Observable and Regime Matrix for Quantum Gravity

Quantum gravity has no regime-independent list of observables. A boundary correlator, a gravitationally dressed bulk field, a surface or horizon entropy, and an asymptotic scattering amplitude require different structures and lose control for different reasons. Small curvature is therefore neither necessary nor sufficient by itself: the observable, state, dressing, approximation, measurement, and limit order must be fixed before an error estimate has meaning.

Helpful background. For the basic taxonomy and scales, use Fixed-Background, Semiclassical, Gravitational-EFT, and Quantum-Gravity Regimes, Effective Field Theory as a Controlled Expansion, and Causal, Killing, Trapping, and Apparent Horizons. For the mean-field problem, The Semiclassical Einstein Equation, Coupled State–Geometry Initial Data, Self-Consistent State–Geometry Solutions, and Quantum-State Evolution on Backreacted Backgrounds supply the equations, admissible data, and evolution tests.

For fluctuation control, consult Renormalizing Stress Fluctuations, Intrinsic and Induced Metric Fluctuations, Gauge-Invariant Stochastic Observables, Large-N Quantum–Stochastic Correspondence, Higher Cumulants and Non-Gaussian Noise, and Validity, Decoherence, and Information Limits. Observable-Specific Validity and Error Contracts, Semiclassical Breakdown Diagnostics, and Cauchy Horizons, Chronology Horizons, and Loss of Global Hyperbolicity provide the handoff tests used below.

An observable contract fixes what is measured

For a proposed observable $X$, record five linked subcontracts,

$$\mathcal C_X =\bigl(\mathcal D_X,\mathcal L_X,\mathcal M_X, \mathcal K_X,\mathcal E_X\bigr).$$

Here $\mathcal D_X$ fixes the definition: theory and global form, sector, Hilbert space or algebra, state, and normalization. The location contract $\mathcal L_X$ fixes the spacetime, region, asymptotics, dressing, or relational reference system. The measurement contract $\mathcal M_X$ fixes preparation, ordering, detector or protocol, and initial and boundary data. The calculation contract $\mathcal K_X$ fixes the regulator, renormalization prescription, operator basis, and approximation used to compute $X$. Finally, the error contract $\mathcal E_X$ fixes expansion parameters, limit order, target tolerance, leading omitted terms, and a stop test. If one piece changes, one must first check whether the same physical observable is still being discussed.

Two expressions carrying the same symbol need not define the same observable. A Lorentzian time-ordered correlator is not a Wightman correlator; an entropy depends on the algebra or factorization being used; a scattering amplitude needs asymptotic states and an infrared prescription. Different regulators or schemes can nevertheless represent the same physical observable after parameters and operators are matched. Complete matched predictions are scheme independent, while residual scheme dependence at finite truncation can diagnose omitted orders Burgess 2004, §2.6.2. Contact terms and separately divergent geometric or matter entropy pieces are intermediate, prescription-dependent quantities whose dependence must cancel in the renormalized observable or remain explicit in its definition.

In gravity, a coordinate label alone does not make $\phi(x)$ diffeomorphism invariant. At leading perturbative order about asymptotically flat spacetime in $D>3$, a compactly supported operator with nonzero Poincaré charges must acquire asymptotic metric dependence Donnelly and Giddings 2016b, §II and §III.A, pp. 3–6. Distinct physical dressings can then yield different nonlocal commutators Donnelly and Giddings 2016a, §§III–IV. These results are precise perturbative statements in that setting, not by themselves theorems excluding all exact relational constructions in AdS, cosmology, or a nonperturbative theory.

The contract is also operational. “Measured at late time” must specify a clock or boundary time, “inside” must identify a region or algebra, and “the entropy” must identify the state, surface, and renormalization prescription. These are physical inputs, not bookkeeping.

Four observables with different prerequisites

Boundary correlator. A CFT correlator is a boundary observable once the theory, manifold, state, operators, normalization, and ordering are fixed. Its existence on the boundary does not depend on a semiclassical bulk interpretation.

