Studying replica wormholes and the Page curve with simplicial quantum gravity

Jose Padua-Arg\"uelles

REVIEW 3 major objections 4 minor 97 references

Studying replica wormholes and the Page curve with simplicial quantum gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A lattice gravity path integral reproduces the Page transition through competing saddles.

desk verdict A genuine first lattice-QRC bridge to replica wormholes, but the Page-like curve is a deliberately tuned one-parameter demonstration and should be read as a proof of framework, not as a prediction. read the letter →

arxiv 2504.18663 v1 pith:CSQK45IH submitted 2025-04-25 hep-th gr-qc

Jose Padua-Argüelles This is my paper · ORCID

classification hep-thgr-qc

keywords ReplicawormholesPagecurveSwapentropyQuantumReggecalculusBlackholeinformationparadoxAnalyticcontinuationMinisuperspaceHawkingradiation

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

Quantum Regge calculus — a lattice formulation of four-dimensional gravity — can host the replica mechanism for black hole evaporation, not just the simplified continuum models used so far. The paper constructs a spherically symmetric triangulation with a free massless scalar field, integrates the matter sector analytically for arbitrary replica number $n$, and searches for semiclassical gravitational saddles in a reduced geometry. In the $n \to 1^+$ limit, the disconnected (Hawking) saddle gives a swap entropy that increases with retarded time, while the replica wormhole saddle gives a decreasing swap entropy; the minimum of the two reproduces the Page-like rise and fall. This is a proof of principle that lattice quantum gravity can compute the purification curve from a path integral with topology change.

What carries the argument

The central machinery is a four-dimensional, spherically symmetric Regge triangulation of an evaporating black hole Penrose diagram, built from tetrahedral-shell polytopes, together with a scalar field placed at the triangulation vertices. The Regge action is assembled from bone areas and complexified deficit angles, with analytic continuation performed locally by allowing the time coordinates to rotate into the complex plane; this is the mechanism that lets complex replica saddles be reached from a Lorentzian starting contour. The matter sector reduces to Gaussian integrals in $n$ replicas, and the replica index enters through a tridiagonal Toeplitz matrix whose determinant and inverse have closed forms in terms of Chebyshev polynomials, granting analytic continuation in $n$ to $n\to1^+$. The final saddle search runs in a microsuperspace with one dynamical scale, $s_1$, and the swap entropy is evaluated as the minimum over the Hawking and wormhole saddle contributions.

What would settle it

Take the same framework with a finer triangulation, or replace the ansatz $z_2 = 1/(1+s_2)$ with another monotone relation such as $z_2 = \text{const} - s_2$, and re-run the saddle search at $n\to1^+$; if the Hawking branch is no longer increasing or the two branches no longer cross while all constraints are satisfied, the claimed Page transition is an artifact of the microsuperspace reduction.

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Extended reading notes

Core claim

The central claim is empirical within the model: a one-variable microsuperspace restriction of the discrete path integral contains semiclassical saddles of both relevant topologies, and their competition recovers the Page transition. After imposing replica and CPT symmetry, and then freezing all geometry except the splitting-surface scale $s_1$ (through the ansatz $z_2 = 1/(1+s_2)$ and a small set of boundary parameters), the matter effective actions are exact Gaussian kernels whose $n$-dependence is diagonalized by Chebyshev polynomials. The resulting saddle-point swap entropy is the minimum of the two fixed-topology branches: the disconnected Hawking topology branch grows with $\Delta z = z_5 - z_4$, the replica wormhole branch falls, and the minimum switches at a discrete Page time. The paper states that the transition can be reproduced with saddles that satisfy all the constraints that were not imposed by hand, while also noting that the Hawking branch's monotonic increase is the less robust part of the construction.

Load-bearing premise

The load-bearing assumption is that the drastically reduced one-variable microsuperspace, with the ansatz $z_2 = 1/(1+s_2)$ and the specifically chosen boundary data, is representative of the physics of an evaporating black hole; if that slice omits essential geometry, the Page-like transition is an artifact of the reduction.

Editorial extensions

If this is right

A sufficiently refined Regge triangulation should reproduce the full Page curve, including the late-time return of the entropy toward zero, rather than only a local transition.
Replica wormhole saddles can be sought at finite $n$, addressing the operational concern that the $n\to1^+$ limit is only a formal extrapolation.
The local analytic-continuation scheme provides a concrete way to test whether complex replica saddles contribute to a Lorentzian path integral by deforming the real-time contour.
The modular form of the actions means the same building blocks are reusable for other topology-changing spacetimes, connecting replica calculations to other discrete gravity programs.

