pith. sign in

arxiv: 2512.23656 · v2 · pith:265PXO5Tnew · submitted 2025-12-29 · ✦ hep-th · gr-qc· quant-ph

Ordering-Independent Wheeler-DeWitt Equation for Flat Minisuperspace Models

Victor Franken , Eftychios Kaimakkamis , Herv\'e Partouche , Nicolaos Toumbas This is my paper

Pith reviewed 2026-05-21 16:36 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph
keywords minisuperspaceWheeler-DeWitt equationoperator orderingpath integralquantum cosmologyJackiw-Teitelboim gravityStarobinsky model
0
0 comments X

The pith

Different path-integral measures each select a distinct operator ordering for the Wheeler-DeWitt equation, yet all produce the same physical predictions.

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

The authors study the quantization of minisuperspace models in quantum cosmology where the target space is flat. They demonstrate that the path-integral formulation constrains the operator ordering in the Wheeler-DeWitt equation to a specific class. Each path-integral measure corresponds to one such ordering through the Jacobian of a field redefinition. All these orderings turn out to be equivalent, giving identical observables at every order in the Planck constant and permitting a positive definite inner product. This equivalence means that the choice of measure does not affect the final quantum theory for these models.

Core claim

In flat minisuperspace models with two-derivative kinetic terms and a closed universe, canonical quantization yields a Wheeler-DeWitt equation whose operator ordering is fixed by the choice of path-integral measure through the corresponding Jacobian. All such orderings are physically equivalent to all orders in ħ, producing the same observables and allowing a positive definite inner product on the Hilbert space.

What carries the argument

The one-to-one correspondence between path-integral measures and operator orderings, mediated by Jacobians from redefining the canonical fields.

If this is right

  • Physical observables remain unchanged regardless of which compatible operator ordering is chosen.
  • A positive definite inner product can be defined for the Hilbert space in each case.
  • The approach applies directly to models such as de Sitter Jackiw-Teitelboim gravity and the Starobinsky model.
  • The Wheeler-DeWitt equation's physical content becomes independent of the specific ordering chosen within the allowed class.

Where Pith is reading between the lines

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

  • Similar measure-ordering correspondences could be explored in models with curved target spaces to see if equivalence persists.
  • This result may help in selecting convenient orderings for numerical computations of wave functions in quantum cosmology.
  • Connections could be drawn to other quantization ambiguities in gravitational theories beyond minisuperspace.

Load-bearing premise

The models must have a flat target space and strictly two-derivative kinetic terms.

What would settle it

Explicit computation of a physical observable, such as the wave function or a transition probability, in the de Sitter Jackiw-Teitelboim gravity model using two different path-integral measures, checking for agreement to all orders in ħ.

read the original abstract

We consider minisuperspace models with two-derivative kinetic terms, assuming a flat target space and a closed Universe. We show that, upon canonical quantization of the Hamiltonian, only a restricted class of operator orderings is compatible with the path-integral formulation. Remarkably, these orderings are physically equivalent to all orders in $\hbar$. More precisely, each choice of path-integral measure in the definition of the wavefunction path integral uniquely determines an operator ordering, and hence a corresponding Wheeler-DeWitt equation. These orderings are in one-to-one correspondence with the Jacobians arising from field redefinitions of a set of canonical fields. For each operator ordering consistent with a path-integral measure, we identify a positive definite Hilbert-space inner product. All such prescriptions define the same quantum theory, in the sense that they yield identical physical observables. We illustrate our formalism by applying it to de Sitter Jackiw-Teitelboim gravity and to the Starobinsky model.

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

2 major / 2 minor

Summary. The paper claims that in minisuperspace models with flat target spaces, strictly two-derivative kinetic terms, and closed universes, each choice of path-integral measure uniquely determines an admissible operator ordering in the Wheeler-DeWitt equation via the Jacobian of a canonical field redefinition. These orderings are shown to be physically equivalent to all orders in ħ, yielding identical observables and positive-definite Hilbert-space inner products. The formalism is illustrated by explicit application to de Sitter Jackiw-Teitelboim gravity and the Starobinsky model.