Dressed bulk field. A perturbative bulk field can be tied to an asymptotic boundary, a geodesic construction, or material reference fields. Different dressings can agree at leading order yet differ at the same order at which gravitational nonlocality matters. Relational, Boundary, and Asymptotic Observables develops the constructions.

Surface or horizon entropy. Several related quantities must be kept distinct. The Bekenstein–Hawking area term $A/(4G)$ is the leading two-derivative result. For a diffeomorphism-covariant Lagrangian, the Wald Noether-charge entropy applies to stationary black holes with a bifurcate Killing horizon under the theorem’s stated hypotheses; extending it to a general dynamical slice carries known ambiguities Iyer and Wald 1994, §6, pp. 16–18, and §7. For a specified codimension-two surface $\Sigma$ and side called “out,” the semiclassical generalized entropy has the form

$$S_{\mathrm{gen}}[\Sigma] =S_{\mathrm{grav,ren}}[\Sigma]+S_{\mathrm{out,ren}}[\Sigma], \qquad S_{\mathrm{grav,ren}} =\frac{\operatorname{Area}(\Sigma)}{4G_{\mathrm{ren}}} +\text{higher-curvature terms}.$$

The split into geometric and matter pieces is scheme dependent while the properly renormalized sum is the candidate physical quantity. The order-$G_N^0$ correction to holographic entanglement entropy contains bulk entanglement together with the local one-loop and counterterm contributions needed for a renormalized result Faulkner, Lewkowycz, and Maldacena 2013, §1, pp. 1–3. Extremizing the generalized entropy gives the quantum-extremal-surface (QES) proposal Engelhardt and Wall 2015, §1, pp. 2–3, and §3, p. 11. A boundary fine-grained entropy, its semiclassical geometric representation, a thermodynamic entropy, and a microscopic state count are not interchangeable without a dictionary and a domain argument.

Asymptotic S-matrix. An S-matrix requires a specified asymptotic theory, in and out state spaces or algebras, and an infrared prescription such as inclusive resolution, coherent dressing, or an algebraic formulation. Faddeev–Kulish-type constructions provide important perturbative candidates in asymptotically flat gravity Ware, Saotome, and Akhoury 2013, but their sufficiency is not universal: proposed asymptotic constructions face additional infrared and memory constraints Prabhu, Satishchandran, and Wald 2022. An ordinary S-matrix need not exist in a generic cosmology. Global AdS instead has a timelike reflecting boundary and a discrete spectrum, so a flat-space S-matrix is obtained only through an additional large-radius and wave-packet or Mellin limit; the massless prescription in the original Mellin construction is explicitly conjectural Penedones 2011, §1, p. 3, and §3, pp. 16–20.

Observable × regime matrix

Read the following two tables together as one compact $4\times6$ matrix. The labels mean D, directly defined once the stated data are fixed; P, perturbative or approximate; C, conditional on extra structure, a limit, or a dictionary; and A, an analogue rather than the same observable. Labels qualify the named row observable, not merely a nearby quantity. When a cell also mentions a distinct exact boundary observable and its bulk representation, it labels them separately. The regime columns are overlapping calculational frameworks, not mutually exclusive phases of nature.

Four observables from fixed-background QFT through gravitational EFT

Observable	Fixed-background QFT	Semiclassical gravity	Gravitational EFT
Boundary correlator	C. Defined when the prescribed nongravitational QFT actually has the stated boundary algebra. No bulk interpretation follows without a dictionary.	C/P. A boundary limit on a self-consistent mean geometry. Stop when metric or state–geometry fluctuations are not controlled.	C/P. A low-energy boundary or asymptotic correlator when such an algebra exists. Errors arise from omitted operators, loops, dressing, and infrared treatment.
Dressed bulk field	A. A coordinate-local field exists on the prescribed metric, but it is not the gravitationally dressed observable.	A/P. A field may be located relative to the mean geometry; the construction omits the full quantum spread of that reference geometry.	P. Relational or boundary dressing is constructed order by order. Stop at the cutoff, large backreaction, or loss of perturbative control.
Surface or horizon entropy	A. Matter entropy is defined after choosing a region or algebra and a regulator, but there is no dynamical gravitational area term.	P. A specified stationary-horizon or generalized-entropy construction is meaningful only within its semiclassical hypotheses.	P. Higher-curvature entropy terms and bulk loops enter systematically. The first omitted EFT term sets the precision ceiling.
Asymptotic S-matrix	C. Defined when the prescribed background admits suitable asymptotic states and an infrared prescription.	C/P. Matter scattering on a prescribed or mean geometry can be computed when an asymptotic region exists; quantum geometry is only partly retained.	P/C(IR). When the background admits asymptotic states, dressed graviton–matter amplitudes are controlled below the cutoff only with a declared soft prescription and verified cancellation of collinear singularities for massless legs.

