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Algebras, Entanglement Islands, and Observers

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Pith's one-line read Entanglement islands carry Type II∞ von Neumann algebras when operators are dressed to a Goldstone 'observer'.

desk verdict A conditional but serious construction: the observer is a Goldstone mode, the island algebra is Type II∞ if the geometric modular flow conjecture and a gauge choice hold, and the paper deserves a real referee. read the letter →

arxiv 2506.12127 v1 pith:AVB4YUPX submitted 2025-06-13 hep-th gr-qc

Hao Geng , Yikun Jiang , Jiuci Xu This is my paper
classification hep-thgr-qc
keywords entanglementislandsTypeIIvonNeumannalgebrascrossedproductGoldstonevectorfieldobservergeneralizedentropyspontaneousdiffeomorphismbreakinggeometricmodularflow
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the operator algebra of an entanglement island becomes an emergent Type II∞ von Neumann algebra once it is enlarged by a mode of the Goldstone vector field produced by spontaneously broken diffeomorphism symmetry. That Goldstone mode is the 'observer' that earlier work postulated by hand, and dressing island operators to it makes them invariant under all diffeomorphisms. Because Type II∞ algebras admit a trace, the paper can define density matrices and an entropy, and it shows this entropy coincides with the generalized gravitational entropy of the island (area term plus matter entropy plus an observer contribution). The result matters because it turns a postulated ingredient of gravitational subregion algebras into a derived, composite object arising from the bath coupling, and it supplies a finite entropy to a closed gravitational region. The derivation assumes the geometric modular flow conjecture.

What carries the argument

The central object is the Goldstone vector field $V^\mu(x)$, the Stückelberg mode of the spontaneously broken diffeomorphisms, transforming nonlinearly as $V^\mu(x) \to V^\mu(x) - \epsilon^\mu(x)$; the combination $x^\mu + \sqrt{16\pi G_N} V^\mu(x)$ is diffeomorphism invariant and serves as the dressing that makes island operators physical. Conjoining the algebra with a mode $\Pi[\xi]$ of its conjugate momentum produces the crossed product $A = A_{I,QFT} \rtimes A_{\Pi[\xi_\Psi] + H_{QFT}[\xi_\Psi]}$, and the crossed-product duality theorem converts the trace-free Type III$_1$ QFT algebra into a Type II$_\infty$ factor with a trace. The entropy calculation is carried by the linearized Hamiltonian and momentum constraints, together with the gauge choice $F_\Psi[h,\dot h]=0$ that identifies the modular Hamiltonian integral with $\delta A(\partial I)/(4G_N)$.

What would settle it

Compute the modular flow of a concrete state in the free scalar island model and find a state for which no geometric vector field $\xi_\Psi$ exists, or for which $\xi_\Psi$ moves $\partial I$; then the unitary equivalence to the modular crossed product breaks and the Type II$\infty$ conclusion is not established. Alternatively, evaluate the leftover term $F_\Psi[h,\dot h]$ for a non-Killing $\xi_\Psi$ and show it cannot be removed by a gauge choice, which would sever the identification of the algebra entropy with the generalized entropy.

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Extended reading notes

Core claim

In the island model—a gravitational asymptotically AdS spacetime coupled to a non-gravitational bath—the transparent coupling spontaneously breaks the AdS diffeomorphism symmetry and gives the graviton a one-loop Stückelberg mass, with a composite Goldstone vector field $V^\mu(x)$ whose holographic dual is the composite operator $O_2 \partial_\mu O_1$. Consistency of entanglement wedge reconstruction requires operators inside the island to be dressed to this field, as $x^\mu + \sqrt{16\pi G_N} V^\mu(x)$, so that they obey the Hamiltonian and momentum constraints. The paper's central construction conjoins to the island algebra a particular mode $\Pi[\xi_\Psi]$ of the Goldstone conjugate momentum; after a unitary transformation that undresses the operators, the enlarged algebra is unitarily equivalent to the crossed product $A_{I,QFT} \rtimes A_{\Pi[\xi_\Psi] + H_{QFT}[\xi_\Psi]}$. Assuming the geometric modular flow conjecture, $\Pi[\xi_\Psi] + H_{QFT}[\xi_\Psi]$ is the modular Hamiltonian of the QFT state, so this is the crossed product of a Type III$_1$ algebra by its modular automorphism group and is a Type II$_\infty$ factor by the crossed-product duality theorem. The trace constructed on the algebra yields a density matrix whose entropy, up to an observer contribution and a state-independent constant, is the generalized entropy of the island.