Significance. If the equivalence result holds within the stated domain, the work resolves the operator-ordering ambiguity for this restricted but physically relevant class of models by linking path-integral and canonical approaches. It supplies a concrete mechanism (Jacobians) that renders the quantum theory independent of ordering choice while preserving a positive inner product, which is a notable technical advance for quantum cosmology and for solvable models such as JT gravity.

major comments (2)
  1. [applications to de Sitter JT gravity and Starobinsky model] The central claim that observables agree to all orders in ħ is load-bearing for the physical equivalence statement. The manuscript should supply at least one explicit higher-order calculation (e.g., expectation value of a simple observable beyond the leading semiclassical term) in either the JT or Starobinsky example to substantiate the 'all orders' assertion.
  2. [general formalism for the inner product] The identification of a positive-definite inner product for each ordering is essential to the Hilbert-space construction. An explicit verification or proof that the inner product remains positive for a generic Jacobian-induced ordering (rather than only for the two illustrated cases) would strengthen the result.
minor comments (2)
  1. Notation for the canonical fields and their redefinitions could be made more uniform across sections to improve readability of the Jacobian correspondence.
  2. A brief comparison table listing the different measures, corresponding orderings, and resulting WdW operators for the two example models would help the reader track the equivalence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. We address the two major comments below and will incorporate revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [applications to de Sitter JT gravity and Starobinsky model] The central claim that observables agree to all orders in ħ is load-bearing for the physical equivalence statement. The manuscript should supply at least one explicit higher-order calculation (e.g., expectation value of a simple observable beyond the leading semiclassical term) in either the JT or Starobinsky example to substantiate the 'all orders' assertion.

    Authors: Our general argument establishes equivalence to all orders in ħ by showing that distinct operator orderings are related by unitary transformations induced by the Jacobians of canonical field redefinitions; this maps the entire quantum theory (including all ħ corrections) between prescriptions while preserving observables. Nevertheless, we agree that an explicit higher-order check in one of the examples would make the claim more concrete. In the revised manuscript we will add a calculation of the next-to-leading-order correction (O(ħ)) to the expectation value of a simple observable, such as the spatial volume, in the de Sitter JT gravity model and verify that it agrees for two different Jacobian-induced orderings. revision: yes

  2. Referee: [general formalism for the inner product] The identification of a positive-definite inner product for each ordering is essential to the Hilbert-space construction. An explicit verification or proof that the inner product remains positive for a generic Jacobian-induced ordering (rather than only for the two illustrated cases) would strengthen the result.

    Authors: The inner product is constructed by inserting the Jacobian factor of the field redefinition into the standard L² measure; because the Jacobian is positive by construction for the class of models considered (flat target space, two-derivative kinetics), positivity is preserved under the redefinition. This argument is already general and does not rely on the specific examples. To address the referee’s request we will expand the relevant section with a short general proof that positivity holds for an arbitrary Jacobian-induced ordering within our assumptions, and we will state the result more explicitly before turning to the illustrations. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from path-integral Jacobians

full rationale

The paper derives a one-to-one correspondence between path-integral measures and admissible operator orderings directly from the definition of the wavefunction path integral together with the Jacobians of canonical field redefinitions, under the explicit restrictions of flat target space and strictly two-derivative kinetic terms. The claimed physical equivalence of all such orderings (identical observables to all orders in ħ and positive-definite inner products) follows by explicit construction within those assumptions rather than by fitting parameters, self-referential definitions, or load-bearing self-citations. No step in the abstract or described formalism reduces the central result to its own inputs by construction; the derivation remains independent of prior fitted quantities or unverified uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard rules of canonical quantization and path-integral formulation in minisuperspace, plus the assumption of a flat target space; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Canonical quantization of the Hamiltonian constraint in minisuperspace with flat target space
    Invoked in the first sentence of the abstract as the starting point for considering operator orderings.
  • domain assumption Path-integral formulation defines the wave function via a specific measure
    Used to select the restricted class of orderings compatible with the path integral.

