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arxiv: 2509.05412 · v3 · pith:DDCFSF6Onew · submitted 2025-09-05 · ✦ hep-th · gr-qc

Gravitational Hilbert spaces: invariant and co-invariant states, inner products, gauge-fixing, and BRST

Jesse Held , Henry Maxfield This is my paper

Pith reviewed 2026-05-18 18:34 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords gravitational Hilbert spacegroup averagingBRST formalismKlein-Gordon inner productWheeler-DeWitt equationgauge fixingphysical statesmini-superspace
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0 comments X

The pith

Group averaging over constraints yields the positive-definite inner product for physical states in gravity.

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

Theories of gravity feature Hamiltonian constraints that generate gauge transformations, making the definition of a physical Hilbert space and its inner product subtle. States can be defined either by requiring annihilation by the constraints, as in the Wheeler-DeWitt equation, or by imposing equivalence relations among wavefunctions, with an inner product relating the two pictures. The paper advocates that group averaging over the gauge group produces the correct physical inner product. The commonly used Klein-Gordon inner product fails to be positive definite because it corresponds to a bad gauge choice, yet it agrees with group averaging in cases without that defect. These relations are systematized within the BRST/BFV formalism, which generates families of equivalent inner products and clarifies the Hilbert-space content of gravitational path integrals, including in semi-classical regimes.

Core claim

A physical Hilbert space arises either by imposing the constraint operators on states or by quotienting wavefunctions under gauge equivalence generated by those constraints; an inner product mediates between the two constructions. Group averaging implements the correct physical inner product by integrating over the gauge group, producing a positive-definite result. The Klein-Gordon inner product is recovered from a particular gauge fixing and coincides with the group-averaged result precisely when that gauge is admissible, otherwise losing positivity. The BRST/BFV formalism embeds all these choices and yields additional physically equivalent inner products, such as those associated with a 1D

What carries the argument

The group averaging procedure, which projects states onto the physical subspace by integrating over the action of the constraint-generated gauge group and thereby defines the physical inner product.

If this is right

  • Multiple inner products, including those from maximal-volume and Gaussian-averaged gauges, are shown to be physically equivalent within the BRST framework.
  • The same group-averaging inner product supplies the Hilbert-space interpretation of certain gravitational path integrals.
  • In the semi-classical regime the construction remains consistent and extends to include non-perturbative gravitational effects.
  • The equivalence between invariant and co-invariant formulations clarifies how canonical and sum-over-histories approaches describe the same physical states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same logic may supply a route to well-defined inner products in higher-dimensional or full quantum-gravity models once the constraints are properly isolated.
  • Observable operators that commute with the constraints can be defined unambiguously once the group-averaged inner product is adopted.
  • Numerical or analytic checks in other constrained systems, such as gauge theories or cosmology with matter, could test whether group averaging systematically outperforms alternate gauge choices.

Load-bearing premise

The conceptual points can be illustrated completely within one-dimensional mini-superspace models.

What would settle it

An explicit computation in a solvable mini-superspace model that produces a different inner-product norm for the same physical state when using group averaging versus direct solution of the constraint equation.

read the original abstract

Hilbert spaces in theories of gravity are notoriously subtle due to the Hamiltonian constraints, particularly regarding the inner product. To demystify this subject, we review and extend a collection of ideas in canonical gravity, and connect to the sum-over-histories approach by clarifying the Hilbert space interpretation of various gravitational path integrals. We use one-dimensional (or mini-superspace) models as the simplest context to exemplify the conceptual ideas. We emphasise that a physical Hilbert space can be defined either by requiring states to be annihilated by constraint operators (e.g., the Wheeler-DeWitt equation) or by equivalence relations between wavefunctions, and explain that these two approaches are related by an inner product. We advocate that the group averaging procedure constructs the correct physical inner product. The Klein-Gordon inner product is not positive-definite, which we explain as arising from a bad gauge choice; nonetheless, it agrees with group averaging when such a problem is absent. These concepts are all embedded in the BRST/BFV formalism, which provides a systematic way to construct these and other physically equivalent inner products (e.g., from maximal-volume gauge and Gaussian averaged gauges). Finally we discuss the application of these ideas in the semi-classical approximation, including non-perturbative gravitational effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper reviews and extends ideas on defining physical Hilbert spaces for gravitational theories subject to Hamiltonian constraints. It connects constraint annihilation (e.g., Wheeler-DeWitt equation) with equivalence relations on wavefunctions via an inner product, advocates the group-averaging procedure as yielding the correct positive-definite physical inner product, interprets the non-positive-definiteness of the Klein-Gordon inner product as a bad gauge choice, and embeds these constructions in the BRST/BFV formalism to obtain physically equivalent inner products (including maximal-volume and Gaussian-averaged gauges). One-dimensional mini-superspace models are used to illustrate the concepts, with additional discussion of semi-classical and non-perturbative applications and links to path-integral formulations.