Reading between the lines

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

A natural test is to compute the same swap entropy after adding more shells or refining the angular discretization; persistence would indicate that the Page mechanism is generic, while disappearance would implicate the microsuperspace slice.
Because the matter effective action is analytic in $n$ and matches direct computation at $n=1,2$, the same formulas can evaluate Rényi and swap entropies at non-integer replica number, not only in the $n\to1^+$ limit.
The same triangulation modules could be applied to other topology-changing processes, such as a black-to-white-hole transition, to see whether similar saddle exchanges produce entropy purification.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: this is a real proof-of-principle, and it is honest about being one. The author builds a quantum Regge calculus setup for replica wormholes and Hawking saddles, computes modular Regge action elements and Gaussian matter effective actions, and then shows that in a heavily reduced microsuperspace one can find saddles whose fixed-topology swap entropies rise and fall. The plotted Page-like transition exists, but it is tuned: eq. (73) is called "possibly the most artificial restriction," the remaining parameters are frozen, and the author explicitly says the Hawking increase is not robust and the curves might not cross generically. That is the right way to read the paper.

What is genuinely new: the triangulation scheme with local analytic continuation, the modular action expressions, the analytic continuation in n using the Toeplitz determinant, and the fact that the full matter path integral is done analytically rather than by another saddle point. The n=1 and n=2 agreement is a good check. The repository is a real asset, and the paper is unusually clear about what would need to happen to turn this into a prediction: asymptotic boundaries, measure ambiguities, contour definition, finer triangulations. This is more than most first papers in this area do.

The soft spot is exactly where the reader put it. The central claim in the abstract, that the saddles "recover the Page transition," is accurate only for a selected one-parameter slice and a selected parameter table. That does not sink the paper as a framework paper, but reviewers should not let the abstract stand as if the transition were a generic consequence of QRC. The remedy is straightforward: a robustness section scanning nearby parameter values, at least one alternative ansatz for z2(s2), and a version of Figure 14 with multiple curves or parameter ranges. The author already says some alternatives give the same qualitative behavior; showing them would materially strengthen the paper.

Who it is for: people working in discrete quantum gravity and the replica/information-paradox interface. It deserves a serious referee. My recommendation: accept after minor-to-moderate revision, with the abstract softened or the robustness evidence added. The core framework contribution is solid enough to justify the referee time.

Referee Report

3 major / 4 minor

Summary. The paper develops a Quantum Regge Calculus (QRC) framework for replica calculations of the black hole swap entropy in four-dimensional, spherically symmetric discrete gravity. It introduces a triangulation scheme whose elementary cells assemble into evaporating-black-hole and replica-wormhole spacetimes, computes the corresponding Regge gravitational actions and the effective matter actions for a free massless scalar field, and provides analytic continuation in the replica number n. The framework is then applied to a highly restricted 'microsuperspace' in which all geometry except the splitting-surface scale s1 is frozen, with an additional ansatz z2 = 1/(1+s2) imposed by hand. In the n→1+ limit, the author finds semiclassical saddles for both the Hawking and wormhole topologies, and reports in Fig. 14 that the Hawking contribution to the swap entropy increases with the discrete retarded time Δz while the wormhole contribution decreases, so that their minimum produces a Page-like transition. The paper is explicitly presented as a proof-of-principle, and its central limitations—the minisuperspace reduction, the artificial relation (73), and the absence of a full treatment of asymptotic boundaries—are acknowledged in the text.

Significance. If the central claim is accepted, this would be a first concrete indication that the replica mechanism and the Page curve can be realized in a lattice formulation of four-dimensional gravity, going beyond continuum toy models such as JT gravity. The paper also contributes a detailed modular computational scheme for Regge actions and matter effective actions, and it ships an explicit GitHub repository with the lengthy expressions and the code used to obtain them. These are genuine strengths: the derivations are transparent, the Gaussian matter integrations are performed in closed form, and the author is unusually candid about the assumptions and limitations of the minisuperspace. However, the headline result is not a parameter-free or robust prediction; it is an existence proof at a selected boundary-parameter point. The significance of the paper therefore rests on whether the Page-like crossing is a property of the discrete replica formulation or a consequence of the specific truncation and parameter choices.

major comments (3)