The same observables in perturbative string theory, large-N holography, and proposed exact constructions

Observable	Perturbative string theory	Large-N holography	Proposed nonperturbative theory
Boundary correlator	C/P. Computed in a chosen background as genus and string-scale expansions. An incomplete series is not the full answer.	D (boundary); C/P (bulk representation). The boundary correlator is directly defined; its local-bulk interpretation carries $1/N$, gap, and state restrictions.	C. Defined only if the proposal supplies a boundary algebra and state. A bulk-looking result alone does not do so.
Dressed bulk field	C/P. Requires an explicit relational or asymptotic dressing in the chosen background. Becchi–Rouet–Stora–Tyutin (BRST) cohomology or a gauge-fixed string-field representative alone does not supply a localized gravitational observable.	C/P. Reconstructed only in a declared code subspace and region, with finite-$N$ and high-energy corrections.	C. A relational observable must be constructed from the proposal’s exact degrees of freedom and reference data.
Surface or horizon entropy	C/P. Genus and string-scale corrections are computed about a chosen saddle. A thermodynamic saddle value is not automatically a microstate count.	C/P for the surface observable. Boundary fine-grained entropy is a distinct, directly defined observable; its identification with a Ryu–Takayanagi (RT), Hubeny–Rangamani–Takayanagi (HRT), or QES surface is conditional and approximate.	C. The microscopic entropy and its map to a semiclassical area, Wald, or generalized entropy must be defined and checked.
Asymptotic S-matrix	C/P. In backgrounds admitting suitable asymptotic string states, on-shell amplitudes are computed at a specified genus order and with a declared infrared prescription.	C. Extracted only through a controlled flat-space limit of boundary data; global-AdS correlators are not themselves an S-matrix.	C. Available only if the proposal has the needed asymptotic states, dynamics, and scattering limit; some cosmological constructions may not.

Cell-by-cell evidence and uncertainty

The compact matrix says what kind of claim is available. The following companion records what supports each cell and what limits it. “Evidence basis” names the kind of support, not a verdict that the strongest possible theory exists.

Boundary correlator.

Regime	Status	Domain and control	Evidence basis	Leading uncertainty or stop
Fixed-background QFT	C	Specified theory with a boundary algebra, manifold, state, operators, normalization, and ordering	Direct QFT definition plus analytic, perturbative, lattice, or other theory-specific calculations	QFT truncation, regulator, volume, and continuation errors; no bulk conclusion without a dictionary
Semiclassical gravity	C/P	Boundary conditions and a self-consistent renormalized mean geometry with controlled fluctuations	Renormalized QFT and semiclassical-response calculations	State–geometry and induced-metric fluctuations beyond tolerance
Gravitational EFT	C/P	A low-energy boundary or asymptotic algebra with all physical scales below the cutoff	EFT matching, power counting, and loop calculations	First omitted operator or loop, dressing ambiguity, and infrared error
Perturbative string theory	C/P	A chosen background, boundary condition, state, and controlled genus and string-scale expansions	Worldsheet, string-field, or low-energy matching calculations	Omitted genera or $\alpha'$ terms and background instability
Large-$N$ holography	D for the boundary observable; C/P for its bulk representation	Fully specified boundary CFT; a bulk interpretation additionally needs the dictionary, state or code sector, gap, and limit order	Direct boundary definition plus protected, perturbative, numerical, or saddle-level dictionary tests	Finite-$N$, finite-gap, strong-coupling, high-energy, and state-dependence errors in the bulk interpretation
Proposed nonperturbative theory	C	The proposal must construct the boundary algebra, state space, and dynamics	Proposal-specific constructive, numerical, or consistency evidence	Existence, completeness, continuum recovery, or comparison with the intended boundary theory remains unshown

Dressed bulk field.