Significance. If the conceptual framework and claimed equivalences hold, the manuscript provides a useful clarification of subtle issues in gravitational Hilbert spaces, bridging canonical and sum-over-histories approaches. The explicit advocacy for group averaging, the gauge-choice explanation for Klein-Gordon issues, and the systematic BRST/BFV treatment of multiple equivalent inner products are strengths that could aid future work on quantum gravity. The choice of mini-superspace models as a tractable setting for exemplifying ideas is appropriate and helps make the discussion concrete.

major comments (1)
  1. [one-dimensional models section] § on one-dimensional models and group averaging: the central claim that group averaging yields a positive-definite inner product physically equivalent to BRST constructions (maximal-volume, Gaussian-averaged) is asserted via conceptual relations between constraint annihilation and equivalence classes, yet the manuscript does not supply an explicit algebraic or numerical computation of the resulting norms for Wheeler-DeWitt solutions in a concrete 1D model. Such a worked example is needed to confirm positivity and cross-gauge agreement when the Klein-Gordon choice is problematic, as this equivalence is load-bearing for the advocated procedure.
minor comments (1)
  1. [Introduction] Notation for invariant versus co-invariant states could be introduced with a brief table or explicit definitions early in the text to aid readability for readers less familiar with the BRST/BFV literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We appreciate the positive assessment of the conceptual framework and its potential to bridge canonical and path-integral approaches. We address the major comment below.

read point-by-point responses

  1. Referee: [one-dimensional models section] § on one-dimensional models and group averaging: the central claim that group averaging yields a positive-definite inner product physically equivalent to BRST constructions (maximal-volume, Gaussian-averaged) is asserted via conceptual relations between constraint annihilation and equivalence classes, yet the manuscript does not supply an explicit algebraic or numerical computation of the resulting norms for Wheeler-DeWitt solutions in a concrete 1D model. Such a worked example is needed to confirm positivity and cross-gauge agreement when the Klein-Gordon choice is problematic, as this equivalence is load-bearing for the advocated procedure.

    Authors: We agree that an explicit algebraic computation of the norms would strengthen the presentation and make the positivity and cross-gauge equivalence more concrete. The current manuscript employs the one-dimensional models primarily to illustrate the conceptual relations between constraint annihilation, equivalence classes of wavefunctions, and the construction of inner products via group averaging and BRST. While these relations are derived generally and applied to the models, we acknowledge the absence of a specific worked example computing norms for Wheeler-DeWitt solutions. In the revised manuscript we will add such an explicit example, for instance by solving the Wheeler-DeWitt equation in a simple harmonic mini-superspace model, applying the group-averaging procedure to obtain the physical inner product, and verifying agreement with the maximal-volume and Gaussian-averaged BRST gauges, including cases where the Klein-Gordon inner product is not positive definite. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained against external literature.

full rationale

The paper reviews established BRST/BFV and group-averaging procedures from prior literature to connect constraint annihilation with equivalence classes of wavefunctions via an inner product. Mini-superspace models serve only to exemplify these conceptual relations rather than to fit parameters or derive the central claims by construction. No equations reduce the advocated physical inner product (or its positivity) to a self-referential definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The Klein-Gordon inner product is presented as agreeing with group averaging except in bad gauges, with the framework embedded in standard BRST formalism; all load-bearing steps remain independent of the present paper's own inputs and are externally falsifiable in the cited literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper draws on standard frameworks of constrained Hamiltonian systems and BRST quantization without introducing new free parameters or postulated entities; its contribution is organizational and connective rather than foundational.

axioms (2)
  • standard math The BRST/BFV formalism systematically constructs physically equivalent inner products for constrained systems.
    Invoked as the embedding structure that generates group-averaged, maximal-volume, and Gaussian gauges.
  • domain assumption Mini-superspace models capture the essential conceptual features of full gravitational constraints and gauge fixing.
    Stated explicitly as the simplest context used to exemplify the ideas.

