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Lectures on Quantum Extremal Surfaces and the Page Curve

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Pith's one-line read Quantum extremal surfaces with disconnected islands reproduce the eternal black hole Page curve.

desk verdict Lecture notes with zero new results but a genuinely careful, well-flagged derivation of the eternal-black-hole Page curve; worth reading for the pedagogy, not for novelty. read the letter →

arxiv 2502.01933 v1 pith:UWADWBAM submitted 2025-02-04 hep-th

Raghu Mahajan This is my paper · ORCID
classification hep-th
keywords quantumextremalsurfacesPagecurveentanglementislandseternalblackholeHawkingradiationKMSconditionJTgravityentropy
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

These lectures try to establish that the black hole information paradox has a controlled semiclassical resolution in a simple setting: an eternal two-sided black hole coupled to two flat-space reservoirs. The claimed resolution is that the von Neumann entropy of the radiation grows linearly for a while, $S_{\mathrm{rad}}(t)\simeq \frac{c}{3}\frac{2\pi t}{\beta}$, and then saturates at $2S_{\mathrm{BH}}(\beta)$ because a disconnected quantum extremal surface, an island inside the black hole, enters the entanglement wedge of the radiation. The paper lays out the full chain of reasoning, from a KMS-based derivation of Hawking radiation to the quantum extremal surface computation in two-dimensional gravity. It matters because it shows that the information paradox can be addressed with semiclassical gravity plus the island rule, without needing the full theory of quantum gravity.

What carries the argument

The quantum extremal surface (QES) prescription with islands. The generalized entropy functional $S_{\mathrm{gen}}(R)=\mathrm{Area}(\partial R)/4G+S_{\mathrm{bulk}}(R)$ is extremized over candidate regions $R$ homologous to the radiation region, and the entropy is the smallest extremum. The island is a disconnected component of the entanglement wedge that lies in the gravitating region, namely the black hole interior. The concrete computation uses the two-dimensional dilaton-gravity model known as JT gravity, where the dilaton value plays the role of the area, together with the thermal CFT entropy formula and the Weyl-factor correction; at late times the two-interval entropy is approximated as the sum of two single-interval entropies, which corresponds to identity-operator dominance in the four-point twist correlator.

What would settle it

Compute the exact two-interval entanglement entropy in a free-fermion conformal field theory with partially transmitting boundary conditions and check whether, at late times, the identity operator dominates the four-point twist correlator; if a different conformal block dominates, the plateau at $2S_{\mathrm{BH}}(\beta)$ would shift.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the entropy of Hawking radiation in an eternal black hole obeys a Page curve: it rises linearly at early times and then flattens at $2S_{\mathrm{BH}}(\beta)$, twice the Bekenstein-Hawking entropy. The mechanism is a competition between two saddle points of the generalized entropy. The no-island extremum gives $S_{\mathrm{rad}}(t)\simeq \frac{c}{3}\frac{2\pi t}{\beta}$ at early times (Eq. 76), while at late times the island configuration, with the quantum extremal surface near the horizon, solves the extremization condition (Eq. 78) and yields the plateau (Eq. 82). The paper presents this as a derivation rather than a postulate, because the island saddle is itself justified by the boundary replica path integral with replica wormholes.

Load-bearing premise

The whole late-time answer depends on the assumption that the entropy of the two radiation intervals is just the sum of their individual entropies, with all cross-correlations ignored; if that factorization is wrong, the predicted plateau at $2S_{\mathrm{BH}}(\beta)$ is not established.

Editorial extensions

If this is right

  • Hawking radiation from an eternal black hole has a Page curve: its entropy rises as $S_{\mathrm{rad}}(t)\simeq \frac{c}{3}\frac{2\pi t}{\beta}$ and then saturates at $2S_{\mathrm{BH}}(\beta)$, so the information paradox is resolved in this model.
  • At late times the entanglement wedge of the radiation reaches into the black hole interior, meaning bulk operators in the island region are encoded in the radiation.
  • The quantum extremal surface prescription is a derived consequence of a boundary replica path integral with replica wormholes, not a fundamental postulate.
  • The spectral paradox disappears because the continuous-spectrum argument is only valid in the strict $N=\infty$ limit, and wormhole contributions repair the late-time behavior.