Load-bearing premise

The load-bearing premise is the geometric modular flow conjecture—that for the QFT state $|\Psi\rangle$ the modular flow is generated geometrically by a vector field $\xi_\Psi$ that leaves the island boundary $\partial I$ invariant, with $\Delta_\Psi = e^{-H_{QFT}[\xi_\Psi]}$; if that fails, the conjoined algebra is not shown to be the crossed product by the modular automorphism group, and the Type II$\infty$ classification and entropy interpretation do not follow.

Editorial extensions

If this is right

  • Entanglement islands are holographically dual to emergent Type II∞ von Neumann algebras, so a closed gravitational subregion can carry a well-defined, trace-bearing algebra without an externally postulated observer.
  • The 'observer' Hamiltonian linear in a phase space variable is realized as a Goldstone mode and need not be bounded from below; the projection used in earlier Type II1 constructions therefore requires justification.
  • The algebra entropy equals the generalized entropy of the island, up to a state-independent constant and an observer contribution, and is UV finite because it is the entropy of a Type II∞ algebra.
  • Dressing island operators to the Goldstone field makes them invariant under all diffeomorphisms, including local ones, resolving the apparent conflict with gravitational Gauss' law and entanglement wedge reconstruction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the central claim is right, the state-independent entropy constant should be fixed by the microscopic bath dynamics, since the trace normalization is ultimately determined by the underlying CFT-plus-bath description.
  • The geometric modular flow conjecture can be tested inside this free-field island model by explicitly constructing states whose modular flow is geometric and checking that the vector field leaves the island boundary invariant; a counterexample would isolate exactly where the Type II∞ argument fails.
  • The same Goldstone mechanism suggests that observers in closed universes such as the de Sitter static patch should also be composite modes of spontaneously broken diffeomorphisms, making their Type II1 versus Type II∞ status depend on whether the relevant mode is bounded from below.
  • Extending the dressing to all orders in $G_N$ would show whether the crossed-product structure and the generalized-entropy identification survive beyond leading order with renormalized data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies operator algebras in the island model, where a gravitational asymptotically AdS spacetime is coupled to a non-gravitational bath, with transparent boundary conditions that spontaneously break AdS diffeomorphisms and give the graviton a Stückelberg mass. The authors construct the algebra A_I of island-localized, diffeomorphism-invariant operators dressed with the Goldstone vector field V^μ, and then argue that adjoining a particular integrated momentum mode Π[ξΨ] makes the algebra unitarily equivalent to the crossed product A = A_{I,QFT} ⋊ A_{Π[ξΨ]+H_QFT[ξΨ]}. Assuming the geometric modular flow conjecture, Eq. (3.5), they identify the crossed product as the one by the modular automorphism group, invoke Takesaki's theorem to conclude that A is Type II∞, construct a trace and density matrix, compute the entropy, and claim it equals the generalized entropy of the island under the gauge choice FΨ[h,ḣ]=0, Eq. (3.33). They also question the projection used in earlier Type II1 constructions, arguing that the observer Hamiltonian is linear in a phase-space variable and need not be bounded below.

Significance. If the main claim holds, the paper provides a concrete microscopic realization of the 'observer' as a composite Goldstone vector mode arising from spontaneous diffeomorphism breaking, rather than as an external input. The derivation is parameter-free in the sense that the entropy is not fitted, and it connects the recent crossed-product program to the island literature, giving a sharp reason to doubt the bounded-below projection that produces Type II1 algebras. The paper also makes good use of standard modular theory and Takesaki duality, and it is transparent about its main assumption, the geometric modular flow conjecture. However, the advertised conclusion that entanglement islands correspond to emergent Type II∞ algebras is strictly conditional: the central classification step and the later entropy interpretation both rest on assumptions that are not proven in this model. The paper is therefore a valuable conditional construction rather than a complete derivation.