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Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages · 24 internal anchors

  1. [1]

    Replica Wormholes and the Entropy of Hawking Radiation

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini,Replica Wormholes and the Entropy of Hawking Radiation,JHEP05(2020) 013, [1911.12333]. – 93 –

    work page internal anchor Pith review arXiv 2020

  2. [2]

    Replica wormholes and the black hole interior

    G. Penington, S. H. Shenker, D. Stanford and Z. Yang,Replica wormholes and the black hole interior,JHEP03(2022) 205, [1911.11977]

    work page internal anchor Pith review arXiv 2022

  3. [3]

    P. Saad, S. H. Shenker and D. Stanford,JT gravity as a matrix integral,1903.11115

    work page internal anchor Pith review Pith/arXiv arXiv 1903

  4. [4]

    L. V. Iliesiu, S. Murthy and G. J. Turiaci,Black hole microstate counting from the gravitational path integral,2209.13602

    work page arXiv

  5. [5]

    Held and H

    J. Held and H. Maxfield,The Hilbert space of de Sitter JT: a case study for canonical methods in quantum gravity,2410.14824

    work page arXiv

  6. [6]

    P. A. M. Dirac,Generalized Hamiltonian dynamics,Can. J. Math.2(1950) 129–148

    work page 1950

  7. [7]

    B. S. DeWitt,Quantum Theory of Gravity. 1. The Canonical Theory,Phys. Rev.160(1967) 1113–1148

    work page 1967

  8. [8]

    An Algebra of Observables for de Sitter Space

    V. Chandrasekaran, R. Longo, G. Penington and E. Witten,An algebra of observables for de Sitter space,JHEP02(2023) 082, [2206.10780]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  9. [9]

    Quantization of diffeomorphism invariant theories of connections with local degrees of freedom

    A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann,Quantization of diffeomorphism invariant theories of connections with local degrees of freedom,J. Math. Phys.36(1995) 6456–6493, [gr-qc/9504018]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  10. [10]

    On the Generality of Refined Algebraic Quantization

    D. Giulini and D. Marolf,On the generality of refined algebraic quantization,Class. Quant. Grav.16(1999) 2479–2488, [gr-qc/9812024]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  11. [11]

    D. Marolf,Group averaging and refined algebraic quantization: Where are we now?, in9th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG 9), 7, 2000.gr-qc/0011112

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  12. [12]

    R. M. Wald,A Proposal for solving the ’problem of time’ in canonical quantum gravity,Phys. Rev. D48(1993) R2377–R2381, [gr-qc/9305024]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  13. [13]

    Quantum Observables and Recollapsing Dynamics

    D. Marolf,Quantum observables and recollapsing dynamics,Class. Quant. Grav.12(1995) 1199–1220, [gr-qc/9404053]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  14. [14]

    Higuchi,Quantum linearization instabilities of de Sitter space-time

    A. Higuchi,Quantum linearization instabilities of de Sitter space-time. 1,Class. Quant. Grav.8(1991) 1961–1981

    work page 1991

  15. [15]

    Witten,A note on the canonical formalism for gravity,Adv

    E. Witten,A note on the canonical formalism for gravity,Adv. Theor. Math. Phys.27 (2023) 311–380, [2212.08270]

    work page arXiv 2023

  16. [16]

    J. J. Halliwell and J. B. Hartle,Wave functions constructed from an invariant sum over histories satisfy constraints,Phys. Rev. D43(1991) 1170–1194

    work page 1991

  17. [17]

    Araujo-Regado, R

    G. Araujo-Regado, R. Khan and A. C. Wall,Cauchy slice holography: a new AdS/CFT dictionary,JHEP03(2023) 026, [2204.00591]

    work page arXiv 2023

  18. [18]