Reading between the lines

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

  • Editorial inference: if the plateau at $2S_{\mathrm{BH}}$ is robust, the same phase transition should appear in Rényi entropies at slightly different times, which is a concrete check through replica computations.
  • Editorial inference: the early-time slope $\frac{c\pi}{3\beta}$ is fixed purely by the thermal CFT formula, so a measurement of the early-time entropy growth in any reservoir-coupled two-dimensional CFT would test the model without invoking gravity.
  • Editorial inference: the factorization assumption could be tested in a free-fermion reservoir with partially transmitting boundary conditions, where exact two-interval entropies are available; if the identity conformal block does not dominate at late times, the saturation value would shift.
  • Editorial inference: the lectures imply that islands are not a sign of nonlocal information transport but an artifact of evaluating a boundary quantity by gravitational saddle points, so analogous island contributions should appear for other boundary-defined quantities.
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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

0 major / 4 minor

Summary. These lecture notes present a pedagogical derivation of Hawking radiation from the KMS property of thermal correlators, a statement of the entropy paradox for the eternal black hole, and a step-by-step QES/island computation of the Page curve in JT gravity coupled to a CFT and flat-space reservoirs. The central calculation is the entropy of the radiation region R1∪R2: at early times it grows linearly as S_rad(t)≈(c/3)(2πt/β) (Eq. (76)), and at late times a disconnected island entanglement wedge dominates, giving saturation S_rad(t)≈2S_BH(β) (Eq. (82)). The extremization leading to the QES condition (Eq. (78)) and the slow-emission solution (Eq. (81)) is worked out explicitly, and the manuscript consistently attributes the derivation to [29,30,34]. The paper makes no claim to new research results and is framed throughout as a write-up of lectures.

Significance. If the exposition is accurate, the notes have real pedagogical value: the QES computation is unusually complete and self-contained, the KMS-based derivation of Hawking radiation is concise and clearly connected to the Unruh effect, and the limitations of the argument are openly acknowledged (the 'very roughly' collapse derivation in §2.5, the factorization approximation in §3.5.2, and the slow-emission limit in Eq. (79)). I checked the algebra leading from the generalized entropy (Eq. (77)) to the QES condition (Eq. (78)) and to the entropy estimate in Eq. (82); it is internally consistent. The manuscript is not a research contribution but a competent pedagogical review of a known derivation, with explicit references to the original literature. For a journal that publishes lecture-style reviews, this is a publishable contribution after minor corrections.

minor comments (4)
  1. [§3.2, Eq. (39)] The mapping for x_- is printed as x_- = -e^{y_-}, but the correct embedding of the thermal CFT on the right Rindler wedge is x_- = -e^{-y_-}. The table in Eq. (42) and all later formulas use the correct relation, so no derived result is affected, but the displayed equation is inconsistent and should be fixed.
  2. [§3.5.2, footnote 12 and text before Eq. (77)] The approximation that the two-interval entropy is the sum of single-interval entropies is the most delicate step in the saturation argument. The footnote mentions OPE dominance, but a student reader would benefit from one explicit sentence stating that the intervals are separated by a large Lorentzian time 2t, making the relevant cross ratio exponentially small and the identity-channel contribution dominant.
  3. [Introduction and §2.4] There are several typographical errors: 'A rotating black holes' and 'It's non-rotating limit' in the Introduction, and 'black hold' instead of 'black hole' in §2.4. A careful proofreading pass is recommended.
  4. [§2.5] The derivation of Hawking radiation from the Fredenhagen-Haag argument is explicitly schematic ('very roughly'). Since the paper is pedagogical, adding a short remark that the full rigorous argument is presented in [31] and that the exponential redshift encoded in Eq. (28) is the physical origin of the KMS property would make the logical status of the derivation clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lecture derives the Page curve from the QES prescription and standard CFT entropy formulas, with the late-time factorization assumption explicitly flagged rather than hidden.