major comments (4)
  1. [Sec. 3.1, Eq. (3.5)] The Type II∞ classification rests entirely on the geometric modular flow conjecture, which is stated but not established for the island model considered here. For generic subregions in QFT, modular flow is not geometric, and the paper provides no evidence that there exists a cyclic separating state |Ψ⟩ with ΔΨ = e^{-H_QFT[ξΨ]} for a vector field ξΨ leaving ∂I invariant. If Eq. (3.5) fails, then the algebra in Eq. (3.6) is not the crossed product by the modular automorphism group of A_{I,QFT}, Takesaki's theorem does not apply, and the central claim that the gravitational island algebra is Type II∞ is unsupported. Since this is the load-bearing step of the paper, the authors should either prove the conjecture in this model, identify a class of islands where it is known to hold, or reformulate the abstract and conclusions so that the result is explicitly presented as conditional on this conjecture.
  2. [Sec. 3.4, Eq. (3.33)] The identification of the computed entropy with the generalized gravitational entropy of the island relies on the gauge choice FΨ[h,ḣ]=0, which the authors state is automatically satisfied only when ξΨ is Killing. For a general geometric modular flow vector field, this condition is not justified. This is not a cosmetic point: Eqs. (3.34)–(3.37) are the only derivation of S(ρ_{bΦ}) = ⟨A(∂I)/4G_N⟩ + S(I)_{QFT,Φ} + S_{obs,f} − c, so if Eq. (3.33) cannot be imposed, the entropy does not reduce to the generalized entropy. The manuscript should either justify this gauge choice for non-Killing ξΨ, argue that the final trace result is independent of it, or explicitly state the entropy identification as a conjecture with the same status as the geometric modular flow assumption.
  3. [Sec. 3.3, footnote 21 and Eq. (3.24)] The evaluation of the third term in Eq. (3.23) uses the factorization ΔΦ|Ψ = ρΦ ⊗ ρ'^{-1}_Ψ, which the authors themselves label as 'sloppy' and note is not a well-defined splitting for a Type III1 algebra. This step is load-bearing for the physical interpretation of the entropy: it is precisely this factorization that converts the algebraically well-defined quantity −⟨bΦ|log ΔΦ|Ψ|bΦ⟩ into S(I)_{QFT,Φ} plus an integral over the complement of the island. The paper should clarify why the final expression in Eq. (3.27) is independent of the ill-defined splitting, or provide a rigorous regularization that reproduces the same terms.
  4. [Sec. 3.1, after Eq. (3.5)] The Type III1 algebra A_I alone does not yield a crossed product; the Type II∞ result is obtained only after the algebra is enlarged by conjoining Π[ξΨ]. The manuscript states that it is 'well-motivated' to add such an element, but it does not derive from the island model that this particular mode, rather than some other mode of the phase space, must be included in the physical algebra. Since the paper's central claim is that the emergent gravitational algebra of island operators is Type II∞, the step of adjoining Π[ξΨ] should be justified from the diffeomorphism-invariant construction of observables, for example by showing that the gauge-invariant algebra generated by the dressed operators and the Goldstone momentum naturally contains Π[ξΨ] and no other independent modes.
minor comments (6)
  1. [Eq. (3.7)] The notation Y = Π[ξΨ] = −H_obs is introduced without connecting it to the earlier Hamiltonian in Eq. (1.2); the reader must infer that H_obs is a particular linear mode of the Goldstone momentum, and this correspondence should be made explicit.
  2. [Sec. 3.2, after Eq. (3.22)] The 'faithful island condition' is invoked to justify dropping O(ϵ) terms, but no precise criterion is given for when a semiclassical state satisfies this condition; since the entropy calculation depends on the validity of this approximation, a quantitative statement would strengthen the argument.
  3. [Sec. 3.4, after Eq. (3.28)] The phrase 'global (pass directed) time translation' appears to contain a typo and should read 'past directed'; please check the direction of the vector field near the asymptotic boundary.
  4. [Sec. 3.4, Eq. (3.36)] The observer entropy S_{obs,f} combines an expectation value of ∫_{\bar I} ξΨ πV μ with a term from f(ϵY), but the physical interpretation of the first term as an 'observer contribution' is not explained; a few clarifying sentences would help.
  5. [Sec. 3.2, Eq. (3.14)] The trace is defined only on a trace-class ideal of A, but the paper does not specify this domain before using the trace in Eq. (3.20); naming the ideal and noting which operators in A are trace-class would avoid ambiguity.
  6. [Abstract] The abstract says the paper 'establishes' that entanglement islands correspond to Type II∞ algebras, while the same paragraph and the main text state that the result relies on assuming the geometric modular flow conjecture; the wording should make the conditional nature of the claim explicit, for example 'we show, conditional on the geometric modular flow conjecture, that...'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the crossed-product equivalence is an algebraic identity, and the Type II∞ claim is explicitly conditional on the stated geometric modular flow conjecture rather than a reduction to input.