pith-pipeline@v0.9.0 · 5715 in / 1536 out tokens · 54977 ms · 2026-05-21T16:36:30.644395+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 12 internal anchors

  1. [1]

    The exact Wheeler–DeWitt equation for the scale-factor minisuperspace model,

    E. Kaimakkamis, H. Partouche, K. Sil and N. Toumbas, “The exact Wheeler–DeWitt equation for the scale-factor minisuperspace model,” Class. Quant. Grav.42(2025) no.11, 115003 [arXiv:2412.18532 [hep-th]]

    work page arXiv 2025

  2. [2]

    Quantum theory of gravity. 1. The canonical theory,

    B. S. DeWitt, “Quantum theory of gravity. 1. The canonical theory,” Phys. Rev.160 (1967), 1113-1148

    work page 1967

  3. [3]

    Operator Ordering in Quantum Mechanics and Quantum Gravity,

    T. Christodoulakis and J. Zanelli, “Operator Ordering in Quantum Mechanics and Quantum Gravity,” Nuovo Cim. B93(1986), 1-21. 20

    work page 1986

  4. [4]

    The origin of structure in the Universe,

    J. J. Halliwell and S. W. Hawking, “The origin of structure in the Universe,” Phys. Rev. D31(1985), 1777

    work page 1985

  5. [5]

    Operator ordering and the flatness of the Universe,

    S. W. Hawking and D. N. Page, “Operator ordering and the flatness of the Universe,” Nucl. Phys. B264(1986), 185-196

    work page 1986

  6. [6]

    Unique factor ordering in the continuum limit of LQC

    W. Nelson and M. Sakellariadou, “Unique factor ordering in the continuum limit of LQC,” Phys. Rev. D78(2008), 024006 [arXiv:0806.0595 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  7. [7]

    Dynamical interpretation of the wavefunction of the universe

    D. He, D. Gao and Q. y. Cai, “Dynamical interpretation of the wavefunction of the Universe,” Phys. Lett. B748(2015), 361-365 [arXiv:1507.06727 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  8. [8]

    Wavefunction of the Universe: Reparametrization invariance and field redefinitions of the minisuperspace path- integral,

    H. Partouche, N. Toumbas and B. de Vaulchier, “Wavefunction of the Universe: Reparametrization invariance and field redefinitions of the minisuperspace path- integral,” Nucl. Phys. B973(2021), 115600 [arXiv:2103.15168 [hep-th]]

    work page arXiv 2021

  9. [9]

    Wavefunction of the Universe: Diffeomeorphism invariance and field redefinitions,

    H. Partouche, N. Toumbas and B. de Vaulchier, “Wavefunction of the Universe: Diffeomeorphism invariance and field redefinitions,” PoSCORFU2021(2022), 159

    work page 2022

  10. [10]

    Gauge fixing and field redefini- tions of the Hartle-Hawking wavefunction path-integral,

    H. Partouche, N. Toumbas and B. de Vaulchier, “Gauge fixing and field redefini- tions of the Hartle-Hawking wavefunction path-integral,” contribution to BSM-2021 [arXiv:2105.04818 [hep-th]]

    work page arXiv 2021

  11. [11]

    Probability distribution for the quan- tum Universe,

    A. Kehagias, H. Partouche and N. Toumbas, “Probability distribution for the quan- tum Universe,” JHEP12(2021), 165 [arXiv:2110.03050 [hep-th]]

    work page arXiv 2021

  12. [12]

    Imprint of Operator Ordering on the Quantum Nature of DeWitt-regular Black Holes,

    H. Singh and M. K. Nandy, “Imprint of Operator Ordering on the Quantum Nature of DeWitt-regular Black Holes,” Int. J. Theor. Phys.64(2025), no.5, 113

    work page 2025

  13. [13]

    Exact path-integrals on half-line in quan- tum cosmology with a fluid clock and aspects of operator ordering ambiguity,