full rationale

This paper is a pedagogical exposition, not a new derivation, and its central results are reproduced in the text rather than imported by citation. The early-time linear growth, Eq. (76), follows directly from the 2d CFT entanglement entropy formula derived earlier in Section 3.2, with no fitted parameters. The late-time saturation value, Eq. (82), is obtained by extremizing the explicit generalized entropy functional in Eq. (77), solving the QES condition in the stated slow-emission limit, and evaluating the entropy at the extremum; this is a self-contained asymptotic calculation. The only delicate input, the replacement of the two-interval entropy by a sum of single-interval entropies in Section 3.5.2, is explicitly announced as an approximation and justified in footnote 12 by identity-OPE dominance, so it is an assumption rather than a circular step. The QES prescription itself is cited from external work [28] and the path-integral justification from [24,25]; the paper explicitly states that the island prescription is not fundamental. The co-authored references [34] and [61] are attributions for the specific model and computation, but since the lecture re-derives the equations, the self-citations are not load-bearing in the argument. No step reduces by construction to its inputs, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new entities or fitted constants are introduced. The calculations use model inputs such as phi_r, beta, c, and b, plus standard approximations. All nontrivial ingredients are imported from prior literature, which is appropriate for a review.

assumptions (4)
  • domain assumption The QES prescription with disconnected entanglement wedges computes the von Neumann entropy of the radiation region.
    Invoked in Section 3.1, Eq. (31), and used throughout Section 3.5; the lectures do not derive it from the replica path integral, citing [28, 29, 30, 34].
  • domain assumption At late times the two-interval entropy factorizes into a sum of single-interval entropies.
    Section 3.5.2 states this approximation; it rests on the identity operator dominating the OPE of twist operators, and the full four-point function is not computed.
  • domain assumption The slow emission and absorption limit phi_r/beta >> c.
    Eq. (79) is used to convert the QES condition into the exponential form of Eq. (80) and to locate the island near the horizon.
  • domain assumption The state of quantum fields near the horizon is smooth and has universal short-distance behavior.
    Section 2.5 uses this in the Hawking-radiation derivation; the paper itself notes the AMPS and firewall challenge to this assumption.

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Cite this review

Pith. "Pith review of Lectures on Quantum Extremal Surfaces and the Page Curve." pith.science (2026). https://pith.science/paper/UWADWBAM

@misc{pith2026250201933,
  author       = {Pith},
  title        = {Pith review of: Lectures on Quantum Extremal Surfaces and the Page Curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWADWBAM}},
  note         = {Machine review of arXiv:2502.01933}
}
read the original abstract

This article is a write-up of pedagogical lectures delivered at the Asia Pacific Center for Theoretical Physics online winter school held in January 2021, and also as part of the Quantum Information, Quantum Field Theory and Gravity program held at ICTS, Bengaluru, in August 2024. The topics covered include a brief derivation of Hawking radiation from the perspective of correlation functions, a description of entropy paradoxes in the eternal black hole, and details of the associated entanglement island computations in two-dimensional models.

Figures

Figures reproduced from arXiv: 2502.01933 by the authors.

Figure 1
Figure 1. The KMS property (for identical scalar Grassmann-even operators). The thermal [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. The right Rindler wedge is the region R` obeying the inequality x 1 ą |x 0 |. A curve of constant ξ is depicted, it is a timelike trajectory with constant proper acceleration. We consider the two-point function of an operator in the Minkowski vacuum state, with the two-points separated purely in the η direction. This correlator obeys the KMS property with β “ 2π and with respect to the generator of η-translations. 2… view at source ↗
Figure 3
Figure 3. The geometry of a black hole formed from collapse. The red shaded region rep [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The left figure shows the Penrose diagram [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: The left figure depicts an SYK quantum mechanical system, with the time direction [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]
Figure 6
Figure 6. Figure 6: The left figure shows an SYK model coupled to a reservoir which is a semi-infinite [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]
Figure 7
Figure 7. Figure 7: Left: We take two copies of the SYK model, each coupled to a reservoir. The [PITH_FULL_IMAGE:figures/full_fig_p030_7.png]
Figure 8
Figure 8. Figure 8: The entanglement wedge of the radiation region contains a disconnected region [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]

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Forward citations

Cited by 2 Pith papers

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  2. JT gravity and deformed CFTs

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    Deformed CFTs on a strip with a stretched-horizon conformal boundary are proposed as UV completions of pure and matter-coupled JT gravity, reproducing its entropy and a Page-curve-like saturation.