full rationale

The paper's central derivation is non-circular. The unitary equivalence in Eq (3.4) follows from the canonical commutation relation between H_QFT[V] and Π[ξ], which shifts Π[ξ] by H_QFT[ξ]; this is a mathematical identity, not a fitted input. The Type II∞ classification uses Takesaki's theorem after the paper explicitly states, in Sec 3.1, 'We will assume the so-called geometric modular flow conjecture [30]', with Δ_Ψ = e^{-H_QFT[ξ_Ψ]}. That conjecture is a genuine assumption, not an input secretly equivalent to the conclusion, and the paper repeatedly acknowledges that the result depends on it (abstract and Sec 5). The entropy identification is also conditional on the stated gauge choice F_Ψ[h,ḣ] = 0 (Eq 3.33), which the authors call crucial and admit is not generally justified; the 'observer contribution' S_obs,f is explicitly defined as the remaining terms, not as a pre-fitted quantity. The graviton mass and composite Goldstone operator are taken from the same group's prior work [43–45], but these are independent one-loop calculations and consistency checks that do not encode the target algebra statement. No parameter is fitted and then renamed as a prediction, and no conclusion is made true by definition. The paper is a conditional construction with clearly labeled assumptions, which is a correctness risk but not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numerical parameters are fitted to data in this paper. The central claim rests on the geometric modular flow conjecture, the prior one-loop graviton mass results, the choice to include the Goldstone mode in the algebra, and a gauge choice needed for the entropy interpretation; these are the costs of the derivation. No new particles, fields, or forces are introduced; the observer is identified with an existing Goldstone vector mode, and the paper emphasizes that the compensating field V^μ is a gauge artifact until it becomes the Stückelberg field.

assumptions (7)
  • domain assumption Geometric modular flow conjecture: for the state |Ψ⟩, modular flow is geometric, generated by a vector field ξΨ that leaves ∂I invariant, with ΔΨ = e^{-H_QFT[ξΨ]}.
    Assumed in Sec 3.1 just before Eq (3.5); without it the crossed product algebra is not shown to be the modular crossed product and the Type II∞ conclusion fails.
  • domain assumption The one-loop effective action for the graviton is the Stückelberg mass term (Eq 2.16) with M^2 given by Eq (2.17), as established in Refs [43-45].
    Underpins the identification of V^μ as the Goldstone/observer field; the paper cites rather than re-derives these results.
  • domain assumption Leading-order perturbative expansion in GN around empty AdS is sufficient for the operator algebra and constraints.
    All dressed operators and constraints are constructed at first nontrivial order in GN; the paper only expects, but does not prove, validity to all orders.
  • domain assumption The physical gravitational algebra should include the Goldstone momentum mode Π[ξ] (Eq 3.1) alongside the dressed island operators.
    This enlargement is what turns the Type III1 QFT algebra into the crossed product; without it the algebra stays Type III1.
  • ad hoc to paper Gauge choice FΨ[h, ḣ] = 0 (Eq 3.33).
    Required to identify the computed entropy with the generalized entropy; authors note it is automatic for Killing ξΨ but not generally justified.
  • standard math Tomita-Takesaki theory, including Takesaki's crossed product theorem (Ref [58], corollary 97).
    Used to classify the crossed product as Type II∞ and to construct the trace and density matrices.
  • domain assumption The quantum extremal surface/island formula Eq (1.8) and the associated entanglement wedge reconstruction are valid.
    The paper treats islands as well-defined subregions in a UV-complete gravity theory, citing [46,47] for derivation.

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Pith. "Pith review of Algebras, Entanglement Islands, and Observers." pith.science (2026). https://pith.science/paper/AVB4YUPX

@misc{pith2026250612127,
  author       = {Pith},
  title        = {Pith review of: Algebras, Entanglement Islands, and Observers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AVB4YUPX}},
  note         = {Machine review of arXiv:2506.12127}
}
abstract

Some recent work has postulated the existence of an "observer" for a consistent definition of subregion algebras in gravitational universes. The subregion algebras consist of operators dressed to this "observer" and are typically Type II von Neumann algebras. Nevertheless, as opposed to standard physical systems, such an "observer" was postulated to have a Hamiltonian $\hat{H}_{\text{obs}}$ linear in phase space variable. This linear form suggests that the complete dynamics of such an "observer" should also be controlled by an external system or some underlying degrees of freedom within the system. In this paper, we show that this is exactly the case in the island model. In the island model, we have a gravitational asymptotically anti-de Sitter (AdS) spacetime coupled with a non-gravitational bath, and the diffeomorphism symmetries in the gravitational AdS are spontaneously broken due to the bath coupling. In this setup, the "observer" is constructed using the Goldstone vector field associated with the spontaneously broken diffeomorphism symmetry, and the external system that also controls the dynamics of the "observer" is the non-gravitational bath. The basic consistency of the entanglement wedge reconstruction requires operators in the entanglement island to be dressed to this "observer". Thus, we establish the result that entanglement islands correspond to emergent Type II$_{\infty}$ von Neumann algebras from the holographic dual perspective. This result relies on assuming the geometric modular flow conjecture. Our study also raises a question for earlier constructions of Type II$_{1}$ von Neumann algebras.