Regime	Status	Domain and control	Evidence basis	Leading uncertainty or stop
Fixed-background QFT	A	Prescribed metric and coordinate chart	Standard local-QFT construction	It is an analogue, not a diffeomorphism-invariant observable once gravity fluctuates
Semiclassical gravity	A/P	Relational locator tied to a controlled mean geometry	Mean-field and linear-response construction	Quantum spread of the reference geometry or induced metric fluctuations
Gravitational EFT	P	Explicit dressing or relational reference, operator normalization, and expansion order	Perturbative constraint solving and dressed-operator calculations	Cutoff violation, dressing dependence, large backreaction, or loss of perturbative locality
Perturbative string theory	C/P	Background plus an explicit relational or asymptotic construction, not BRST data alone	String perturbation theory combined with a stated gauge-invariant construction	Genus and string-scale truncation, background dependence, and incomplete localization
Large-$N$ holography	C/P	Declared boundary region, code subspace, reconstruction prescription, and state	Correlator matching, modular or error-correction arguments, and perturbative bulk reconstruction	Finite-$N$, finite-gap, state, region, and high-energy corrections
Proposed nonperturbative theory	C	Exact degrees of freedom and relational reference data supplied by the proposal	Proposal-specific construction and algebraic checks	Existence, uniqueness, completeness, or a semiclassical localization limit remains unshown

Surface or horizon entropy.

Regime	Status	Domain and control	Evidence basis	Leading uncertainty or stop
Fixed-background QFT	A	Specified region or algebra, state, regulator, and subtraction	QFT entanglement or algebraic-entropy calculation	No dynamical gravitational area term; regulator-sensitive pieces are not standalone observables
Semiclassical gravity	P	Specified stationary horizon or surface, state, side, and renormalized generalized entropy	Black-hole thermodynamics, Noether-charge results, and semiclassical entropy calculations	Quantum fluctuations, dynamical-horizon ambiguity, and renormalization or state dependence
Gravitational EFT	P	Declared effective action, surface prescription, loop order, and boundary terms	Wald-type and replica calculations within the EFT	First omitted higher-curvature or loop term; dynamical ambiguities can exceed tolerance
Perturbative string theory	C/P	Chosen background, saddle, ensemble, and controlled $g_s$ and $\alpha'$ expansions	Genus and string-scale corrections, protected counts, and saddle thermodynamics	Saddle competition, truncation, and the distinction between thermodynamic entropy and microstate count
Large-$N$ holography	C/P for the surface observable; D for the distinct boundary entropy	Specified boundary state and region plus the RT, HRT, or QES domain	Direct boundary definition and semiclassical replica or extremization evidence for the map	Finite-$N$, bulk-loop, string-scale, code-subspace, and competing-surface corrections
Proposed nonperturbative theory	C	Exact entropy or state count and its map to semiclassical geometry must be constructed	Proposal-specific state counting, algebraic, or numerical evidence	Ambiguous factorization, coarse graining, ensemble, or semiclassical recovery

Asymptotic S-matrix.