Reference graph

Works this paper leans on

75 extracted references · 25 canonical work pages · cited by 2 Pith papers

  1. [1]

    Misner, K

    Charles W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973) ISBN 978-0-7167-0344-0, 978-0-691-17779-3 37

    work page 1973

  2. [2]

    Wald, General Relativity (Chicago Univ

    Robert M. Wald, General Relativity (Chicago Univ. Pr., Chicago, USA, 1984)

    1984

  3. [3]

    The Confrontation between General Relativity and Experiment,

    Clifford M. Will, “The Confrontation between General Relativity and Experiment,” Living Rev. Rel. 17, 4 (2014), arXiv:1403.7377 [gr-qc]

    arXiv 2014

  4. [4]

    On Continued gravitational contraction,

    J. R. Oppenheimer and H. Snyder, “On Continued gravitational contraction,” Phys. Rev. 56, 455–459 (1939)

    work page 1939

  5. [5]

    Observation of Gravitational Waves from a Binary Black Hole Merger,

    B. P. Abbott et al. (LIGO Scientific, Virgo), “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    arXiv 2016

  6. [6]

    First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,

    Kazunori Akiyama et al. (Event Horizon Telescope), “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. Lett. 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

    arXiv 2019

  7. [7]

    Polchinski, String theory

    J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2007) ISBN 978-0- 511-25227-3, 978-0-521-67227-6, 978-0-521-63303-1

    work page 2007

  8. [8]

    Polchinski, String theory

    J. Polchinski, String theory. Vol. 2: Superstring theory and beyond , Cambridge Mono- graphs on Mathematical Physics (Cambridge University Press, 2007) ISBN 978-0-511- 25228-0, 978-0-521-63304-8, 978-0-521-67228-3

    work page 2007

Show all 75 references
  1. [9]

    Black hole explosions,

    S. W. Hawking, “Black hole explosions,” Nature 248, 30–31 (1974)

    1974

  2. [10]

    Particle Creation by Black Holes,

    S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43, 199–220 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)]

    1975

  3. [11]

    Black holes and entropy,

    Jacob D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7, 2333–2346 (1973)

    1973

  4. [12]

    The Four laws of black hole me- chanics,

    James M. Bardeen, B. Carter, and S. W. Hawking, “The Four laws of black hole me- chanics,” Commun. Math. Phys. 31, 161–170 (1973)

    1973

  5. [13]

    Action Integrals and Partition Functions in Quan- tum Gravity,

    G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quan- tum Gravity,” Phys. Rev. D 15, 2752–2756 (1977)

    1977

  6. [14]

    Microscopic origin of the Bekenstein-Hawking entropy,

    Andrew Strominger and Cumrun Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Phys. Lett. B 379, 99–104 (1996), arXiv:hep-th/9601029 38

    1996 arXiv

  7. [15]

    Black Hole Entropy Function, Attractors and Precision Counting of Mi- crostates,

    Ashoke Sen, “Black Hole Entropy Function, Attractors and Precision Counting of Mi- crostates,” Gen. Rel. Grav. 40, 2249–2431 (2008), arXiv:0708.1270 [hep-th]

    2008 arXiv

  8. [16]

    Breakdown of Predictability in Gravitational Collapse,

    S. W. Hawking, “Breakdown of Predictability in Gravitational Collapse,” Phys. Rev. D 14, 2460–2473 (1976)

    1976

  9. [17]

