Observable and Regime Matrix for Quantum Gravity

Quantum gravity does not come with one regime-independent list of observables. A boundary correlator, a gravitationally dressed bulk field, a generalized entropy, and an asymptotic scattering amplitude require different structures and remain controlled under different approximations. The right question is therefore not simply whether curvature is small, but whether the chosen observable is defined and its error is bounded in the stated regime.

Helpful background. Fixed-Background, Semiclassical, Gravitational-EFT, and Quantum-Gravity Regimes supplies the low-energy taxonomy, Effective Field Theory as a Controlled Expansion supplies the expansion logic, and Causal, Killing, Trapping, and Apparent Horizons: The QFT Interface distinguishes horizon-dependent observables. For mean backreaction, see The Semiclassical Einstein Equation, Coupled State–Geometry Initial Data, Self-Consistent State–Geometry Solutions, and Quantum-State Evolution on Backreacted Backgrounds. For fluctuation observables, see Renormalizing Stress Fluctuations and Coincident Limits, Intrinsic and Induced Metric Fluctuations, Gauge-Invariant Stochastic Observables, Large-N Quantum–Stochastic Correspondence, Higher Cumulants and Non-Gaussian Noise, and Validity, Decoherence, and What Stochastic Gravity Does Not Capture. For breakdown and handoff, see Observable-Specific Validity and Error Contracts, Semiclassical Breakdown Diagnostics, and Cauchy Horizons, Chronology Horizons, and Loss of Global Hyperbolicity.

The observable contract

For a quantity $X$, record

$$\mathcal C_X= (\text{degrees of freedom},\text{algebra},\text{state},\text{boundary data}, \text{regulator},\text{expansion},\text{error}).$$

Two formulas carrying the same symbol do not define the same observable unless these entries agree or a map between them has been demonstrated. This matters especially in gravity, where a coordinate-labeled field $\phi(x)$ is generally not invariant under diffeomorphisms.

Regime	Controlled objects	Characteristic limitation
QFT on a fixed background	Local or algebraic QFT observables on prescribed geometry	No metric backreaction
Semiclassical gravity	Renormalized matter observables and a mean geometry solving a semiclassical equation	Metric quantum fluctuations are not fully retained
Gravitational EFT	Dressed low-energy observables and scattering amplitudes below a cutoff	Expansion fails near the cutoff or for uncontrolled species and backgrounds
Perturbative string theory	On-shell amplitudes and worldsheet observables about a chosen background	Expansion in $g_s$ and $\alpha'/L^2$ need not define the full theory
Large-$N$ holography	Boundary observables with a conditional bulk interpretation	Bulk locality and precision are code-subspace and order-of-limits dependent
Proposed nonperturbative theory	Objects defined by the proposal’s exact degrees of freedom and dynamics	Completeness and recovery of spacetime must be shown proposal by proposal

Four quantities across the matrix

Boundary correlator. A CFT correlator is a boundary observable once the theory, state, operator normalization, manifold, and ordering are fixed. Its bulk interpretation may be approximate, but the boundary quantity need not be.

Dressed bulk field. A relational or boundary-anchored field can be constructed order by order in the gravitational coupling. Its precision is limited by dressing choice, code subspace, perturbative order, and possible nonperturbative ambiguities; the obstruction to strictly local gauge-invariant gravitational subsystems is analyzed by Donnelly and Giddings 2016.

Horizon entropy. In a fixed classical solution, area is geometric data. In semiclassical gravity, generalized entropy combines renormalized area and matter entropy. In a full quantum theory, the microscopic object whose logarithm is being compared must be specified; a thermodynamic saddle value is not automatically a state count.

Asymptotic S-matrix. When suitable asymptotic states exist, a dressed scattering matrix is an observable. It is not available in a generic cosmology or in global AdS without an additional limiting construction.

Small curvature is not a universal certificate

Gravitational EFT organizes low-energy corrections even though general relativity is perturbatively nonrenormalizable in the ultraviolet Burgess 2004.

The condition

$$\ell_{\mathrm P}^2\lvert R\rvert\ll1$$

controls only one class of local corrections. An observable may still fail because

$$\frac{E}{\Lambda_{\mathrm{EFT}}}\not\ll1, \qquad N_{\mathrm{species}}\frac{E^2}{M_{\mathrm P}^2}\not\ll1, \qquad t\,\Gamma_{\mathrm{small}}\not\ll1,$$

or because topology, entropy, infrared dressing, or state preparation is uncontrolled. Long times can amplify exponentially small spectral effects even when every local curvature invariant remains mild.

The adversarial test is thus to hold curvature fixed while increasing energy, observation time, species number, or topological complexity. A boundary low-point correlator may remain controlled while a late-time bulk reconstruction or topology sum does not. The matrix licenses conclusions cell by cell; it never produces a universal “quantum gravity is negligible” label.

Where the classification stops

Volume XIV owns the fixed-background, semiclassical, stochastic, and gravitational-EFT calculations. Later chapters give concrete holographic realizations. This page supplies only the observable-specific comparison and does not elevate a proposed nonperturbative construction to a completed theory.

Evidence cutoff. The classification and literature examples are fixed to 25 July 2026. Current empirical or program-status conclusions require a dated Research record.

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.
• 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.
• Maldacena, Juan M. 1998. “The Large N Limit of Superconformal Field Theories and Supergravity,” Advances in Theoretical and Mathematical Physics 2, 231–252.