de Sitter Observables, States, and Horizon Patches

de Sitter correlators, relational gravitational quantities, and static-detector responses are distinct observables with different domains. A global or expanding-patch field correlator can be useful asymptotic data without being accessible to one static observer. Gauge choice, gravitational dressing, quantum state, patch, switching time, and infrared prescription must accompany any proposed holographic interpretation.

Required background. de Sitter Infrared Regimes: States, Observables, Gauges, and Limits supplies state and IR control; Relational, Boundary, and Asymptotic Observables supplies the gravitational observable criterion.

Helpful background. Graviton Infrared Claims and Relational Observables treats gauge sensitivity; Bunch–Davies, Euclidean, and Alpha-State Diagnostics fixes state choices; Cosmological and Multiple-Horizon State Obstructions and Euclidean and Bunch–Davies Free Fields in de Sitter supply the patch comparison.

Patches do not define the same access

In the expanding patch,

$$ds^2=\frac{1}{H^2\eta^2}(d\eta^2-d\mathbf x^2), \qquad -\infty<\eta<0,$$

late-time correlators are limits at $\eta\to0^-$. In the static patch,

$$ds^2=(1-H^2r^2)dt^2-\frac{dr^2}{1-H^2r^2}-r^2d\Omega_{d-1}^2,$$

one observer accesses $0\le r<H^{-1}$. Global de Sitter covers both static diamonds and the regions between them. Coordinate overlap does not make the associated operator domains identical.

For a free scalar in the Bunch–Davies state, the Wightman function $G^+(x,x')$ is de Sitter invariant when the mass and zero-mode conditions permit. Its late-time limit yields conformal power laws with weights $\Delta_\pm=d/2\pm\nu$. This is asymptotic field data. In gravity, a coordinate-labeled field insertion is not diffeomorphism invariant; it must be tied to a worldline, boundary condition, geodesic construction, or other relational dressing.

First application: three observables in one state

Late-time scalar data. For fixed comoving points, take $\eta,\eta'\to0^-$ in $G^+(\eta,\mathbf x;\eta',\mathbf x')$. The answer probes the expanding patch and depends on the chosen falloff and state. It need not be measurable by a single finite-lived detector.

Relational quantity. Choose an observer worldline and define the scalar at fixed proper time and geodesic separation from it. Perturbatively, metric and coordinate shifts cancel only after the dressing terms are included. Different dressings can define different physical observables, especially at the horizon.

Static detector. A stationary Unruh–DeWitt detector with energy gap $\omega$ has long-time response

$$\dot{\mathcal F}(\omega)=\int_{-\infty}^{\infty}d\Delta\tau\, e^{-i\omega\Delta\tau}G^+(x(\tau),x(\tau')),$$

and the Bunch–Davies KMS property gives

$$\frac{\dot{\mathcal F}(\omega)}{\dot{\mathcal F}(-\omega)} =e^{-2\pi\omega/H}.$$

This is an operational response in one patch with temperature $T=H/(2\pi)$, the Gibbons–Hawking result Gibbons and Hawking 1977, §§ 2–3. Finite switching replaces the stationary rate by a smeared response and introduces duration-dependent transients.

State, gauge, and infrared qualifications

An alpha-state changes short-distance and antipodal correlations and generally changes detector response; de Sitter symmetry alone does not select Bunch–Davies. Massless minimally coupled scalars have a zero-mode obstruction to the naive invariant Fock state. Graviton propagators can contain gauge-dependent infrared growth, so only relational or asymptotic quantities with a declared dressing can support a physical claim.

These are QFT statements on a background. Gravitational backreaction is controlled only when $G_NH^{d-1}\ll1$ and the measurement energy and duration do not invalidate the patch. A string embedding must separately suppress $\alpha'H^2$, loops, Kaluza–Klein modes, and vacuum decay.

Adversarial control: change one defining datum

Repeat the comparison in an alpha-state, with a nongeodesic observer, and under a gauge transformation that moves the coordinate insertion. The late-time and detector functions change with the state; acceleration changes the detector spectrum; the undressed coordinate field changes under the gauge transformation while the correctly dressed relational quantity does not. Relabel every surviving answer by its actual state, patch, and dressing.

The evidence ceiling is a well-defined set of perturbative QFT and semiclassical relational observables. It does not establish that global correlators are accessible to one observer, that a thermal patch is a finite exact system, or that any boundary proposal reconstructs all three objects. Static algebras and late-time wavefunctions retain separate pages.

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

• Bunch, T. S., and Davies, P. C. W. (1978). “Quantum Field Theory in de Sitter Space: Renormalization by Point-Splitting.” Proceedings of the Royal Society A 360, 117–134. DOI.
• Gibbons, G. W., and Hawking, S. W. (1977). “Cosmological Event Horizons, Thermodynamics, and Particle Creation.” Physical Review D 15, 2738–2751. DOI.