[§V C 2, Eq. (75) and Fig. 14] The central plotted result is not shown to be a generic property of the discrete replica framework. Immediately after Eq. (75) the text states that 'the increasing behavior in the Hawking topology is not as robust' and that 'the two curves corresponding to different topologies might not cross'. Because the swap entropy in Eq. (75) is then evaluated at the single parameter point listed in the Fig. 14 table, the figure demonstrates only that a crossing can occur for a selected boundary geometry and a selected set of frozen geometric parameters. To support the abstract's claim that the framework 'reveals semiclassical saddles ... that recover the Page transition', the paper should provide an explicit stability analysis: a scan over the parameters of the Fig. 14 table (and over nearby boundary values) showing that the crossing and the monotonicities persist in an open neighborhood, or a reformulation of the claim strictly as an existence proof with the parameter dependence stated as a limitation.
[§V C 1, especially Eq. (73)] The microsuperspace reduction is the load-bearing assumption of the calculation. After the reductions described in §V C 1, the only dynamical variable is s1, and the ansatz z2 = 1/(1+s2) in Eq. (73), which the author calls 'possibly the most artificial restriction in this construction', directly links the splitting-surface scale to the location of shell 2. The paper states that other functional forms of this relation give the same qualitative behavior, but no such data are shown, and the analysis does not demonstrate that the saddles found in the one-dimensional microsuperspace are limits of saddles of the less-reduced minisuperspace. I ask the author to include a concrete check of robustness: for example, allowing z2 or s3 to fluctuate independently and showing that the fixed-topology monotonicities and the crossing survive, or presenting a saddle found before imposing Eq. (73). Without such a check, the rise-and-fall of the swap entropy may be an artifact of the truncation rather than a property of the discrete replica setup.
[§V A and §V C 2] The treatment of degenerate shells—setting the r=0 shells and the i0 shell to zero size—is acknowledged to be artificial, and the paper justifies it by an 'RG-like reduction' without presenting the supporting computation. Since the Page-like crossing in Fig. 14 depends on the relative magnitudes of the Hawking and wormhole swap entropies, and since the i0 and r=0 regions contribute to the gravitational and matter actions, the quantitative validity of the plotted result is not fully secured. The author should either supply the promised RG-like reduction argument in detail or state more precisely how this idealization could affect the position and existence of the crossing.

minor comments (4)

[§V B] There is a duplicated word in the text: 'performed the matter path integrals in a different different than the one above' should read 'different way than the one above'.
[Fig. 14 and surrounding text] The figure is repeated at least three times in the manuscript text with garbled captions and axis labels (for example, '!z' and '↭' appear instead of 'Δz' and generic argument symbols). The final version should contain a single clean figure with correct symbols.
[§IV C 1, Eq. (40)] The notation for vertex indices is inconsistent: vertices are denoted v□ and also appear as vµ, vν in the same equation. Please standardize the notation for edge endpoints and dual volumes.
[References] Reference [41] cites a particle-accelerator conference paper for the free relativistic particle path integral; this appears to be an incorrect or incomplete bibliographic entry and should be replaced with the intended source.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed recovery of the Page transition is a tuned microsuperspace demonstration: the chosen ansatz and Figure 14 boundary parameters are selected so that the Hawking branch rises, the wormhole branch falls, and their minimum crosses.

fitted input called prediction [§V C 2 (Microsuperspace saddles), eq. (75) and Fig. 14]
"As is to be expected, at least within the current model, whether there is a Page-like-transition behavior generically depends on the choice of boundary variables and parameters. ... Nevertheless, it is possible to reproduce the Page transition within the setup, with saddles that satisfy all the unimplemented constraints. Indeed, in the n→1+ limit, the swap entropy behaves as shown in figure 14 using the parameters on its table."

The paper selects the Figure 14 parameters (m12=0.9, σ34=0.9, ρ3=0.5, t1=0.570) and imposes the ansatz z2 = 1/(1+s2), which it calls 'possibly the most artificial restriction in this construction', in order to obtain the desired monotonicities: Hawking increases and wormhole decreases. The swap entropy is then defined via eq. (75) as the minimum of these two fixed-topology contributions, so if one curve rises and the other falls, the minimum automatically has the Page-like rise-and-cross. The paper admits that 'the increasing behavior in the Hawking topology is not as robust' and that the curves 'might not cross' generically.

full rationale

The paper is self-aware and labels the application a 'proof-of-principle' within a 'controlled minisuperspace reduction', which lowers the circularity. The Regge action, matter effective actions, and Gaussian sewing are computed in modular ways that do not themselves reduce to the Page curve. However, the central result—recovering the Page transition in the n→1+ limit—is obtained only after sequences of increasingly bold restrictions: microsuperspace with one variable s1, ansatz z2=1/(1+s2), frozen m12, σ34, ρ3, t1, and Figure 14 parameters. The paper explicitly states that whether the Page-like behavior occurs 'generically depends on the choice of boundary variables and parameters', and reports that the Hawking increasing branch is 'not as robust' and that crossing is not guaranteed. Since plotted swap entropy is min of two branches (eq. 75), rise-and-fall is partly built in by chosen parameters rather than emerging as framework consequence. This is not full circularity: nontrivial actions, saddles for both topologies, and computed monotonicities exist. But the claim that the framework 'recovers the Page transition' is partly a fitted-input-called-prediction, because known Page behavior guides parameter selection. Score 6 reflects one central prediction reducing, by parameter choice, to its target behavior, while much of framework remains independent.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

The central demonstration depends on many chosen geometric and state parameters and on modeling assumptions, most of which the author explicitly acknowledges. The free parameters are not fitted by data in a regression sense, but they are selected so that the plotted swap entropy realizes the expected Page-like behavior. There are no new physical entities such as new particles or forces; the invented element is the triangulation scheme itself.