    The Large N limit of superconformal field theories and super- gravity,

    Juan Martin Maldacena, “The Large N limit of superconformal field theories and super- gravity,” Adv. Theor. Math. Phys. 2, 231–252 (1998), arXiv:hep-th/9711200

    1998 arXiv

  10. [18]

    Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,

    Edward Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,” Adv. Theor. Math. Phys. 2, 505–532 (1998), arXiv:hep-th/9803131

    1998 arXiv

  11. [19]

    Eternal black holes in anti-de Sitter,

    Juan Martin Maldacena, “Eternal black holes in anti-de Sitter,” JHEP 04, 021 (2003), arXiv:hep-th/0106112

    2003 arXiv

  12. [20]

    Average entropy of a subsystem,

    Don N. Page, “Average entropy of a subsystem,” Phys. Rev. Lett. 71, 1291–1294 (1993), arXiv:gr-qc/9305007

    1993 arXiv

  13. [21]

    Information in black hole radiation,

    Don N. Page, “Information in black hole radiation,” Phys. Rev. Lett. 71, 3743–3746 (1993), arXiv:hep-th/9306083

    1993 arXiv

  14. [22]

    Time Dependence of Hawking Radiation Entropy,

    Don N. Page, “Time Dependence of Hawking Radiation Entropy,” JCAP 09, 028 (2013), arXiv:1301.4995 [hep-th]

    2013 arXiv

  15. [23]

    Generalized gravitational entropy,

    Aitor Lewkowycz and Juan Maldacena, “Generalized gravitational entropy,” JHEP 08, 090 (2013), arXiv:1304.4926 [hep-th]

    2013 arXiv

  16. [24]

    Replica wormholes and the black hole interior,

    Geoff Penington, Stephen H. Shenker, Douglas Stanford, and Zhenbin Yang, “Replica wormholes and the black hole interior,” JHEP 03, 205 (2022), arXiv:1911.11977 [hep-th]

    2022 arXiv

  17. [25]

    Replica Wormholes and the Entropy of Hawking Radiation,

    Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhos- sein Tajdini, “Replica Wormholes and the Entropy of Hawking Radiation,” JHEP 05, 013 (2020), arXiv:1911.12333 [hep-th]

    2020 arXiv

  18. [26]

    Holographic derivation of entanglement entropy from AdS/CFT,

    Shinsei Ryu and Tadashi Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006), arXiv:hep-th/0603001

    2006 arXiv

  19. [27]

    Quantum corrections to holographic entanglement entropy,

    Thomas Faulkner, Aitor Lewkowycz, and Juan Maldacena, “Quantum corrections to holographic entanglement entropy,” JHEP 11, 074 (2013), arXiv:1307.2892 [hep-th] 39

    2013 arXiv

  20. [28]

    Quantum Extremal Surfaces: Holographic Entan- glement Entropy beyond the Classical Regime,

    Netta Engelhardt and Aron C. Wall, “Quantum Extremal Surfaces: Holographic Entan- glement Entropy beyond the Classical Regime,” JHEP 01, 073 (2015), arXiv:1408.3203 [hep-th]

    2015 arXiv

  21. [29]

    Entanglement Wedge Reconstruction and the Information Para- dox,

    Geoffrey Penington, “Entanglement Wedge Reconstruction and the Information Para- dox,” JHEP 09, 002 (2020), arXiv:1905.08255 [hep-th]

    2020 arXiv

  22. [30]

    The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,

    Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” JHEP 12, 063 (2019), arXiv:1905.08762 [hep-th]

    2019 arXiv

  23. [31]

    On the Derivation of Hawking Radiation Asso- ciated With the Formation of a Black Hole,

    Klaus Fredenhagen and Rudolf Haag, “On the Derivation of Hawking Radiation Asso- ciated With the Formation of a Black Hole,” Commun. Math. Phys. 127, 273 (1990)

    1990

  24. [32]

    Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space,

    Juan Maldacena, Douglas Stanford, and Zhenbin Yang, “Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space,” PTEP 2016, 12C104 (2016), arXiv:1606.01857 [hep-th]

    2016 arXiv

  25. [33]

    What is the dual of two entangled CFTs?