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Reference graph

Works this paper leans on

86 extracted references · 9 canonical work pages · cited by 3 Pith papers

  1. [1]

    H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas et al., Information Transfer with a Gravitating Bath , 2012.04671

    arXiv 2012

  2. [2]

    Balasubramanian and C

    V. Balasubramanian and C. Cummings, The entropy of finite gravitating regions , 2312.08434

    arXiv

  3. [3]

    Susskind and J

    L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [ hep-th/9401070]

    arXiv 1994

  4. [4]

    Black Hole Entropy in the O(N) Model

    D.N. Kabat, S.H. Shenker and M.J. Strassler, Black hole entropy in the O(N) model , Phys. Rev. D 52 (1995) 7027 [ hep-th/9506182]

    work page Pith review arXiv 1995

  5. [5]

    Donnelly and A.C

    W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes , Phys. Rev. Lett. 114 (2015) 111603 [ 1412.1895]

    arXiv 2015

  6. [6]

    Headrick, V.E

    M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality \& holographic entanglement entropy, JHEP 12 (2014) 162 [ 1408.6300]

    arXiv 2014

  7. [7]

    Engelhardt and A.C

    N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime , JHEP 01 (2015) 073 [ 1408.3203]

    arXiv 2015

  8. [8]

    The Gravity Duals of Modular Hamiltonians

    D.L. Jafferis and S.J. Suh, The Gravity Duals of Modular Hamiltonians , JHEP 09 (2016) 068 [1412.8465]

    work page Pith review arXiv 2016

Show all 86 references
  1. [9]

    Jafferis, A

    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [ 1512.06431]

    2016 arXiv

  2. [10]

    X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality , Phys. Rev. Lett. 117 (2016) 021601 [ 1601.05416]

    2016 arXiv

  3. [11]

    H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas et al., Inconsistency of Islands in Theories with Long-Range Gravity , 2107.03390

    arXiv

  4. [12]

    Giddings and A

    S.B. Giddings and A. Kinsella, Gauge-invariant observables, gravitational dressings, and holography in AdS, JHEP 11 (2018) 074 [ 1802.01602]

    2018 arXiv

  5. [13]

    Donnelly and S.B

    W. Donnelly and S.B. Giddings, Diffeomorphism-invariant observables and their nonlocal algebra, Phys. Rev. D93 (2016) 024030 [ 1507.07921]

    2016 arXiv

  6. [14]

    Donnelly and L

    W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity , JHEP 09 (2016) 102 [1601.04744]. – 30 –

    2016 arXiv

  7. [15]

    Donnelly and S.B

    W. Donnelly and S.B. Giddings, Observables, gravitational dressing, and obstructions to locality and subsystems , Phys. Rev. D94 (2016) 104038 [ 1607.01025]

    2016 arXiv

  8. [16]

    Donnelly and S.B

    W. Donnelly and S.B. Giddings, How is quantum information localized in gravity? , Phys. Rev. D96 (2017) 086013 [ 1706.03104]

    2017 arXiv

  9. [17]

    Donnelly and S.B

    W. Donnelly and S.B. Giddings, Gravitational splitting at first order: Quantum information localization in gravity, Phys. Rev. D 98 (2018) 086006 [ 1805.11095]

    2018 arXiv

  10. [18]

    Giddings, Quantum gravity observables: observation, algebras, and mathematical structure , 2505.22708

    S.B. Giddings, Quantum gravity observables: observation, algebras, and mathematical structure , 2505.22708

    arXiv

  11. [19]

    Almheiri, N

    A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole , JHEP 12 (2019) 063 [ 1905.08762]

    2019 arXiv

  12. [20]

    Penington, Entanglement Wedge Reconstruction and the Information Paradox , JHEP 09 (2020) 002 [ 1905.08255]

    G. Penington, Entanglement Wedge Reconstruction and the Information Paradox , JHEP 09 (2020) 002 [ 1905.08255]

    2020 arXiv

  13. [21]

    Geng and A

    H. Geng and A. Karch, Massive islands , JHEP 09 (2020) 121 [ 2006.02438]

    2020 arXiv

  14. [22]

    Giddings, D

    S.B. Giddings, D. Marolf and J.B. Hartle, Observables in effective gravity , Phys. Rev. D 74 (2006) 064018 [ hep-th/0512200]

    2006 arXiv

  15. [23]