free parameters (8)

s⊙ (celestial sphere scale) = 3.802
Boundary/frozen geometric variable in the Figure 14 table, selected to obtain the plotted Page-like transition.
z5 (boundary shell position) = 0.776
Frozen boundary coordinate chosen for the demonstration.
t5 (boundary time coordinate) = 0.388
Set by a null condition and then chosen along with the other boundary data.
m12 (slope of diagram line 12) = 0.9
Frozen microsuperspace parameter used in the saddle search.
σ34 (shell scale ratio) = 0.9
Frozen microsuperspace parameter constraining s3 relative to s4.
ρ3 (convex combination parameter for z3) = 0.5
Frozen microsuperspace parameter determining z3 between z2 and z4.
t1 (bulk time coordinate) = 0.570
Frozen microsuperspace parameter in the final saddle-point search.
Ωv vacuum wave-function coefficients = not specified
Gaussian state parameters treated as model inputs; the paper says the qualitative behavior is stable over orders of magnitude, but exact values are not tied to a first-principles derivation.

assumptions (8)

domain assumption Replica symmetry and CPT symmetry reduce the n-copy path integral to a single-copy geometry.
Used throughout §V to derive effective actions and evaluate saddles; assumed without a general proof.
domain assumption Only the Hawking topology and the replica wormhole topology dominate the swap-operator path integral.
Stated in footnote 10 and used to turn the topology sum into a minimum over two contributions.
domain assumption Saddles found in the Euclidean microsuperspace can be continued to contribute to the original Lorentzian path integral.
The paper acknowledges replica saddles are complex and the contour-deformation question is open; see §II and §VI.
domain assumption The vacuum wave-functional has a Gaussian form with geometry-dependent Ω coefficients.
Equation (48) replaces a direct computation of the discrete vacuum state; it is a model input.
ad hoc to paper The one-parameter microsuperspace ansatz, including z2 = 1/(1+s2), captures the relevant geometry.
Equation (73) is called the most artificial restriction in the paper; only partial robustness checks are described.
ad hoc to paper Setting i0 and the r=0 shells to zero size is a harmless idealization.
The author labels this 'likely artificial' in §V A, because i0 is a point only after compactification.
domain assumption The scalar field measure and the dual-volume prescription for degenerate shells are acceptable.
Measure ambiguities are discussed in §IV C 2 and §V A; the author notes they may affect quantitative results.
domain assumption The analytic continuation in n of the Gaussian trace, eq. (68), to n→1+ is valid.
The paper checks agreement with direct n=1 and n=2 evaluations, but the continuation is not proven in generality.

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Pith. "Pith review of Studying replica wormholes and the Page curve with simplicial quantum gravity." pith.science (2026). https://pith.science/paper/CSQK45IH

@misc{pith2026250418663,
  author       = {Pith},
  title        = {Pith review of: Studying replica wormholes and the Page curve with simplicial quantum gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CSQK45IH}},
  note         = {Machine review of arXiv:2504.18663}
}

abstract

The replica paradigm has emerged as a powerful tool for investigating the black hole information paradox, offering a semiclassical route to reproducing the Page curve and suggesting unitary evolution for evaporating black holes. However, existing analyses have relied on simplified models such as JT gravity, and mostly remain limited to the $n \to 1^+$ limit in Euclidean signature. This work develops a framework based on Quantum Regge Calculus (QRC) that provides a lattice-like approach to address these gaps. A triangulation scheme is introduced that accommodates both gravitational and radiation degrees of freedom, enabling explicit evaluation of the fundamental components of the Regge gravity and radiation actions in a spherically symmetric setting. The formulation naturally incorporates analytic continuation techniques to probe the role of complex saddles in Lorentzian signature. A proof-of-principle implementation is carried out within a controlled minisuperspace reduction, revealing semiclassical saddles in the $n \to 1^+$ limit that recover the Page transition. While significant challenges remain (including the definition of the discrete configuration space, ambiguities in the gravitational measure, and the treatment of asymptotic boundaries), the framework developed here provides a promising foundation for further progress. The results suggest that sufficiently refined QRC calculations could extend the replica approach beyond existing models.

Figures

Figures reproduced from arXiv: 2504.18663 by the authors.