Regime	Status	Domain and control	Evidence basis	Leading uncertainty or stop
Fixed-background QFT	C	Background with asymptotic in and out states plus a declared infrared prescription	Standard scattering theory and theory-specific calculations	Regulator, truncation, finite-volume, continuation, and infrared errors
Semiclassical gravity	C/P	Suitable asymptotic region and a self-consistent mean geometry	Matter scattering and semiclassical-response calculations	Quantum-geometry fluctuations, particle-production ambiguities, or loss of asymptotic states
Gravitational EFT	P/C(IR)	Energies below the cutoff, explicit gravitational dressing, and a soft prescription	EFT amplitude, unitarity, and soft-factor calculations	Omitted operators and loops, unresolved soft radiation, or failed collinear cancellation
Perturbative string theory	C/P	Background admitting asymptotic string states, stated vacuum, genus order, and infrared prescription	Perturbative on-shell string amplitudes	Omitted genera, nonperturbative effects, instability, or absence of the assumed asymptotic states
Large-$N$ holography	C	Controlled flat-space and wave-packet or Mellin limit of boundary data	Boundary correlators plus flat-space-limit derivations and checks	Nonuniform large-$N$, large-radius, energy, gap, and wave-packet limits
Proposed nonperturbative theory	C	Exact asymptotic states, dynamics, and a scattering limit supplied by the proposal	Proposal-specific construction and comparison with controlled limits	Existence, completeness, infrared definition, or cosmological inapplicability

Regime control cards

The matrix identifies the object; the next table identifies the approximation. “Expected error” means the first omitted contribution, not a guarantee that its coefficient is order one.

Regime	Dynamical variables and accessible algebra	Control parameters	Expected error and stop rule
Fixed-background QFT	Quantum matter on prescribed $g_{\mu\nu}$; local or algebraic observables	QFT couplings, regulator or lattice spacing, volume, and measurement scales	QFT truncation and regulator errors; stop when metric backreaction is part of the question
Semiclassical gravity	Classical mean metric coupled to renormalized quantum matter	Curvature, derivatives, and physical energies below the gravitational-EFT cutoff; a self-consistent renormalized solution; controlled connected stress and metric fluctuations	Truncation errors and induced metric fluctuations; stop for response instabilities, runaway solutions, or fluctuations exceeding the observable’s tolerance
Gravitational EFT	Low-energy metric and matter fields; relational and asymptotic algebra	energy, curvature, gradients, loop order, and active species relative to the EFT cutoff	First omitted local operator or loop; stop when no hierarchy suppresses it
Perturbative string theory	String states and worldsheet or string-field observables about a background	$g_s$, $\alpha'$ times curvature and momentum scales, genus, and background stability	Omitted genera and string-scale terms; stop at strong coupling, string-scale curvature, or an uncontrolled background
Large-$N$ holography	Exact boundary algebra with an approximate bulk code-subspace interpretation	$1/N$, spectral gap, coupling, excitation energy, and limit order	Factorization, bulk-loop, and stringy corrections; stop beyond the code subspace or at a nonuniform limit
Proposed nonperturbative theory	Proposal-specific exact degrees of freedom and whatever observable algebra has been constructed	Continuum, large-size, semiclassical, or other proposal-specific controls	Must be supplied by the proposal; stop when existence, completeness, or spacetime recovery has not been shown

For the semiclassical rows, the detailed mean-field and fluctuation tests belong to The Semiclassical Einstein Equation and Validity, Decoherence, and What Stochastic Gravity Does Not Capture. This page uses their outcomes without duplicating the calculations.

Why small scalar curvature is not enough

Gravitational EFT is a derivative and loop expansion, not an expansion in the Ricci scalar alone Donoghue 1994, §III, pp. 5–8 and Burgess 2004, §§3.1–3.3, pp. 23–31. With the site convention $(+---)$, choose a unit timelike detector velocity $u^a$ and form the positive-definite metric

$$q_{ab}=2u_a u_b-g_{ab}.$$

Contracting tensor indices with $q_{ab}$ and its inverse defines detector-frame norms $\lVert\cdot\rVert_u$. Necessary local diagnostics for the curvature and derivative expansion include

$$\epsilon_k(u) =\frac{\lVert\nabla^k R_{abcd}\rVert_u} {\Lambda_{\mathrm{EFT}}^{k+2}}\ll1, \qquad \epsilon_Q=\frac{Q_{\mathrm{phys}}}{\Lambda_{\mathrm{EFT}}}\ll1,$$

for every derivative order and physical momentum or frequency $Q_{\mathrm{phys}}$ relevant to the retained operator basis. The actual error estimate must weight the independent invariants by their Wilson coefficients and by the observable’s response, then compare the leading omitted contribution with a declared tolerance. In Lorentzian geometry, a finite list of scalar invariants need not bound all components measured in a physical frame.