    Marnelius and M

    R. Marnelius and M. Ogren,Symmetric inner products for physical states in BRST quantization,Nucl. Phys. B351(1991) 474–490

    work page 1991

  19. [19]

    Batalin and R

    I. Batalin and R. Marnelius,Solving general gauge theories on inner product spaces,Nucl. Phys. B442(1995) 669–696, [hep-th/9501004]

    work page arXiv 1995

  20. [20]

    O. Y. Shvedov,On correspondence of BRST-BFV, Dirac and refined algebraic quantizations of constrained systems,Annals Phys.302(2002) 2–21, [hep-th/0111270]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  21. [21]

    E. S. Fradkin and G. A. Vilkovisky,QUANTIZATION OF RELATIVISTIC SYSTEMS WITH CONSTRAINTS,Phys. Lett. B55(1975) 224–226. – 94 –

    work page 1975

  22. [22]

    I. A. Batalin and G. A. Vilkovisky,Relativistic S Matrix of Dynamical Systems with Boson and Fermion Constraints,Phys. Lett. B69(1977) 309–312

    work page 1977

  23. [23]

    Henneaux,Hamiltonian Form of the Path Integral for Theories with a Gauge Freedom, Phys

    M. Henneaux,Hamiltonian Form of the Path Integral for Theories with a Gauge Freedom, Phys. Rept.126(1985) 1–66

    work page 1985

  24. [24]

    Henneaux and C

    M. Henneaux and C. Teitelboim,Quantization of Gauge Systems. Princeton University Press, 8, 1994

    work page 1994

  25. [25]

    J. J. Halliwell,Derivation of the Wheeler-De Witt Equation from a Path Integral for Minisuperspace Models,Phys. Rev. D38(1988) 2468

    work page 1988

  26. [26]

    Moss,Quantum Cosmology and the Selfobserving Universe,Ann

    I. Moss,Quantum Cosmology and the Selfobserving Universe,Ann. Inst. H. Poincare Phys. Theor.49(1988) 341–349

    work page 1988

  27. [27]

    Casali, D

    E. Casali, D. Marolf, H. Maxfield and M. Rangamani,Baby universes and worldline field theories,Class. Quant. Grav.39(2022) 134004, [2101.12221]

    work page arXiv 2022

  28. [28]

    Refined Algebraic Quantization: Systems with a single constraint

    D. Marolf,Refined algebraic quantization: Systems with a single constraint,gr-qc/9508015

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

    Linear Wave Equations on Lorentzian Manifolds

    C. B¨ ar,Linear wave equations on lorentzian manifolds,1006.2354

    work page internal anchor Pith review Pith/arXiv arXiv

  30. [30]

    The Factorization Problem in Jackiw-Teitelboim Gravity,

    D. Harlow and D. Jafferis,The Factorization Problem in Jackiw-Teitelboim Gravity,JHEP 02(2020) 177, [1804.01081]

    work page arXiv 2020

  31. [31]

    L. V. Iliesiu, A. Levine, H. W. Lin, H. Maxfield and M. Mezei,On the non-perturbative bulk Hilbert space of JT gravity,JHEP10(2024) 220, [2403.08696]

    work page arXiv 2024

  32. [32]

    R. P. Woodard,Enforcing the Wheeler-de Witt Constraint the Easy Way,Class. Quant. Grav.10(1993) 483–496

    work page 1993

  33. [33]

    A Uniqueness Theorem for Constraint Quantization

    D. Giulini and D. Marolf,A Uniqueness theorem for constraint quantization,Class. Quant. Grav.16(1999) 2489–2505, [gr-qc/9902045]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  34. [34]

    The Spectral Analysis Inner Product for Quantum Gravity

    D. Marolf,The Spectral analysis inner product for quantum gravity, in7th Marcel Grossmann Meeting on General Relativity (MG 7), pp. 851–853, 9, 1994.gr-qc/9409036

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  35. [35]

    Quantum fields in curved spacetime

    S. Hollands and R. M. Wald,Quantum fields in curved spacetime,Phys. Rept.574(2015) 1–35, [1401.2026]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  36. [36]

    R. M. Wald,Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics. Chicago Lectures in Physics. University of Chicago Press, Chicago, IL, 1995

    work page 1995

  37. [37]