    V. Mondal, H. S. Sahota and K. Lochan, “Exact path-integrals on half-line in quan- tum cosmology with a fluid clock and aspects of operator ordering ambiguity,” JHEP 05(2025), 128 [arXiv:2501.11680 [gr-qc]]

    work page arXiv 2025

  14. [14]

    Creation of Universes from nothing,

    A. Vilenkin, “Creation of Universes from nothing,” Phys. Lett. B117(1982), 25-28

    work page 1982

  15. [15]

    The birth of inflationary Universes,

    A. Vilenkin, “The birth of inflationary Universes,” Phys. Rev. D27(1983), 2848

    work page 1983

  16. [16]

    Quantum creation of Universes,

    A. Vilenkin, “Quantum creation of Universes,” Phys. Rev. D30(1984), 509-511

    work page 1984

  17. [17]

    Quantum cosmology and the initial state of the Universe,

    A. Vilenkin, “Quantum cosmology and the initial state of the Universe,” Phys. Rev. D37(1988), 888. 21

    work page 1988

  18. [18]

    Wave function of the Universe,

    J. B. Hartle and S. W. Hawking, “Wave function of the Universe,” Phys. Rev. D28 (1983), 2960-2975

    work page 1983

  19. [19]

    The cosmological constant is probably zero,

    S. W. Hawking, “The cosmological constant is probably zero,” Phys. Lett. B134 (1984), 403

    work page 1984

  20. [20]

    Derivation of the Wheeler–DeWitt equation from a path-integral for minisuperspace models,

    J. J. Halliwell, “Derivation of the Wheeler–DeWitt equation from a path-integral for minisuperspace models,” Phys. Rev. D38(1988), 2468

    work page 1988

  21. [21]

    Steepestdescentcontoursinthepath-integralapproach to quantum cosmology. 1. The de Sitter minisuperspace model,

    J.J.HalliwellandJ.Louko, “Steepestdescentcontoursinthepath-integralapproach to quantum cosmology. 1. The de Sitter minisuperspace model,” Phys. Rev. D39 (1989), 2206

    work page 1989

  22. [22]

    The Classical Universes of the No-Boundary Quantum State

    J. B. Hartle, S. W. Hawking and T. Hertog, “The classical Universes of the no- boundary quantum state,” Phys. Rev. D77(2008), 123537 [arXiv:0803.1663 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  23. [23]

    Lorentzian Quantum Cosmology

    J. Feldbrugge, J. L. Lehners and N. Turok, “Lorentzian quantum cosmology,” Phys. Rev. D95(2017) no.10, 103508 [arXiv:1703.02076 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  24. [24]

    The Real No-Boundary Wave Function in Lorentzian Quantum Cosmology

    J. Diaz Dorronsoro, J. J. Halliwell, J. B. Hartle, T. Hertog and O. Janssen, “The real no-boundary wave function in Lorentzian quantum cosmology,” Phys. Rev. D 96(2017) no.4, 043505 [arXiv:1705.05340 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  25. [25]

    Particle Physics and Inflationary Cosmology

    A. D. Linde, “Particle physics and inflationary cosmology,” Contemp. Concepts Phys.5(1990), 1-362 [arXiv:hep-th/0503203 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 1990

  26. [26]

    Dynamical Structure and Definition of Energy in General Relativity,

    R. L. Arnowitt, S. Deser and C. W. Misner, “Dynamical Structure and Definition of Energy in General Relativity,” Phys. Rev.116(1959), 1322-1330

    work page 1959

  27. [27]

    The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,

    A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D23(1981), 347-356

    work page 1981

  28. [28]

    Renormalization of Higher Derivative Quantum Gravity,

    K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D16(1977), 953-969

    work page 1977

  29. [29]

    Classical Gravity with Higher Derivatives,

    K. S. Stelle, “Classical Gravity with Higher Derivatives,” Gen. Rel. Grav.9(1978), 353-371

    work page 1978

  30. [30]