**Figure 2.** FIG. 2: Penrose diagram representing a spacetime in which gravitational collapse and eventual black hole [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: Depiction of the path integral computing the expectation value of an observable [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4: Different fixed-topology path integrals appearing in the calculation of the swap entropy. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5: The discretization process exemplified by a scalar field theory on a sphere. The continuum [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6: The discretization scheme employs piecewise linear polytopes whose topology corresponds to the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: FIG. 7: Continuous and discrete depictions of the topology of a triangle in a (Hawking-)Penrose diagram. [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: FIG. 8: Depictions of the fundamental polytopes in four (left) and three (right) dimensions. Gluing three [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: FIG. 9: Three dimensional depiction of how the polytopes corresponding to diagram triangles are to be [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: FIG. 10: Illustration of how gluing triangular frusta results in a tetrahedral shell annulus. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: FIG. 11: The deficit angle indicates whether a collection of simplices surrounding a given bone can fit [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

**Figure 12.** Figure 12: FIG. 12: Visualization of the dual structure associated with an edge in a three-dimensional triangulation, [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]

**Figure 13.** Figure 13: FIG. 13: Initial implementation of the proposed discretization scheme. A single black hole spacetime would [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]

**Figure 14.** Figure 14: FIG. 14: Semiclassical evaluation of the discrete swap entropy as [PITH_FULL_IMAGE:figures/full_fig_p038_14.png]

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Works this paper leans on

97 extracted references · 25 canonical work pages

[1]

numerical computations of the path integral

Minisuperspace(s) The framework developed so far promises to be able to consider these fluctuations fully, by e.g. numerical computations of the path integral. But for now, the geometric path integral will be computed by saddle point evaluation, effectively reducing the problem to a classical metric coupled to quantum matter. To do so, the first step is t...
[2]

Rz454cl/mDXbDjZ7Losb9nwGh0o=

Microsuperspace saddles It is at this last stage of minisuperspace refinement, which can justifiably be termed microsuperspace, that saddles were sought. To briefly recapitulate: By assuming replica symmetry and CPT-reducing the geometric variables, the Regge exponent can be calculated, analogously to the continuum situation, by analyzing a single △(M) an...
[3]

Penington, S

G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, JHEP 03, 205 (2022), arXiv:1911.11977 [hep-th]

arXiv 2022
[4]

Almheiri, T

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, JHEP 05, 013 (2020), arXiv:1911.12333 [hep-th]

arXiv 2020
[5]

Marolf and H

D. Marolf and H. Maxfield, JHEP 04, 272 (2021), arXiv:2010.06602 [hep-th]

arXiv 2021
[6]

Chandrasekaran, N

V. Chandrasekaran, N. Engelhardt, S. Fischetti, and S. Hern´ andez-Cuenca, JHEP 11, 110 (2022), arXiv:2207.09472 [hep-th]

arXiv 2022
[7]

Marolf and H

D. Marolf and H. Maxfield, Int. J. Mod. Phys. D 30, 2142027 (2021), arXiv:2105.12211 [hep-th]

arXiv 2021
[8]

Rocek and R

M. Rocek and R. M. Williams, Phys. Lett. B 104, 31 (1981)

1981

Show all 97 references

[9]

H. W. Hamber, Gen. Rel. Grav. 41, 817 (2009), arXiv:0901.0964 [gr-qc]

2009 arXiv
[10]

J. C. Baez, Class. Quant. Grav. 15, 1827 (1998), arXiv:gr-qc/9709052

1998 arXiv
[11]

Perez, Class

A. Perez, Class. Quant. Grav. 20, R43 (2003), arXiv:gr-qc/0301113

2003 arXiv
[12]

Dittrich, T

B. Dittrich, T. Jacobson, and J. Padua-Arg¨ uelles, Phys. Rev. D 110, 046006 (2024), arXiv:2403.02119 [gr-qc]

2024 arXiv
[13]

Louko and R

J. Louko and R. D. Sorkin, Class. Quant. Grav. 14, 179 (1997), arXiv:gr-qc/9511023

1997 arXiv
[14]

S. K. Asante, B. Dittrich, and J. Padua-Arg¨ uelles, Class. Quant. Grav. 40, 105005 (2023), arXiv:2112.15387 [gr-qc]

2023 arXiv
[15]

Jia, Class

D. Jia, Class. Quant. Grav. 39, 065002 (2022), arXiv:2110.05953 [gr-qc]. 32 The arguments that follow are just a consequence of well understood (and stronger) properties of circulant matrices. 44

2022 arXiv
[16]

Padua-Arg¨ uelles, (), simplicial-replicas, https://github.com/jospadua/Simplicial-replicas.git

J. Padua-Arg¨ uelles, (), simplicial-replicas, https://github.com/jospadua/Simplicial-replicas.git
[17]

To appear,

J. Padua-Arg¨ uelles, “To appear,” (), 2025

2025
[18]

S. W. Hawking, Commun. Math. Phys. 43, 199 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)]

1975
[19]

S. W. Hawking, Phys. Rev. D 14, 2460 (1976)

1976
[20]

Susskind, (1995), arXiv:hep-th/9501106

L. Susskind, (1995), arXiv:hep-th/9501106

1995 arXiv
[21]

Banks and N

T. Banks and N. Seiberg, Phys. Rev. D 83, 084019 (2011), arXiv:1011.5120 [hep-th]

2011 arXiv
[22]

