Static-Patch Holography

Static-patch holography asks for a microscopic description of everything accessible to one geodesic observer between that observer’s past and future horizons. The semiclassical patch has a thermal KMS state, but this does not determine whether its exact observables form a finite matrix algebra, an infinite type-II factor, a type-III local algebra, or some gravitational completion. This page fixes $(d+1)$-dimensional de Sitter radius $L$, the Bunch–Davies state, an observer at $r=0$, and observables gravitationally dressed to that observer. Every microscopic identification remains proposal-level as of 2026-08-20.

Required background. Gauge Theory and String-Theory Preparation provides constraints, algebras, and thermal correlation functions.

Helpful background. Quantum Gravity frames relational observables; Quantum Information supplies density matrices and entropy; Causality and Global Structure supplies causal diamonds; and Semiclassical Gravity supplies horizon thermality.

The observer and the KMS state

The static metric is

$$ds^2=-\left(1-\frac{r^2}{L^2}\right)dt_s^2 +\frac{dr^2}{1-r^2/L^2}+r^2d\Omega_{d-1}^2, \qquad 0\leq r\lt L.$$

Euclidean smoothness fixes $\beta=2\pi L$, using time normalized at $r=0$. Assume a matter sector for which the Euclidean/Bunch–Davies state exists; a minimally coupled massless scalar requires separate treatment of its homogeneous mode, as explained on dS/CFT Wave-Function Proposals. For analytic observables $A$ and $B$ evolved by the static generator, the restricted state obeys the Kubo–Martin–Schwinger (KMS) relation

$$\langle A(t_s-i\beta)B(0)\rangle =\langle B(0)A(t_s)\rangle, \qquad T=\beta^{-1}=\frac{1}{2\pi L}.$$

This establishes thermality for a specified algebra and state. It does not show that the patch is an autonomous finite quantum system. In gravity, a local operator must be dressed relationally, and the Hamiltonian constraint ties the observer’s clock energy to the fields. Those facts change the operator algebra itself.

Candidate algebra classes

An ordinary finite system with Hilbert-space dimension $D$ has algebra $M_D(\mathbb C)$, normalized trace, entropy bounded by $\log D$, and discrete recurrences. By contrast, a continuum quantum field theory restricted to a region usually gives a type-III algebra: no trace-class density matrix belongs intrinsically to the region, and entanglement entropy needs a regulator.

Chandrasekaran, Longo, Penington, and Witten 2023, §§2–4, pp. 9–34 include an observer and the gravitational constraint and obtain a type-II$_1$ algebra for the de Sitter static patch in semiclassical gravity. A type-II$_1$ factor has a finite normalized trace and a well-defined entropy relative to an additive convention, yet it is infinite-dimensional. Therefore the result must not be paraphrased as a derivation of $D=e^{A/(4G)}$. It instead supplies a precise observer-dependent algebraic setting in which empty de Sitter is a maximum-entropy state.

Results and limitations through 2026-08-20

The formulation axes are observer worldline, causal diamond, vacuum, time generator, dressing, algebra class, entropy regulator, and which correlators the proposal claims to compute.

Evidence or position	Controlled result	Limitation or adverse discriminator	Date
Static-patch thermality, Gibbons and Hawking 1977, pp. 2738–2743	Euclidean regularity and detector response give $T=1/(2\pi L)$ and horizon thermodynamics	A KMS state does not specify the microscopic algebra or prove finite Hilbert-space dimension	1977
Observer algebra, Chandrasekaran, Longo, Penington, and Witten 2023, §§2–4, pp. 9–34	Gravitational constraints with an observer lead semiclassically to a type-II$_1$ algebra and generalized-entropy relation	The result is perturbative and algebraic, not a unique ultraviolet completion or finite matrix model	2023
Negative-shock test, Chen, Stanford, Tang, and Yang 2026, §§2–4, pp. 8–31	Including recoil and gravitational backreaction yields explicit eikonal OTOCs	The tested Euclidean-path-integral-as-trace conjecture conflicts with cyclicity and positivity; this adverse result does not erase the distinct observer-algebra construction	2026
Finite-state complementarity	Finite entropy motivates a patch Hilbert space with roughly $e^{S_{\mathrm{dS}}}$ states and recurrence time	No controlled map reproduces all semiclassical correlators, locality, dressing, and recurrences while deriving that dimension	2026 cutoff