    A. G. Cohen, G. W. Moore, P. C. Nelson and J. Polchinski,An Off-Shell Propagator for String Theory,Nucl. Phys. B267(1986) 143–157

    work page 1986

  38. [38]

    Polchinski,String theory

    J. Polchinski,String theory. Vol. 1: An introduction to the bosonic string. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 12, 2007, 10.1017/CBO9780511816079

    work page doi:10.1017/cbo9780511816079 2007

  39. [39]

    Erbin, J

    H. Erbin, J. Maldacena and D. Skliros,Two-Point String Amplitudes,JHEP07(2019) 139, [1906.06051]

    work page arXiv 2019

  40. [40]

    Path Integrals and Instantons in Quantum Gravity

    D. Marolf,Path integrals and instantons in quantum gravity: Minisuperspace models,Phys. Rev. D53(1996) 6979–6990, [gr-qc/9602019]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  41. [41]

    J. B. Hartle and D. Marolf,Comparing formulations of generalized quantum mechanics for reparametrization - invariant systems,Phys. Rev. D56(1997) 6247–6257, [gr-qc/9703021]. – 95 –

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  42. [42]

    Applications of a New Proposal for Solving the "Problem of Time" to Some Simple Quantum Cosmological Models

    A. Higuchi and R. M. Wald,Applications of a new proposal for solving the ’problem of time’ to some simple quantum cosmological models,Phys. Rev. D51(1995) 544–561, [gr-qc/9407038]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  43. [43]

    B. S. DeWitt,The Quantization of geometry,

    work page

  44. [44]

    S. B. Giddings, D. Marolf and J. B. Hartle,Observables in effective gravity,Phys. Rev. D74 (2006) 064018, [hep-th/0512200]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  45. [45]

    V. N. Gribov,Quantization of Nonabelian Gauge Theories,Nucl. Phys. B139(1978) 1

    work page 1978

  46. [46]

    Arisue, T

    H. Arisue, T. Fujiwara, T. Inoue and K. Ogawa,GENERALIZED SCHRODINGER REPRESENTATIONS AND ITS APPLICATION TO GAUGE THEORIES., in1981 INS Symposium on Quark and Lepton Physics, 1981

    work page 1981

  47. [47]

    Cvitanovi´ c, R

    P. Cvitanovi´ c, R. Artuso, R. Mainieri, G. Tanner and G. Vattay,Chaos: Classical and Quantum. Niels Bohr Inst., Copenhagen, 2016

    work page 2016

  48. [48]

    S. A. Hartnoll,Wheeler-DeWitt states of the AdS-Schwarzschild interior,JHEP01(2023) 066, [2208.04348]

    work page arXiv 2023

  49. [49]

    G. W. Gibbons, S. W. Hawking and M. J. Perry,Path Integrals and the Indefiniteness of the Gravitational Action,Nucl. Phys. B138(1978) 141–150

    work page 1978

  50. [50]

    Louko,A Feynman Prescription for the Hartle-hawking Proposal,Class

    J. Louko,A Feynman Prescription for the Hartle-hawking Proposal,Class. Quant. Grav.5 (1988) L181

    work page 1988

  51. [51]

    A proper-time cure for the conformal sickness in quantum gravity

    A. Dasgupta and R. Loll,A Proper time cure for the conformal sickness in quantum gravity, Nucl. Phys. B606(2001) 357–379, [hep-th/0103186]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  52. [52]

    Marolf,Gravitational thermodynamics without the conformal factor problem: partition functions and Euclidean saddles from Lorentzian path integrals,JHEP07(2022) 108, [2203.07421]

    D. Marolf,Gravitational thermodynamics without the conformal factor problem: partition functions and Euclidean saddles from Lorentzian path integrals,JHEP07(2022) 108, [2203.07421]

    work page arXiv 2022

  53. [53]

    Banihashemi and T

    B. Banihashemi and T. Jacobson,On the lapse contour in the gravitational path integral, Phys. Rev. D111(2025) 066014, [2405.10307]

    work page arXiv 2025

  54. [54]

    Kontsevich and G

    M. Kontsevich and G. Segal,Wick Rotation and the Positivity of Energy in Quantum Field Theory,Quart. J. Math. Oxford Ser.72(2021) 673–699, [2105.10161]

    work page arXiv 2021

  55. [55]