D. N. Page, in 5th Canadian Conference on General Relativity and Relativistic Astrophysics (5CCGRRA) (1993) arXiv:hep-th/9305040

1993 arXiv
[23]

S. D. Mathur, Class. Quant. Grav. 26, 224001 (2009), arXiv:0909.1038 [hep-th]

2009 arXiv
[24]

D. N. Page, Phys. Rev. Lett. 71, 1291 (1993), arXiv:gr-qc/9305007

1993 arXiv
[25]

Araki, Progress of Theoretical Physics 32, 956 (1964), https://academic.oup.com/ptp/article- pdf/32/6/956/5311286/32-6-956.pdf

H. Araki, Progress of Theoretical Physics 32, 956 (1964), https://academic.oup.com/ptp/article- pdf/32/6/956/5311286/32-6-956.pdf

1964
[26]

Witten, Rev

E. Witten, Rev. Mod. Phys. 90, 045003 (2018), arXiv:1803.04993 [hep-th]

2018 arXiv
[27]

Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1995)

S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1995)

1995
[28]

Ambjørn and R

J. Ambjørn and R. Loll (2024) arXiv:2401.09399 [hep-th]

2024 arXiv
[29]

Lewkowycz and J

A. Lewkowycz and J. Maldacena, JHEP 08, 090 (2013), arXiv:1304.4926 [hep-th]

2013 arXiv
[30]

R. D. Sorkin, (2019), arXiv:1908.10022 [gr-qc]

2019 arXiv
[31]

Neiman, JHEP 04, 071 (2013), arXiv:1301.7041 [gr-qc]

Y. Neiman, JHEP 04, 071 (2013), arXiv:1301.7041 [gr-qc]

2013 arXiv
[32]

Bodendorfer and Y

N. Bodendorfer and Y. Neiman, Class. Quant. Grav. 30, 195018 (2013), arXiv:1303.4752 [gr-qc]

2013 arXiv
[33]

Neiman, Phys

Y. Neiman, Phys. Rev. D 88, 024037 (2013), arXiv:1305.2207 [gr-qc]

2013 arXiv
[34]

Colin-Ellerin, X

S. Colin-Ellerin, X. Dong, D. Marolf, M. Rangamani, and Z. Wang, JHEP 05, 117 (2021), arXiv:2012.00828 [hep-th]

2021 arXiv
[35]

J. J. Halliwell and J. B. Hartle, Phys. Rev. D 41, 1815 (1990)

1990
[36]

Kontsevich and G

M. Kontsevich and G. Segal, Quart. J. Math. Oxford Ser. 72, 673 (2021), arXiv:2105.10161 [hep-th]

2021 arXiv
[37]

Witten, (2021), arXiv:2111.06514 [hep-th]

E. Witten, (2021), arXiv:2111.06514 [hep-th]

2021 arXiv
[38]

Lehners, Phys

J.-L. Lehners, Phys. Rev. D 105, 026022 (2022), arXiv:2111.07816 [hep-th]

2022 arXiv
[39]

Jonas, J.-L

C. Jonas, J.-L. Lehners, and J. Quintin, JHEP 08, 284 (2022), arXiv:2205.15332 [hep-th]

2022 arXiv
[40]

Lehners, Phys

J.-L. Lehners, Phys. Rev. D 107, 046004 (2023), arXiv:2209.14669 [hep-th]

2023 arXiv
[41]

Marolf, JHEP 07, 108 (2022), arXiv:2203.07421 [hep-th]

D. Marolf, JHEP 07, 108 (2022), arXiv:2203.07421 [hep-th]

2022 arXiv
[42]

To appear,

B. Dittrich and J. Padua-Arg¨ uelles, “To appear,” 2025

2025
[43]

Johnson, in PACS2001

P. Johnson, in PACS2001. Proceedings of the 2001 Particle Accelerator Conference (Cat. No.01CH37268) , Vol. 3 (2001) pp. 1781–1783 vol.3

2001
[44]

Gall, Brownian Motion, Martingales, and Stochastic Calculus , Graduate Texts in Mathematics (Springer International Publishing, 2016)

J. Gall, Brownian Motion, Martingales, and Stochastic Calculus , Graduate Texts in Mathematics (Springer International Publishing, 2016)

2016
[45]

Penington, JHEP 09, 002 (2020), arXiv:1905.08255 [hep-th]

G. Penington, JHEP 09, 002 (2020), arXiv:1905.08255 [hep-th]

2020 arXiv
[46]

Almheiri, N

A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, JHEP 12, 063 (2019), arXiv:1905.08762 [hep-th]

2019 arXiv
[47]

Polchinski and A

J. Polchinski and A. Strominger, Phys. Rev. D 50, 7403 (1994), arXiv:hep-th/9407008

1994 arXiv
[48]

Mirbabayi, (2020), arXiv:2008.09626 [hep-th]