    The Cauchy problem for theR+R2 theories of gravity without torsion,

    P. Teyssandier and P. Tourrenc, “The Cauchy problem for theR+R2 theories of gravity without torsion,” J. Math. Phys.24(1983), 2793. 22

    work page 1983

  31. [31]

    Fourth Order Gravity as General Relativity Plus Matter,

    B. Whitt, “Fourth Order Gravity as General Relativity Plus Matter,” Phys. Lett. B 145(1984), 176-178

    work page 1984

  32. [32]

    Inflation and the Conformal Structure of Higher Order Gravity Theories,

    J. D. Barrow and S. Cotsakis, “Inflation and the Conformal Structure of Higher Order Gravity Theories,” Phys. Lett. B214(1988), 515-518

    work page 1988

  33. [33]

    ThePrematureRecollapseProbleminClosedInflationaryUniverses,

    J.D.Barrow, “ThePrematureRecollapseProbleminClosedInflationaryUniverses,” Nucl. Phys. B296(1988), 697-709

    work page 1988

  34. [34]

    1/R gravity and Scalar-Tensor Gravity

    T. Chiba, “1/R gravity and scalar - tensor gravity,” Phys. Lett. B575(2003), 1-3 [arXiv:astro-ph/0307338 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  35. [35]

    The Perturbation Spectrum Evolving from a Nonsingular Ini- tially de-Sitter Cosmology and the Microwave Background Anisotropy,

    A. A. Starobinsky, “The Perturbation Spectrum Evolving from a Nonsingular Ini- tially de-Sitter Cosmology and the Microwave Background Anisotropy,” Sov. Astron. Lett.9(1983), 302

    work page 1983

  36. [36]

    Quantum Fluctuations and a Nonsingular Universe,

    V. F. Mukhanov and G. V. Chibisov, “Quantum Fluctuations and a Nonsingular Universe,” JETP Lett.33(1981), 532-535

    work page 1981

  37. [37]

    The Fourth Order Gravitational Action for Manifolds With Bound- aries,

    N. H. Barth, “The Fourth Order Gravitational Action for Manifolds With Bound- aries,” Class. Quant. Grav.2(1985), 497

    work page 1985

  38. [38]

    On boundary terms and conformal transformations in curved space-times

    R. Casadio and A. Gruppuso, “On boundary terms and conformal transformations in curved space-times,” Int. J. Mod. Phys. D11(2002), 703-714 [arXiv:gr-qc/0107077 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  39. [39]

    Boundary Terms, Variational Principles and Higher Derivative Modified Gravity

    E. Dyer and K. Hinterbichler, “Boundary Terms, Variational Principles and Higher Derivative Modified Gravity,” Phys. Rev. D79(2009), 024028 [arXiv:0809.4033 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  40. [40]

    Gibbons-Hawking Boundary Terms and Junction Conditions for Higher-Order Brane Gravity Models

    A. Balcerzak and M. P. Dabrowski, “Gibbons–Hawking Boundary Terms and Junc- tion Conditions for Higher-Order Brane Gravity Models,” JCAP01(2009), 018 [arXiv:0804.0855 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  41. [41]

    Lower Dimensional Gravity,

    R. Jackiw, “Lower Dimensional Gravity,” Nucl. Phys. B252(1985), 343-356

    work page 1985

  42. [42]

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

    C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimen- sions,” Phys. Lett. B126(1983), 41-45

    work page 1983

  43. [43]

    A New Type of Isotropic Cosmological Models Without Singu- larity,

    A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singu- larity,” Phys. Lett. B91(1980), 99-102. 23

    work page 1980

  44. [44]

    Selfadjointness and Spontaneously Broken Symmetry,

    A.Z. Capri, “Selfadjointness and Spontaneously Broken Symmetry,” Am. J. Phys. 45(1977), 823-825

    work page 1977

  45. [45]

    An Algebra of Observables for de Sitter Space

    V. Chandrasekaran, R. Longo, G. Penington and E. Witten, “An algebra of observ- ables for de Sitter space,” JHEP02(2023), 082 [arXiv:2206.10780 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  46. [46]