Competing positions interpret the patch as a finite quantum system, an observer-dependent algebra, a stretched-horizon model, or one complementary encoding of global physics. The controlled results are the KMS property and specific semiclassical algebraic constructions. A unique microscopic dual remains conjectural and unresolved.

What the negative-shock calculation tests

The 2026 calculation studies a proposed equality between observer-worldline correlators computed by a Euclidean gravitational path integral and a Hilbert-space trace. Positive-energy shocks at a cosmological horizon create a time advance rather than the familiar black-hole time delay. In the eikonal OTOC, two resummations are exchanged by cyclicity, while the leading correction has the wrong sign for the assumed positivity property.

This is a sharp formulation test because cyclicity and positivity are minimal trace requirements. Its scope is equally important: it targets the stated Euclidean trace conjecture, its analyticity assumptions, and the correlator class studied. It is not a theorem excluding every static-patch algebra, every modified contour, or every finite-state model. Any repair must say which assumption changes and recompute the same adverse observable.

Finite entropy, locality, and recurrences

If the exact patch Hilbert space were finite-dimensional, generic exact correlators would be quasiperiodic and exhibit recurrences. Semiclassical quantum field theory instead predicts decay over a long but finite calculational regime and uses type-III local subalgebras. A candidate must state its error tolerance and time window, then show how the finite theory approximates these correlators before recurrences become visible.

Nested observers add another discriminator. Two worldlines have overlapping but nonidentical diamonds; a proposal must give consistent embeddings for their observable algebras without cloning independently accessible information. Gravitational dressing may prevent naïve tensor factorization. These requirements are stronger than matching temperature or entropy alone.

What could change these conclusions

The conclusions could change if a theorem fixes the exact observer algebra and state space, if an independent calculation reproduces or overturns the negative-shock cyclicity and positivity test, or if a controlled microscopic model derives both semiclassical correlators and recurrences. The conclusions also depend on whether a new operational protocol, dressing prescription, algebra class, or Euclidean contour changes which observables are claimed to be encoded.

Exercises

Compare a finite matrix algebra, a type-II$_1$ factor, and a type-III local algebra as candidates for the static patch. Which facts follow from the KMS equation alone?

Solution

The KMS equation fixes thermal analyticity relative to a chosen time evolution, but it does not determine the algebra class. A finite matrix algebra has a normalized trace, bounded entropy, and recurrences; a type-II$_1$ factor also has a normalized finite trace but remains infinite-dimensional; a type-III algebra has no intrinsic trace or density matrix. Therefore temperature alone cannot establish $D=e^{S_{\mathrm{dS}}}$. Comparing candidates requires additional checks of dressing, entropy normalization, correlators, nested observers, and long-time behavior. The negative-shock result is especially discriminating because it directly tests trace cyclicity and positivity for a specific proposed path-integral dictionary.

References

• Chandrasekaran, Venkatesa, Roberto Longo, Geoff Penington, and Edward Witten. “An Algebra of Observables for de Sitter Space.” Journal of High Energy Physics 02 (2023) 082. arXiv:2206.10780.
• Chen, Yiming, Douglas Stanford, Haifeng Tang, and Zhenbin Yang. “Negative Shocks versus Static Patch Holography.” Preprint (2026). arXiv:2607.14042.
• Gibbons, G. W., and S. W. Hawking. “Cosmological Event Horizons, Thermodynamics, and Particle Creation.” Physical Review D 15 (1977) 2738–2751. DOI: 10.1103/PhysRevD.15.2738.