    Candelas, Vacuum polarization in schwarzschild spacetime , Phys

    P. Candelas, Vacuum polarization in schwarzschild spacetime , Phys. Rev. D 21 (1980) 2185

    1980

  16. [24]

    Rovelli, What is observable in classical and quantum gravity? , Classical and Quantum Gravity 8 (1991) 297

    C. Rovelli, What is observable in classical and quantum gravity? , Classical and Quantum Gravity 8 (1991) 297

    1991

  17. [25]

    Rovelli, Quantum reference systems, Classical and Quantum Gravity 8 (1991) 317

    C. Rovelli, Quantum reference systems, Classical and Quantum Gravity 8 (1991) 317

    1991

  18. [26]

    Chandrasekaran, R

    V. Chandrasekaran, R. Longo, G. Penington and E. Witten, An algebra of observables for de Sitter space, JHEP 02 (2023) 082 [ 2206.10780]

    2023 arXiv

  19. [27]

    Witten, Algebras, Regions, and Observers , 2303.02837

    E. Witten, Algebras, Regions, and Observers , 2303.02837

    arXiv

  20. [28]

    Witten, A Background Independent Algebra in Quantum Gravity , 2308.03663

    E. Witten, A Background Independent Algebra in Quantum Gravity , 2308.03663

    arXiv

  21. [29]

    Goeller, P.A

    C. Goeller, P.A. Hoehn and J. Kirklin, Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance , 2206.01193

    arXiv

  22. [30]

    Jensen, J

    K. Jensen, J. Sorce and A.J. Speranza, Generalized entropy for general subregions in quantum gravity, JHEP 12 (2023) 020 [ 2306.01837]

    2023 arXiv

  23. [31]

    Alexandre, A

    B. Alexandre, A. Etkin and F.-S. Rassouli, Unimodular JT gravity and de Sitter quantum cosmology, 2501.17213

    arXiv

  24. [32]

    De Vuyst, S

    J. De Vuyst, S. Eccles, P.A. Hoehn and J. Kirklin, Linearization (in)stabilities and crossed products, 2411.19931

    arXiv

  25. [33]

    Kolchmeyer and H

    D.K. Kolchmeyer and H. Liu, Chaos and the Emergence of the Cosmological Horizon , 2411.08090

    arXiv

  26. [34]

    Hoehn, A

    P.A. Hoehn, A. Russo and A.R.H. Smith, Matter relative to quantum hypersurfaces , Phys. Rev. D 109 (2024) 105011 [ 2308.12912]

    2024 arXiv

  27. [35]

    De Vuyst, S

    J. De Vuyst, S. Eccles, P.A. Hoehn and J. Kirklin, Gravitational entropy is observer-dependent , 2405.00114. – 31 –

    arXiv

  28. [36]

    Witten, A Background Independent Algebra in Quantum Gravity , Gen

    E. Witten, A Background Independent Algebra in Quantum Gravity , Gen. Rel. Grav. 57 (2025) 17

    2025

  29. [37]

    Chen and G

    C.-H. Chen and G. Penington, A clock is just a way to tell the time: gravitational algebras in cosmological spacetimes, 2406.02116

    arXiv

  30. [38]

    Maldacena, Real observers solving imaginary problems , 2412.14014

    J. Maldacena, Real observers solving imaginary problems , 2412.14014

  31. [39]

    Z. Yang, Y. Zhang and W. Zheng, Comments on the de Sitter Double Cone , 2505.08647

  32. [40]

    Geng, Quantum Rods and Clock in a Gravitational Universe , 2412.03636

    H. Geng, Quantum Rods and Clock in a Gravitational Universe , 2412.03636

    arXiv

  33. [41]

    Porrati, Higgs phenomenon for the graviton in ADS space , Mod

    M. Porrati, Higgs phenomenon for the graviton in ADS space , Mod. Phys. Lett. A 18 (2003) 1793 [hep-th/0306253]

    2003 arXiv

  34. [42]

    Aharony, A.B

    O. Aharony, A.B. Clark and A. Karch, The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture, Phys. Rev. D 74 (2006) 086006 [ hep-th/0608089]

    2006 arXiv

  35. [43]

    Geng, Open AdS/CFT via a Double Trace Deformation , 2311.13633

    H. Geng, Open AdS/CFT via a Double Trace Deformation , 2311.13633

    arXiv

  36. [44]

    Geng, Graviton Mass and Entanglement Islands in Low Spacetime Dimensions , 2312.13336

    H. Geng, Graviton Mass and Entanglement Islands in Low Spacetime Dimensions , 2312.13336

  37. [45]