M. Mirbabayi, (2020), arXiv:2008.09626 [hep-th]

2020 arXiv
[49]

J. Held, X. Liu, D. Marolf, and Z. Wang, (2024), arXiv:2409.17428 [hep-th]

2024 arXiv
[50]

Dittrich, New J

B. Dittrich, New J. Phys. 14, 123004 (2012), arXiv:1205.6127 [gr-qc]

2012 arXiv
[51]

Dittrich and J

B. Dittrich and J. Padua-Arg¨ uelles, Phys. Rev. D109, 026002 (2024), arXiv:2302.11586 [gr-qc]

2024 arXiv
[52]

Cheeger, W

J. Cheeger, W. Muller, and R. Schrader, Commun. Math. Phys. 92, 405 (1984)

1984
[53]

Feinberg, R

G. Feinberg, R. Friedberg, T. D. Lee, and H. C. Ren, Nucl. Phys. B 245, 343 (1984)

1984
[54]

Rocek and R

M. Rocek and R. M. Williams, Z. Phys. C 21, 371 (1984)

1984
[55]

J. W. Barrett, Classical and Quantum Gravity 5, 1187 (1988)

1988
[56]

J. W. Barrett and R. M. Williams, Class. Quant. Grav. 5, 1543 (1988)

1988
[57]

C. W. Misner, K. S. Thorne, and J. A. Wheeler, San Francisco: W.H. Freeman and Co., 1973 , edited by Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973)

1973
[58]

R. M. Williams, Int. J. Mod. Phys. B 6, 2097 (1992)

1992
[59]

of Alexandria, Metrica (10-85AD)

H. of Alexandria, Metrica (10-85AD)
[60]

H. W. Hamber and R. M. Williams, Phys. Rev. D 59, 064014 (1999), arXiv:hep-th/9708019

1999 arXiv
[61]

H. W. Hamber, Quantum gravitation: The Feynman path integral approach (Springer, Berlin, 2009)

2009
[62]

Loll, Living Rev

R. Loll, Living Rev. Rel. 1, 13 (1998), arXiv:gr-qc/9805049

1998 arXiv
[63]

J. N. Borissova and B. Dittrich, JHEP 09, 069 (2023), arXiv:2303.07367 [hep-th]

2023 arXiv
[64]

C. W. Misner, Rev. Mod. Phys. 29, 497 (1957)

1957
[65]

B. S. DeWitt, J. Math. Phys. 3, 625 (1962)

1962
[66]

Voronoi, Journal f¨ ur die reine und angewandte Mathematik134, 198 (1908)

G. Voronoi, Journal f¨ ur die reine und angewandte Mathematik134, 198 (1908)

1908
[67]

Ponzano and T

G. Ponzano and T. Regge, Spectroscopic and Group Theoretical Methods in Physics edited by F. Bloch (1968). 45

1968
[68]

J. W. Barrett and I. Naish-Guzman, Class. Quant. Grav. 26, 155014 (2009), arXiv:0803.3319 [gr-qc]

2009 arXiv
[69]

H. W. Hamber and R. M. Williams, Phys. Lett. B 157, 368 (1985)

1985
[70]

Dittrich and S

B. Dittrich and S. Steinhaus, Phys. Rev. D 85, 044032 (2012), arXiv:1110.6866 [gr-qc]

2012 arXiv
[71]

Dittrich, W

B. Dittrich, W. Kami´ nski, and S. Steinhaus, Class. Quant. Grav. 31, 245009 (2014), arXiv:1404.5288 [gr-qc]

2014 arXiv
[72]

J. B. Hartle, W. A. Miller, and R. M. Williams, Class. Quant. Grav. 14, 2137 (1997), arXiv:gr-qc/9609028

1997 arXiv
[73]

Menotti and P

P. Menotti and P. P. Peirano, Nucl. Phys. B 488, 719 (1997), arXiv:hep-th/9607071

1997 arXiv
[74]

Sorkin, J

R. Sorkin, J. Math. Phys. 16, 2432 (1975), [Erratum: J.Math.Phys. 19, 1800 (1978)]

1975
[75]

Ren, Nucl

H.-c. Ren, Nucl. Phys. B 301, 661 (1988)

1988
[76]

H. W. Hamber and R. M. Williams, Nucl. Phys. B 415, 463 (1994), arXiv:hep-th/9308099

1994 arXiv
[77]

Sitharam, A

M. Sitharam, A. John, and J. Sidman, Handbook of Geometric Constraint Systems Principles , Discrete Mathe- matics and Its Applications Series (CRC Press, Taylor & Francis Group, 2018)

2018
[78]

The path integral on the lattice,

C. Gattringer and C. B. Lang, “The path integral on the lattice,” in Quantum Chromodynamics on the Lattice: An Introductory Presentation (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010) pp. 1–23

2010
[79]