Schwarzschild spacetime gives an elementary counterexample to the criterion $\ell_{\mathrm P}^2\lvert R\rvert\ll1$. In vacuum,

$$R=0, \qquad R_{\mu\nu}R^{\mu\nu}=0, \qquad R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} =\frac{48G_4^2M^2}{r^6}.$$

The Ricci-scalar test passes trivially even where tidal curvature becomes large. This is an invariant check that the old one-number criterion cannot certify a matrix cell.

Many light species give a different failure mode. For $D>2$, choose a convention with $G_D=M_{\mathrm{P},D}^{-(D-2)}$. Dimensional loop power counting then gives the parametric diagnostic

$$\epsilon_{\mathrm{species}}(E) \sim c_D\,N_{\mathrm{eff}}(E)\,G_D E^{D-2} =c_D\,N_{\mathrm{eff}}(E) \left(\frac{E}{M_{\mathrm{P},D}}\right)^{D-2}\ll1,$$

where $c_D$ contains loop, spin, and geometric weights and must be fixed by the calculation. If $N_{\mathrm{eff}}$ is approximately constant, the estimated species scale behaves as

$$\Lambda_{\mathrm{species}} \sim\frac{M_{\mathrm{P},D}} {\bigl(c_DN_{\mathrm{eff}}\bigr)^{1/(D-2)}}.$$

When species have thresholds, $N_{\mathrm{eff}}$ must be evaluated self-consistently near the inferred scale. Large multiplicity can therefore lower the useful loop cutoff even when the background curvature is mild. This scaling is a power-counting estimate, not a universal theorem with a fixed coefficient; black-hole and string arguments provide additional, assumption-dependent evidence Dvali and Gomez 2010, §§1–2.

Long times and topology supply independent stop tests

Suppose a truncated calculation omits an energy shift $\delta E$ and a decay width $\Gamma$. A contribution evolves as

$$\mathcal A(t) =\mathcal A(0) \exp\!\left[-i(E+\delta E)t-\frac{\Gamma t}{2}\right].$$

Relative to the leading oscillation, the correction factor and its short-time expansion are

$$\frac{\mathcal A(t)}{\mathcal A_0(t)} =e^{-i\delta E t-\Gamma t/2} =1-\left(i\delta E+\frac{\Gamma}{2}\right)t+\cdots, \qquad \mathcal A_0(t)=\mathcal A(0)e^{-iEt}.$$

Dropping the corrections requires $t\lvert\delta E\rvert\ll1$ and $t\Gamma\ll1$. A survival probability is first sensitive to $\Gamma t$, whereas interference with a phase reference can be sensitive to $\delta E t$. Reaching $\Gamma t\sim1$ invalidates the approximation that neglects the width, not necessarily a controlled resummation of the exponential. After resummation, uncertainties $\sigma_\Gamma$ and $\sigma_{\delta E}$ instead require $t\sigma_\Gamma$ and $t\sigma_{\delta E}$ to remain below the target tolerance.

More generally, perturbation theory can generate secular terms $g^p t^q$ that require reorganization even when $g\ll1$. An exponentially small splitting can spoil a phase prediction long before decay is appreciable, or the reverse. Calling $1/\lvert\delta E\rvert$ a Heisenberg time is justified only when $\delta E$ is the relevant mean level spacing; otherwise it is simply the phase-resolution time.

Topology and saddle competition are not captured by local curvature either. A low-point correlator about one geometry can remain perturbatively controlled while a sum over topologies, a recurrence-scale observable, or an entropy comparison is not. The correct adversarial test holds the local geometry fixed and varies energy, duration, species multiplicity, state complexity, or the set of allowed saddles. The answer is then downgraded cell by cell rather than declaring the whole regime valid or invalid.