    Samir D. Mathur, “What is the dual of two entangled CFTs?.” (2 2014), arXiv:1402.6378 [hep-th]

    2014 arXiv

  26. [34]

    Islands outside the horizon,

    Ahmed Almheiri, Raghu Mahajan, and Juan Maldacena, “Islands outside the horizon,” (10 2019), arXiv:1910.11077 [hep-th]

    2019 arXiv

  27. [35]

    The entropy of Hawking radiation,

    Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhos- sein Tajdini, “The entropy of Hawking radiation,” Rev. Mod. Phys. 93, 035002 (2021), arXiv:2006.06872 [hep-th]

    2021 arXiv

  28. [36]

    Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems,

    Ryogo Kubo, “Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems,” Journal of the physical society of Japan 12, 570–586 (1957)

    1957

  29. [37]

    Theory of many-particle systems. i,

    Paul C Martin and Julian Schwinger, “Theory of many-particle systems. i,” Physical Review 115, 1342 (1959)

    1959

  30. [38]

    KMS Conditions and Local Thermodynamical Stability of Quantum Lattice Systems,

    H. Araki and G. L. Sewell, “KMS Conditions and Local Thermodynamical Stability of Quantum Lattice Systems,” Commun. Math. Phys. 52, 103–109 (1977) 40

    1977

  31. [39]

    Kms conditions and local thermodynamical stability of quantum lattice systems. ii,

    Geoffrey L Sewell, “Kms conditions and local thermodynamical stability of quantum lattice systems. ii,” Communications in Mathematical Physics 55, 53–61 (1977)

    1977

  32. [40]

    Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking- Unruh Effect in Rindler Manifold of Arbitrary Dimension,

    Shin Takagi, “Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking- Unruh Effect in Rindler Manifold of Arbitrary Dimension,” Prog. Theor. Phys. Suppl. 88, 1–142 (1986)

    1986

  33. [41]

    Kruskal Space and the Uniformly Accelerated Frame,

    W. Rindler, “Kruskal Space and the Uniformly Accelerated Frame,” Am. J. Phys. 34, 1174 (1966)

    1966

  34. [42]

    Notes on black hole evaporation,

    W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D 14, 870 (1976)

    1976

  35. [43]

    On the Duality Condition for Quantum Fields,

    J. J Bisognano and E. H. Wichmann, “On the Duality Condition for Quantum Fields,” J. Math. Phys. 17, 303–321 (1976)

    1976

  36. [44]

    Quantum Field Theory, Horizons and Thermodynamics,

    D. W. Sciama, P. Candelas, and D. Deutsch, “Quantum Field Theory, Horizons and Thermodynamics,” Adv. Phys. 30, 327–366 (1981)

    1981

  37. [45]

    APS Medal for Exceptional Achievement in Research: Invited arti- cle on entanglement properties of quantum field theory,

    Edward Witten, “APS Medal for Exceptional Achievement in Research: Invited arti- cle on entanglement properties of quantum field theory,” Rev. Mod. Phys. 90, 045003 (2018), arXiv:1803.04993 [hep-th]

    2018 arXiv

  38. [46]

    A semiclassical ramp in SYK and in gravity,

    Phil Saad, Stephen H. Shenker, and Douglas Stanford, “A semiclassical ramp in SYK and in gravity,” (6 2018), arXiv:1806.06840 [hep-th]

    2018 arXiv

  39. [47]

    Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity,

    Phil Saad, “Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity,” (10 2019), arXiv:1910.10311 [hep-th]

    2019 arXiv

  40. [48]

    On Quantum Field Theory in Gravi- tational Background,

    Rudolf Haag, Heide Narnhofer, and Ulrich Stein, “On Quantum Field Theory in Gravi- tational Background,” Commun. Math. Phys. 94, 219 (1984)

    1984

  41. [49]

    Generally Covariant Quantum Field Theory and Scaling Limits,

    Klaus Fredenhagen and Rudolf Haag, “Generally Covariant Quantum Field Theory and Scaling Limits,” Commun. Math. Phys. 108, 91 (1987)

    1987

  42. [50]

    Why does quantum field theory in curved spacetime make sense? And what happens to the algebra of observables in the thermodynamic limit?