    Geng, The Mechanism behind the Information Encoding for Islands , 2502.08703

    H. Geng, The Mechanism behind the Information Encoding for Islands , 2502.08703

    arXiv

  38. [46]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation , JHEP 05 (2020) 013 [ 1911.12333]

    2020 arXiv

  39. [47]

    Geng, Replica wormholes and entanglement islands in the Karch-Randall braneworld , JHEP 01 (2025) 063 [ 2405.14872]

    H. Geng, Replica wormholes and entanglement islands in the Karch-Randall braneworld , JHEP 01 (2025) 063 [ 2405.14872]

    2025 arXiv

  40. [48]

    Leutheusser and H

    S.A.W. Leutheusser and H. Liu, Emergent Times in Holographic Duality , Phys. Rev. D 108 (2023) 086020 [ 2112.12156]

    2023 arXiv

  41. [49]

    Leutheusser and H

    S. Leutheusser and H. Liu, Causal connectability between quantum systems and the black hole interior in holographic duality , Phys. Rev. D 108 (2023) 086019 [ 2110.05497]

    2023 arXiv

  42. [50]

    Leutheusser and H

    S. Leutheusser and H. Liu, Subregion-subalgebra duality: Emergence of space and time in holography, Phys. Rev. D 111 (2025) 066021 [ 2212.13266]

    2025 arXiv

  43. [51]

    Soni, A type I approximation of the crossed product , JHEP 01 (2024) 123 [ 2307.12481]

    R.M. Soni, A type I approximation of the crossed product , JHEP 01 (2024) 123 [ 2307.12481]

    2024 arXiv

  44. [52]

    Gesteau, Large N von Neumann Algebras and the Renormalization of Newton ’s Constant , Commun

    E. Gesteau, Large N von Neumann Algebras and the Renormalization of Newton ’s Constant , Commun. Math. Phys. 406 (2025) 40 [ 2302.01938]

    2025 arXiv

  45. [53]

    Kudler-Flam, S

    J. Kudler-Flam, S. Leutheusser and G. Satishchandran, Generalized black hole entropy is von Neumann entropy, Phys. Rev. D 111 (2025) 025013 [ 2309.15897]

    2025 arXiv

  46. [54]

    Kudler-Flam, S

    J. Kudler-Flam, S. Leutheusser and G. Satishchandran, Algebraic Observational Cosmology, 2406.01669

    arXiv

  47. [55]

    Gesteau and H

    E. Gesteau and H. Liu, Toward stringy horizons, 2408.12642

    arXiv

  48. [56]

    Faulkner and A.J

    T. Faulkner and A.J. Speranza, Gravitational algebras and the generalized second law , 2405.00847

    arXiv

  49. [57]

    Jensen, S

    K. Jensen, S. Raju and A.J. Speranza, Holographic observers for time-band algebras , 2412.21185

    arXiv

  50. [58]

    Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Mathematica 131 (1973) 249

    M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Mathematica 131 (1973) 249 . – 32 –

    1973

  51. [59]

    Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory , Rev

    E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory , Rev. Mod. Phys. 90 (2018) 045003 [1803.04993]

    2018 arXiv

  52. [60]

    Witten, Gravity and the crossed product , JHEP 10 (2022) 008 [ 2112.12828]

    E. Witten, Gravity and the crossed product , JHEP 10 (2022) 008 [ 2112.12828]

    2022 arXiv

  53. [61]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [ 2006.06872]

    2021 arXiv

  54. [62]

    Maldacena, The Large N limit of superconformal field theories and supergravity , Int

    J.M. Maldacena, The Large N limit of superconformal field theories and supergravity , Int. J. Theor. Phys. 38 (1999) 1113 [ hep-th/9711200]

    1999 arXiv

  55. [63]

    Gubser, I.R

    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [ hep-th/9802109]

    1998 arXiv

  56. [64]

    Witten, Anti-de Sitter space and holography , Adv

    E. Witten, Anti-de Sitter space and holography , Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150]

    1998 arXiv

  57. [65]

    Witten, Multitrace operators, boundary conditions, and AdS / CFT correspondence , hep-th/0112258

    E. Witten, Multitrace operators, boundary conditions, and AdS / CFT correspondence , hep-th/0112258

    arXiv

  58. [66]

    Karch and L

    A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008 [ hep-th/0011156]

    2001 arXiv

  59. [67]

    Hinterbichler, Theoretical Aspects of Massive Gravity , Rev

    K. Hinterbichler, Theoretical Aspects of Massive Gravity , Rev. Mod. Phys. 84 (2012) 671 [1105.3735]

    2012 arXiv

  60. [68]