The promised adversarial test gives the following explicit downgrades:

Change while local curvature stays mild	Cells most directly affected	Failed hypothesis or control	Downgrade and strongest surviving claim
Raise the physical energy or resolution	Gravitational-EFT, perturbative-string, and large-$N$ bulk representations of correlators, dressed fields, and scattering	$Q_{\mathrm{phys}}/\Lambda_{\mathrm{EFT}}\ll1$, small $\alpha'Q_{mathrm{phys}}^2$, low backreaction, or code-subspace support	The bulk calculation becomes uncontrolled. A directly defined boundary correlator still exists, but its low-energy local-bulk representation does not survive
Increase the observation time	Perturbative correlator, dressed-field, decay, and scattering predictions in every framework using a truncated frequency or width	$t\lvert\delta E\rvert\ll1$, $t\Gamma\ll1$, or more general secular control	The unresummed approximation is downgraded; after justified resummation, only the result with propagated spectral uncertainties survives
Increase the active species count	Semiclassical and gravitational-EFT cells, especially metric response and entropy loops	$N_{\mathrm{eff}}(E)G_DE^{D-2}\ll1$ and controlled connected stress fluctuations	The gravitational loop or mean-field inference is lost at the lowered cutoff. The corresponding fixed-background matter observable may remain defined, but it is not a controlled quantum-gravity prediction
Enlarge the allowed saddle or topology set	Semiclassical, string-saddle, and holographic bulk representations of correlators, entropies, and state maps	Assumed saddle dominance, contour completeness, or exponential suppression of omitted sectors	A one-saddle answer becomes conditional or incomplete. An exact boundary observable survives when independently defined; its proposed geometric interpretation may not

Common pitfalls

Confusing a related quantity with the same observable. A coordinate-local QFT field on a prescribed metric is not the gravitationally dressed field obtained after the metric becomes dynamical. A Wald entropy, generalized entropy, boundary fine-grained entropy, and microscopic state count require different contracts.

Using one cutoff for every matrix cell. A local derivative expansion, a genus expansion, a large-$N$ expansion, an infrared expansion, and a long-time approximation have different parameters. The smallest applicable ceiling controls the stated result.

Forgetting measurement and limit order. An operator identity does not fix Lorentzian ordering, a detector protocol, or the order of the flat-space, large-$N$, late-time, and continuum limits. Those choices can change both the object and its error.

Exercises

1. Stress-test the Ricci-scalar criterion. For Schwarzschild spacetime, use the curvature invariant above to find the radius at which an EFT length $\ell_*$ ceases to satisfy $\ell_*^4 R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\ll1$. Explain why the test based only on $R$ gives no warning.

Solution

The invariant criterion is

$$\frac{48G_4^2M^2\ell_*^4}{r^6}\ll1.$$

It requires

$$r\gg48^{1/6}(G_4M)^{1/3}\ell_*^{2/3}.$$

By contrast, $R=0$ for the vacuum Schwarzschild solution at every $r$ away from the singularity, so $\ell_*^2\lvert R\rvert\ll1$ is identically satisfied. The latter condition is therefore not a sufficient curvature diagnostic.

2. Separate phase and decay control. Let an omitted correction obey $\delta E=\varepsilon E$ and $\Gamma=\varepsilon^2E$, with $0<\varepsilon\ll1$. At what times does each omission become order one?

Solution

The phase error becomes order one when $t\lvert\delta E\rvert\sim1$, so

$$t_{\mathrm{phase}}\sim\frac{1}{\varepsilon E}.$$

The magnitude error becomes order one when $t\Gamma\sim1$, so

$$t_{\mathrm{decay}}\sim\frac{1}{\varepsilon^2E}.$$

Thus $t_{\mathrm{phase}}/t_{\mathrm{decay}}\sim\varepsilon$: an approximation that drops both corrections loses phase accuracy parametrically earlier. If the exponential has instead been derived and resummed, these times do not alone invalidate it; one must propagate the uncertainties in $\delta E$ and $\Gamma$.

3. Derive the species-scale exponent. In $D>2$, assume $N$ light species with approximately equal loop weights below a scale $\Lambda$ and $G_D=M_{\mathrm{P},D}^{-(D-2)}$. Ignoring the order-one factor $c_D$, solve $N G_D\Lambda^{D-2}\sim1$ and specialize to $D=4$.