    Edward Witten, “Why does quantum field theory in curved spacetime make sense? And what happens to the algebra of observables in the thermodynamic limit?.” (2022) arXiv:2112.11614 [hep-th] 41

    2022 arXiv

  43. [51]

    The Information paradox: A Pedagogical introduction,

    Samir D. Mathur, “The Information paradox: A Pedagogical introduction,” Class. Quant. Grav. 26, 224001 (2009), arXiv:0909.1038 [hep-th]

    2009 arXiv

  44. [52]

    Black Holes: Complementarity or Firewalls?

    Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully, “Black Holes: Complementarity or Firewalls?.” JHEP 02, 062 (2013), arXiv:1207.3123 [hep-th]

    2013 arXiv

  45. [53]

    Gravitational vacuum polarization. 4: Energy conditions in the Unruh vacuum,

    Matt Visser, “Gravitational vacuum polarization. 4: Energy conditions in the Unruh vacuum,” Phys. Rev. D 56, 936–952 (1997), arXiv:gr-qc/9703001

    1997 arXiv

  46. [54]

    Ingoing Eddington- Finkelstein Metric of an Evaporating Black Hole,

    Shohreh Abdolrahimi, Don N. Page, and Christos Tzounis, “Ingoing Eddington- Finkelstein Metric of an Evaporating Black Hole,” Phys. Rev. D 100, 124038 (2019), arXiv:1607.05280 [hep-th]

    2019 arXiv

  47. [55]

    Evanescent black holes,

    Curtis G. Callan, Jr., Steven B. Giddings, Jeffrey A. Harvey, and Andrew Strominger, “Evanescent black holes,” Phys. Rev. D 45, R1005 (1992), arXiv:hep-th/9111056

    1992 arXiv

  48. [56]

    Les Houches lectures on black holes,

    Andrew Strominger, “Les Houches lectures on black holes,” in NATO Advanced Study Institute: Les Houches Summer School, Session 62: Fluctuating Geometries in Statistical Mechanics and Field Theory (1994) arXiv:hep-th/9501071

    1994 arXiv

  49. [57]

    Time Evolution of Entanglement Entropy from Black Hole Interiors,

    Thomas Hartman and Juan Maldacena, “Time Evolution of Entanglement Entropy from Black Hole Interiors,” JHEP 05, 014 (2013), arXiv:1303.1080 [hep-th]

    2013 arXiv

  50. [58]

    Evolution of entanglement entropy in one- dimensional systems,

    Pasquale Calabrese and John L. Cardy, “Evolution of entanglement entropy in one- dimensional systems,” J. Stat. Mech. 0504, P04010 (2005), arXiv:cond-mat/0503393

    2005 arXiv

  51. [59]

    Lessons from the information paradox,

    Suvrat Raju, “Lessons from the information paradox,” Phys. Rept. 943, 1–80 (2022), arXiv:2012.05770 [hep-th]

    2022 arXiv

  52. [60]

    Generalised Hydrodynamics description of the Page curve-like dynamics of a freely expanding fermionic gas,

    Madhumita Saha, Manas Kulkarni, and Abhishek Dhar, “Generalised Hydrodynamics description of the Page curve-like dynamics of a freely expanding fermionic gas,” (2 2024), arXiv:2402.18422 [quant-ph]

    2024 arXiv

  53. [61]

    The Page curve of Hawking radiation from semiclassical geometry,

    Ahmed Almheiri, Raghu Mahajan, Juan Maldacena, and Ying Zhao, “The Page curve of Hawking radiation from semiclassical geometry,” JHEP03, 149 (2020), arXiv:1908.10996 [hep-th] 42

    2020 arXiv

  54. [62]

    Entropy, Extremality, Euclidean Variations, and the Equations of Motion,

    Xi Dong and Aitor Lewkowycz, “Entropy, Extremality, Euclidean Variations, and the Equations of Motion,” JHEP 01, 081 (2018), arXiv:1705.08453 [hep-th]

