomorphism is not automatically redundant. Transformations with nonzero asymptotic charges act on physical states and observables; only the declared small transformations are quotiented as gauge.

An infrared-finite perturbative example is not a universal completion. Its process, perturbative order, asymptotic data, and soft-memory prescription remain part of the claim.

Exercises

Leading-order dressing cancellation

Using the transformations in the first section, verify the gauge variation of $\Phi_V$ through first order. If $V_1^\mu$ and $V_2^\mu$ obey the same transformation law, what is the status of $V_1^\mu-V_2^\mu$?

Solution

Expanding the dressed field and retaining first order gives

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

Because both dressings shift by $\kappa\xi^\mu$, their difference has zero gauge variation at this order. It is gauge-invariant dressing data, not another redundancy; physically it can describe a different radiative or Coulombic gravitational field.

When does the rescaled boundary limit exist?

Insert the source/response expansion into $z^{-\Delta}\varphi$. For $\Delta>d/2$, determine what happens when $\varphi_{(0)}\ne0$ and state the correct observable.

Solution

The rescaled field is

$$z^{-\Delta}\varphi =z^{d-2\Delta}\varphi_{(0)}+A+\cdots.$$

Since $d-2\Delta<0$, the source term diverges as $z\to0$. The simple extrapolate limit selects $A$ only when the source vanishes or after all source-determined pieces, including logarithms when present, are subtracted. The observable is the renormalized response, obtained from the renormalized on-shell action; with the notation above it is $\mathcal N_\varphi(2\Delta-d)A$ plus local or contact terms.

Small gauge transformations versus asymptotic symmetries

In asymptotically AdS spacetime, let $\xi_{\mathrm{AdS}}$ be the identity in a boundary collar and let $\eta_{\mathrm{AdS}}$ approach an allowed nonzero boundary symmetry. In a separate asymptotically flat spacetime, let $\xi_{\mathrm{flat}}$ be the identity near $\mathscr I^\pm$ and let $\eta_{\mathrm{BMS}}$ approach a nonzero BMS symmetry. Compare each pair’s action on the corresponding boundary or asymptotic observable $\mathcal A$.

Solution

Each compactly supported vector field changes only its bulk representative. The AdS boundary observable and the asymptotically flat radiative observable are invariant under their respective small transformations because their defining data are untouched. Each nonzero asymptotic transformation instead has a surface charge $Q_\eta$ and, in the convention used above, generally acts by $\delta_\eta\mathcal A=i[Q_\eta,\mathcal A]$. A nonzero result is the representation of a physical symmetry, not gauge dependence.

For detailed low-energy clock-and-rod constructions, continue to Relational and Gauge-Invariant Gravitational Observables. For explicit AdS dressings and their interactions, use Relational Bulk Observables and Dressing Choices and Interactions, Dressing, and Microcausality. Infrared Dressing and Gravitational Scattering States develops the asymptotic problem, while Haag–Kastler Nets and Locality supplies the rigorous local-net comparison. Dated claims and competing infrared proposals belong in Research.

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

• de Haro, Sebastian, Kostas Skenderis, and Sergey N. Solodukhin. 2001. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217: 595–622. DOI. Open PDF.
• Donnelly, William, and Steven B. Giddings. 2016. “Diffeomorphism-Invariant Observables and Their Nonlocal Algebra.” Physical Review D 93, 024030. DOI. Open PDF.
• Giddings, Steven B., and Alex Kinsella. 2018. “Gauge-Invariant Observables, Gravitational Dressings, and Holography in AdS.” Journal of High Energy Physics 2018 (11): 074. DOI. Open PDF.
• Giddings, Steven B., Donald Marolf, and James B. Hartle. 2006. “Observables in Effective Gravity.” Physical Review D 74, 064018. DOI. Open PDF.
• Marolf, Donald. 2015. “Comments on Microcausality, Chaos, and Gravitational Observables.” Classical and Quantum Gravity 32, 245003. DOI. Open PDF.
• Oertel, Brett. 2026. “Finite-Time Memory Detectors and Fully Constraining Faddeev–Kulish Dressings in QED and Gravity.” arXiv:2605.06774 [hep-th]. arXiv. Open PDF.
• Prabhu, Kartik, and Gautam Satishchandran. 2024. “Infrared Finite Scattering Theory: Amplitudes and Soft Theorems.” Physical Review D 110, 085022. DOI. Open PDF.
• Prabhu, Kartik, Gautam Satishchandran, and Robert M. Wald. 2022. “Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity.” Physical Review D 106, 066005. DOI. Open PDF.
• Ware, John, Ryo Saotome, and Ratindranath Akhoury. 2013. “Construction of an Asymptotic S Matrix for Perturbative Quantum Gravity.” Journal of High Energy Physics 2013 (10): 159. DOI. Open PDF.