Solution

The threshold equation gives

$$N\left(\frac{\Lambda}{M_{\mathrm{P},D}}\right)^{D-2}\sim1, \qquad \Lambda_{\mathrm{species}} \sim M_{\mathrm{P},D}N^{-1/(D-2)}.$$

In four spacetime dimensions, $\Lambda_{\mathrm{species}}\sim M_{\mathrm{P},4}/\sqrt N$. This conclusion is only parametric: unequal spin weights, thresholds, and convention-dependent loop factors modify the coefficient and make $N_{\mathrm{eff}}(\Lambda)$ implicit.

4. Downgrade a matrix cell. A fully specified large-$N$ boundary CFT has a well-defined time-ordered four-point function. A researcher represents it by one low-energy bulk saddle, raises the wave-packet energy to $Q_{\mathrm{phys}}\simeq\Lambda_{\mathrm{EFT}}$, and waits until an omitted shift obeys $t\lvert\delta E\rvert\simeq1$. State the observable contract, initial cell label, failed controls, required downgrade, and strongest surviving claim.

Solution

The definition contract fixes the CFT, global sector, state, four normalized operators, Lorentzian ordering, and boundary time. The location and measurement contracts fix the boundary insertions and wave packets. The calculation and error contracts add the chosen bulk saddle, EFT operator basis, cutoff, large-$N$ and gap assumptions, limit order, and target tolerance.

The initial matrix label is D for the boundary observable; C/P for its bulk representation. The energy change violates $Q_{\mathrm{phys}}/\Lambda_{\mathrm{EFT}}\ll1$, and the late-time change violates the truncation condition $t\lvert\delta E\rvert\ll1$. The one-saddle low-energy prediction is therefore uncontrolled unless a new high-energy description and a justified late-time resummation with propagated uncertainties are supplied. The strongest surviving claim is that the boundary four-point function remains a directly defined CFT observable; this calculation no longer provides its controlled local-bulk representation.

The holographic-claim contract supplies the theory and parameter maps used by the large-$N$ columns. Semiclassical Breakdown Diagnostics provides the calculation-level handoff when mean-field gravity fails.

Status boundary. This page classifies durable observables and approximation tests; it does not supply a current empirical verdict. Dated program assessments, contrary evidence, and specialist review belong in Research.

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

• Burgess, C. P. 2004. “Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory,” Living Reviews in Relativity 7, 5. Open PDF.
• Donnelly, William, and Steven B. Giddings. 2016a. “Diffeomorphism-Invariant Observables and Their Nonlocal Algebra,” Physical Review D 93, 024030. Open PDF.
• Donnelly, William, and Steven B. Giddings. 2016b. “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems,” Physical Review D 94, 104038. Open PDF.
• Donoghue, John F. 1994. “General Relativity as an Effective Field Theory: The Leading Quantum Corrections,” Physical Review D 50, 3874–3888. Open PDF.
• Dvali, Gia, and Cesar Gomez. 2010. “Species and Strings,” arXiv:1004.3744 [hep-th]. Open PDF.
• Engelhardt, Netta, and Aron C. Wall. 2015. “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime,” Journal of High Energy Physics 2015, 73. Open PDF.
• Faulkner, Thomas, Aitor Lewkowycz, and Juan Maldacena. 2013. “Quantum Corrections to Holographic Entanglement Entropy,” Journal of High Energy Physics 2013, 74. Open PDF.
• Iyer, Vivek, and Robert M. Wald. 1994. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy,” Physical Review D 50, 846–864. Open PDF.
• Penedones, João. 2011. “Writing CFT Correlation Functions as AdS Scattering Amplitudes,” Journal of High Energy Physics 2011, 25. 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. Open PDF.
• Ware, Jonathan, Ryo Saotome, and Ratindranath Akhoury. 2013. “Construction of an Asymptotic S Matrix for Perturbative Quantum Gravity,” Journal of High Energy Physics 2013, 159. Open PDF.