    2018 arXiv

  55. [63]

    Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,

    Xi Dong, Daniel Harlow, and Aron C. Wall, “Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Phys. Rev. Lett. 117, 021601 (2016), arXiv:1601.05416 [hep-th]

    2016 arXiv

  56. [64]

    Entanglement Wedge Reconstruction via Universal Recovery Chan- nels,

    Jordan Cotler, Patrick Hayden, Geoffrey Penington, Grant Salton, Brian Swingle, and Michael Walter, “Entanglement Wedge Reconstruction via Universal Recovery Chan- nels,” Phys. Rev. X 9, 031011 (2019), arXiv:1704.05839 [hep-th]

    2019 arXiv

  57. [65]

    Entanglement Wedge Recon- struction using the Petz Map,

    Chi-Fang Chen, Geoffrey Penington, and Grant Salton, “Entanglement Wedge Recon- struction using the Petz Map,” JHEP 01, 168 (2020), arXiv:1902.02844 [hep-th]

    2020 arXiv

  58. [66]

    Geometric and renormalized entropy in conformal field theory,

    Christoph Holzhey, Finn Larsen, and Frank Wilczek, “Geometric and renormalized entropy in conformal field theory,” Nucl. Phys. B 424, 443–467 (1994), arXiv:hep- th/9403108

    1994

  59. [67]

    Entanglement entropy and quantum field the- ory,

    Pasquale Calabrese and John L. Cardy, “Entanglement entropy and quantum field the- ory,” J. Stat. Mech. 0406, P06002 (2004), arXiv:hep-th/0405152

    2004 arXiv

  60. [68]

    Entanglement entropy and conformal field the- ory,

    Pasquale Calabrese and John Cardy, “Entanglement entropy and conformal field the- ory,” J. Phys. A 42, 504005 (2009), arXiv:0905.4013 [cond-mat.stat-mech]

    2009 arXiv

  61. [69]

    Lower Dimensional Gravity,

    R. Jackiw, “Lower Dimensional Gravity,” Nucl. Phys. B 252, 343–356 (1985)

    1985

  62. [70]

    Gravitation and Hamiltonian Structure in Two Space-Time Dimen- sions,

    C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimen- sions,” Phys. Lett. B 126, 41–45 (1983)

    1983

  63. [71]

    Gapless spin fluid ground state in a random, quantum Heisenberg magnet,

    Subir Sachdev and Jinwu Ye, “Gapless spin fluid ground state in a random, quantum Heisenberg magnet,” Phys. Rev. Lett. 70, 3339 (1993), arXiv:cond-mat/9212030

    1993 arXiv

  64. [72]

    Remarks on the Sachdev-Ye-Kitaev model,

    Juan Maldacena and Douglas Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Phys. Rev. D 94, 106002 (2016), arXiv:1604.07818 [hep-th]

    2016 arXiv

  65. [73]

    The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual,

    Alexei Kitaev and S. Josephine Suh, “The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual,” JHEP 05, 183 (2018), arXiv:1711.08467 [hep-th] 43

    2018 arXiv

  66. [74]

    Free fermion entanglement with a semitransparent interface: the effect of graybody factors on entanglement is- lands,

    Jorrit Kruthoff, Raghu Mahajan, and Chitraang Murdia, “Free fermion entanglement with a semitransparent interface: the effect of graybody factors on entanglement is- lands,” SciPost Phys. 11, 063 (2021), arXiv:2106.10287 [hep-th]

    2021 arXiv

  67. [75]

    Black Holes and Random Matrices,

    Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, Joseph Polchinski, Phil Saad, Stephen H. Shenker, Douglas Stanford, Alexandre Streicher, and Masaki Tezuka, “Black Holes and Random Matrices,” JHEP 05, 118 (2017), [Erratum: JHEP 09, 002 (2018)], arXiv:1611.04650 [hep-th] 44

    2017 arXiv

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Reviewed August 9, 2026 · model on record in the stance chip above.