    Witten,A Note On Complex Spacetime Metrics,2111.06514

    E. Witten,A Note On Complex Spacetime Metrics,2111.06514

    work page arXiv

  56. [56]

    Henneaux,Wheeler-DeWitt Equation and Bondi-Metzner-Sachs (BMS) Symmetry,Phys

    M. Henneaux,Wheeler-DeWitt Equation and Bondi-Metzner-Sachs (BMS) Symmetry,Phys. Rev. Lett.135(2025) 061501, [2506.02240]

    work page arXiv 2025

  57. [57]

    Chowdhury, V

    C. Chowdhury, V. Godet, O. Papadoulaki and S. Raju,Holography from the Wheeler-DeWitt equation,JHEP03(2022) 019, [2107.14802]

    work page arXiv 2022

  58. [58]

    Moncrief,Spacetime symmetries and linearization stability of the Einstein equations

    V. Moncrief,Spacetime symmetries and linearization stability of the Einstein equations. I,J. Math. Phys.16(1975) 493

    work page 1975

  59. [59]

    Group Averaging for de Sitter free fields

    D. Marolf and I. A. Morrison,Group Averaging for de Sitter free fields,Class. Quant. Grav. 26(2009) 235003, [0810.5163]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  60. [60]

    Chakraborty, J

    T. Chakraborty, J. Chakravarty, V. Godet, P. Paul and S. Raju,The Hilbert space of de Sitter quantum gravity,JHEP01(2024) 132, [2303.16315]

    work page arXiv 2024

  61. [61]

    Chakraborty, J

    T. Chakraborty, J. Chakravarty, V. Godet, P. Paul and S. Raju,Holography of information in de Sitter space,JHEP12(2023) 120, [2303.16316]

    work page arXiv 2023

  62. [62]

    Cotler and K

    J. Cotler and K. Jensen,Norm of the no-boundary state,2506.20547. – 96 –

    work page arXiv

  63. [63]

    Cotler and K

    J. Cotler and K. Jensen,Isometric Evolution in de Sitter Quantum Gravity,Phys. Rev. Lett. 131(2023) 211601, [2302.06603]

    work page arXiv 2023

  64. [64]

    Balasubramanian, A

    V. Balasubramanian, A. Lawrence, J. M. Magan and M. Sasieta,Microscopic Origin of the Entropy of Black Holes in General Relativity,Phys. Rev. X14(2024) 011024, [2212.02447]

    work page arXiv 2024

  65. [65]

    Maxfield,Counting states in a model of replica wormholes,2311.05703

    H. Maxfield,Counting states in a model of replica wormholes,2311.05703

    work page arXiv

  66. [66]

    J. M. Maldacena and L. Maoz,Wormholes in AdS,JHEP02(2004) 053, [hep-th/0401024]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  67. [67]

    Marolf and H

    D. Marolf and H. Maxfield,Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information,JHEP08(2020) 044, [2002.08950]

    work page arXiv 2020

  68. [68]

    Complex actions in two-dimensional topology change

    J. Louko and R. D. Sorkin,Complex actions in two-dimensional topology change,Class. Quant. Grav.14(1997) 179–204, [gr-qc/9511023]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  69. [69]

    X. Dong, A. Lewkowycz and M. Rangamani,Deriving covariant holographic entanglement, JHEP11(2016) 028, [1607.07506]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  70. [70]

    Marolf and H

    D. Marolf and H. Maxfield,The page curve and baby universes,Int. J. Mod. Phys. D30 (2021) 2142027, [2105.12211]

    work page arXiv 2021

  71. [71]

    Colin-Ellerin, X

    S. Colin-Ellerin, X. Dong, D. Marolf, M. Rangamani and Z. Wang,Real-time gravitational replicas: Formalism and a variational principle,JHEP05(2021) 117, [2012.00828]

    work page arXiv 2021

  72. [72]

    Dittrich, T

    B. Dittrich, T. Jacobson and J. Padua-Arg¨ uelles,de Sitter horizon entropy from a simplicial Lorentzian path integral,Phys. Rev. D110(2024) 046006, [2403.02119]. – 97 –

    work page arXiv 2024