    Self-Adjoint Wheeler-DeWitt Operators, the Problem of Time and the Wave Function of the Universe

    J. Feinberg and Y. Peleg, “Selfadjoint Wheeler–DeWitt operators, the problem of time and the wave function of the universe,” Phys. Rev. D52(1995), 1988-2000 [arXiv:hep-th/9503073 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  47. [47]

    Maldacena, G

    J.Maldacena, G.J. Turiaciand Z.Yang, “Twodimensional Nearly deSittergravity,” JHEP01(2021), 139 [arXiv:1904.01911 [hep-th]]

    work page arXiv 2021

  48. [48]

    Emergent unitarity in de Sitter from matrix integrals,

    J. Cotler and K. Jensen, “Emergent unitarity in de Sitter from matrix integrals,” JHEP12(2021), 089 [arXiv:1911.12358 [hep-th]]

    work page arXiv 2021

  49. [49]

    Cotler, K

    J. Cotler, K. Jensen and A. Maloney, “Low-dimensional de Sitter quantum gravity,” JHEP06(2020), 048 [arXiv:1905.03780 [hep-th]]

    work page arXiv 2020

  50. [50]

    JT gravity in de Sitter space and the problem of time,

    K. K. Nanda, S. K. Sake and S. P. Trivedi, “JT gravity in de Sitter space and the problem of time,” JHEP02(2024), 145 [arXiv:2307.15900 [hep-th]]

    work page arXiv 2024

  51. [51]

    Non-perturbative de Sitter Jackiw–Teitelboim gravity,

    J. Cotler and K. Jensen, “Non-perturbative de Sitter Jackiw–Teitelboim gravity,” JHEP12(2024), 016 [arXiv:2401.01925 [hep-th]]

    work page arXiv 2024

  52. [52]

    DeWitt wave functions for de Sitter JT gravity,

    W. Buchmuller, A. Hebecker and A. Westphal, “DeWitt wave functions for de Sitter JT gravity,” JHEP06(2025), 049 [arXiv:2412.09211 [hep-th]]

    work page arXiv 2025

  53. [53]

    JT Gravity in de Sitter Space and Its Extensions,

    I. Dey, K. K. Nanda, A. Roy, S. K. Sake and S. P. Trivedi, “JT Gravity in de Sitter Space and Its Extensions,” [arXiv:2501.03148 [hep-th]]

    work page arXiv

  54. [54]

    Held and H

    J. Held and H. Maxfield, “The Hilbert space of de Sitter JT: a case study for canonical methods in quantum gravity,” [arXiv:2410.14824 [hep-th]]

    work page arXiv

  55. [55]

    Phase space of Jackiw–Teitelboim gravity with positive cosmological constant,

    E. Alonso-Monsalve, D. Harlow and P. Jefferson, “Phase space of Jackiw–Teitelboim gravity with positive cosmological constant,” [arXiv:2409.12943 [hep-th]]

    work page arXiv

  56. [56]

    Exact Dirac quantization of all 2-D dilaton gravity theories,

    D. Louis-Martinez, J. Gegenberg and G. Kunstatter, “Exact Dirac quantization of all 2-D dilaton gravity theories,” Phys. Lett. B321(1994), 193-198 [arXiv:gr- qc/9309018 [gr-qc]]. 24

    work page arXiv 1994

  57. [57]

    Quantization of gauge systems,

    M. Henneaux and C. Teitelboim,“Quantization of gauge systems,” Princeton Uni- versity Press (1992)

    work page 1992

  58. [58]

    Unimodular Jackiw–Teitelboim grav- ity and de Sitter quantum cosmology,

    B. Alexandre, A. Etkin and F. S. Rassouli, “Unimodular Jackiw–Teitelboim grav- ity and de Sitter quantum cosmology,” Phys. Rev. D111(2025), no.12, 124050 [arXiv:2501.17213 [gr-qc]]. 25

    work page arXiv 2025