Static-Patch Algebras and Observer Dependence

A static-patch observer has access to an algebra of operations in one causal diamond, not to a tensor factor containing every global de Sitter mode. In background QFT the local algebra is type III and the Bunch–Davies restriction is thermal. Including gravitational constraints and a physical observer clock can instead produce a crossed-product type-II$_1$ algebra in semiclassical perturbation theory. This is an observer-relative algebraic construction, not proof of a finite exact Hilbert space.

Required background. de Sitter Observables, States, and Horizon Patches fixes the patch; Cosmological and Multiple-Horizon State Obstructions fixes the state restrictions.

Helpful background. Redshift, Restricted Access, and Effective Channel Noise supplies the operational channel; Operator Algebras and Positive Functionals: a Bridge supplies the algebraic language.

Accessible algebra and KMS state

The static Killing field $\xi=\partial_t$ is timelike only for $r<H^{-1}$. Let $\mathcal A_{\mathrm{QFT}}(\mathcal D)$ be the algebra generated by bounded functions of fields smeared inside the diamond $\mathcal D$. The Bunch–Davies state restricted to this algebra satisfies the KMS relation at

$$\beta_{\mathrm{dS}}=\frac{2\pi}{H}$$

with respect to static time. KMS is an analytic property of correlation functions; it does not require writing a trace-class density matrix on a type-III factor.

Gravity changes the question. A local field must be dressed to an observer worldline or another relational anchor, and the Hamiltonian constraint ties matter, horizon area, and observer energy. Introducing an observer with clock variable and imposing the constraint leads to a crossed-product algebra. In the semiclassical construction of Chandrasekaran, Longo, Penington, and Witten, the resulting static-patch algebra is type II$_1$, possesses a normalized trace, and assigns empty de Sitter the maximum entropy Chandrasekaran et al. 2023, §§ 2–5.

First application: field algebra and detector response

For a free scalar, smear $\phi$ with test functions supported away from the horizon and form the Weyl operators

$$W(f)=e^{i\phi(f)}, \qquad W(f)W(g)=e^{-\frac i2\sigma(f,g)}W(f+g).$$

The causal propagator determines $\sigma(f,g)$. Restrict the Bunch–Davies two-point function to the static patch. Along the central geodesic, its imaginary-time periodicity yields the KMS detailed-balance condition for an Unruh–DeWitt detector,

$$\dot{\mathcal F}(\omega)=e^{-\beta_{\mathrm{dS}}\omega} \dot{\mathcal F}(-\omega).$$

Global modes whose Cauchy data lie beyond the observer horizon are not generated by smearings in $\mathcal D$. Correlations with the complementary patch exist in the global state but are not independent accessible operators for the observer.

Now dress $W(f)$ to the observer’s proper clock and impose the gravitational constraint. The algebra includes relational time translations and area/energy information, while entropy differences agree semiclassically with generalized-entropy differences up to a state-independent additive convention. This result is stronger than background QFT thermality and weaker than a microscopic finite-state model.

Observer and duration dependence

An accelerated worldline has a different proper-time flow and detector response. A finite-time observer uses switched smearings and does not measure an exact KMS rate. Different clock dressings can differ by gravitational soft or horizon data. The type-II$_1$ construction assumes a semiclassical code of excitations and controlled backreaction; it is not an all-orders algebra for arbitrary energies or durations.

The regime requires $G_NH^{d-1}\ll1$, excitation energy below the scale that significantly shifts the horizon, and a relational clock accurate within the perturbative expansion. It contains no automatic string, $\alpha'$, Kaluza–Klein, or metastability control.

Adversarial control: change observer and clock

Repeat the detector calculation on an accelerated trajectory and with a compact switching function. Then change the relational dressing while keeping the undressed coordinate point fixed. If the response or algebra changes, the result is observer- or dressing-specific and must be labeled so. Demand also a global $\mathcal I^+$ coefficient: the patch algebra does not supply it without an additional reconstruction map.

The evidence ceiling is a precise semiclassical observer algebra, KMS structure, and generalized-entropy relation. It does not establish a finite-dimensional exact Hilbert space, global factorization, or equivalence to late-time dS/CFT. Entropy interpretation and horizon state counting remain separate questions.

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

• Chandrasekaran, V., Longo, R., Penington, G., and Witten, E. (2023). “An Algebra of Observables for de Sitter Space.” Journal of High Energy Physics 2023(2), 082. DOI.