    Arnowitt, S

    R.L. Arnowitt, S. Deser and C.W. Misner, The Dynamics of general relativity , Gen. Rel. Grav. 40 (2008) 1997 [ gr-qc/0405109]

    2008 arXiv

  61. [69]

    Dirac, Lectures on Quantum mechanics , Dover (2001)

    P.A.M. Dirac, Lectures on Quantum mechanics , Dover (2001)

    2001

  62. [70]

    Hawking and G.T

    S.W. Hawking and G.T. Horowitz, The Gravitational Hamiltonian, action, entropy and surface terms, Class. Quant. Grav. 13 (1996) 1487 [ gr-qc/9501014]

    1996 arXiv

  63. [71]

    Chowdhury, V

    C. Chowdhury, V. Godet, O. Papadoulaki and S. Raju, Holography from the Wheeler-DeWitt equation, 2107.14802

    arXiv

  64. [72]

    Chandrasekaran, G

    V. Chandrasekaran, G. Penington and E. Witten, Large N algebras and generalized entropy , JHEP 04 (2023) 009 [ 2209.10454]

    2023 arXiv

  65. [73]

    Karch, Autolocalization in de Sitter space , JHEP 07 (2003) 050 [ hep-th/0305192]

    A. Karch, Autolocalization in de Sitter space , JHEP 07 (2003) 050 [ hep-th/0305192]

    2003 arXiv

  66. [74]

    Alishahiha, A

    M. Alishahiha, A. Karch, E. Silverstein and D. Tong, The dS/dS correspondence, AIP Conf. Proc. 743 (2004) 393 [ hep-th/0407125]

    2004 arXiv

  67. [75]

    X. Dong, E. Silverstein and G. Torroba, De Sitter Holography and Entanglement Entropy , JHEP 07 (2018) 050 [ 1804.08623]

    2018 arXiv

  68. [76]

    H. Geng, S. Grieninger and A. Karch, Entropy, Entanglement and Swampland Bounds in DS/dS, JHEP 06 (2019) 105 [ 1904.02170]

    2019 arXiv

  69. [77]

    Geng, Some Information Theoretic Aspects of De-Sitter Holography , JHEP 02 (2020) 005 [1911.02644]

    H. Geng, Some Information Theoretic Aspects of De-Sitter Holography , JHEP 02 (2020) 005 [1911.02644]

    2020 arXiv

  70. [78]

    Geng, Non-local entanglement and fast scrambling in de-Sitter holography , Annals Phys

    H. Geng, Non-local entanglement and fast scrambling in de-Sitter holography , Annals Phys. 426 (2021) 168402 [ 2005.00021]

    2021 arXiv

  71. [79]

    H. Geng, Y. Nomura and H.-Y. Sun, Information paradox and its resolution in de Sitter holography, Phys. Rev. D 103 (2021) 126004 [ 2103.07477]. – 33 –

    2021 arXiv

  72. [80]

    Gibbons and S.W

    G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752

    1977

  73. [81]

    Takesaki, Tomita’s theory of modular Hilbert algebras and its applications , vol

    M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications , vol. 128, Springer (2006)

    2006

  74. [82]

    Jensen and P

    B.P. Jensen and P. Candelas, The Schwarzschild Radial Functions , Phys. Rev. D33 (1986) 1590

    1986

  75. [83]

    Sorce, Analyticity and the Unruh effect: a study of local modular flow , JHEP 24 (2024) 040 [2403.18937]

    J. Sorce, Analyticity and the Unruh effect: a study of local modular flow , JHEP 24 (2024) 040 [2403.18937]

    2024 arXiv

  76. [84]

    Caminiti, F

    J. Caminiti, F. Capeccia, L. Ciambelli and R.C. Myers, Geometric modular flows in 2d CFT and beyond, 2502.02633

    arXiv

  77. [85]

    Takesaki, Tomita’s Theory of Modular Hilbert Algebras and its Applications , Lecture Notes in Mathematics, Springer-Verlag (1970), 10.1007/bfb0065832

    M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and its Applications , Lecture Notes in Mathematics, Springer-Verlag (1970), 10.1007/bfb0065832

    1970 doi

  78. [86]

    Connes, Une classification des facteurs de type {iii}, Annales scientifiques de l’ ´Ecole Normale Sup´ erieure6 (1973) 133

    A. Connes, Une classification des facteurs de type {iii}, Annales scientifiques de l’ ´Ecole Normale Sup´ erieure6 (1973) 133. – 34 –

    1973

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