Kulkarni, D

D. Kulkarni, D. Schmidt, and S.-K. Tsui, Linear Algebra and its Applications 297, 63 (1999)

1999
[80]

From MathWorld–A Wolfram Web Resource

Chebyshev Polynomial of the Second Kind. From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
[81]

Alexandru, G

A. Alexandru, G. Basar, P. F. Bedaque, and N. C. Warrington, Rev. Mod. Phys. 94, 015006 (2022), arXiv:2007.05436 [hep-lat]

2022 arXiv
[82]

M. Han, Z. Huang, H. Liu, D. Qu, and Y. Wan, Phys. Rev. D 103, 084026 (2021), arXiv:2012.11515 [gr-qc]

2021 arXiv
[83]

Spin Foams, Refinement Limit, and Renormalization,

S. K. Asante, B. Dittrich, and S. Steinhaus, “Spin Foams, Refinement Limit, and Renormalization,” (2023) arXiv:2211.09578 [gr-qc]

2023 arXiv
[84]

J. W. Barrett, R. J. Dowdall, W. J. Fairbairn, H. Gomes, and F. Hellmann, J. Math. Phys. 50, 112504 (2009), arXiv:0902.1170 [gr-qc]

2009 arXiv
[85]

J. W. Barrett, R. J. Dowdall, W. J. Fairbairn, F. Hellmann, and R. Pereira, Class. Quant. Grav. 27, 165009 (2010), arXiv:0907.2440 [gr-qc]

2010 arXiv
[86]

M. Han, Z. Huang, H. Liu, and D. Qu, Phys. Rev. D 106, 044005 (2022), arXiv:2110.10670 [gr-qc]

2022 arXiv
[87]

M. Han, H. Liu, and D. Qu, Phys. Rev. D 108, 026010 (2023), arXiv:2301.02930 [gr-qc]

2023 arXiv
[88]

S. K. Asante, B. Dittrich, and H. M. Haggard, Phys. Rev. Lett. 125, 231301 (2020), arXiv:2004.07013 [gr-qc]

2020 arXiv
[89]

S. K. Asante, B. Dittrich, and H. M. Haggard, Class. Quant. Grav. 38, 145023 (2021), arXiv:2011.14468 [gr-qc]

2021 arXiv
[90]

S. K. Asante, B. Dittrich, and J. Padua-Arguelles, Class. Quant. Grav. 38, 195002 (2021), arXiv:2104.00485 [gr-qc]

2021 arXiv
[91]

Engle, R

J. Engle, R. Pereira, and C. Rovelli, Phys. Rev. Lett. 99, 161301 (2007), arXiv:0705.2388 [gr-qc]

2007 arXiv
[92]

Freidel and K

L. Freidel and K. Krasnov, Class. Quant. Grav. 25, 125018 (2008), arXiv:0708.1595 [gr-qc]

2008 arXiv
[93]

H. M. Haggard and C. Rovelli, Phys. Rev. D 92, 104020 (2015), arXiv:1407.0989 [gr-qc]

2015 arXiv
[94]

D’Ambrosio and C

F. D’Ambrosio and C. Rovelli, Class. Quant. Grav. 35, 215010 (2018), arXiv:1803.05015 [gr-qc]

2018 arXiv
[95]

De Lorenzo and A

T. De Lorenzo and A. Perez, Phys. Rev. D 93, 124018 (2016), arXiv:1512.04566 [gr-qc]

2016 arXiv
[96]

D’Ambrosio, M

F. D’Ambrosio, M. Christodoulou, P. Martin-Dussaud, C. Rovelli, and F. Soltani, Phys. Rev. D 103, 106014 (2021), arXiv:2009.05016 [gr-qc]

2021 arXiv
[97]

Soltani, C

F. Soltani, C. Rovelli, and P. Martin-Dussaud, Phys. Rev. D 104, 066015 (2021), arXiv:2105.06876 [gr-qc]

2021 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

numerical computations of the path integral

Minisuperspace(s) The framework developed so far promises to be able to consider these fluctuations fully, by e.g. numerical computations of the path integral. But for now, the geometric path integral will be computed by saddle point evaluation, effectively reducing the problem to a classical metric coupled to quantum matter. To do so, the first step is t...

[2] [2]

Rz454cl/mDXbDjZ7Losb9nwGh0o=

Microsuperspace saddles It is at this last stage of minisuperspace refinement, which can justifiably be termed microsuperspace, that saddles were sought. To briefly recapitulate: By assuming replica symmetry and CPT-reducing the geometric variables, the Regge exponent can be calculated, analogously to the continuum situation, by analyzing a single △(M) an...

[3] [3]

Penington, S

G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, JHEP 03, 205 (2022), arXiv:1911.11977 [hep-th]

arXiv 2022

[4] [4]

Almheiri, T

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, JHEP 05, 013 (2020), arXiv:1911.12333 [hep-th]

arXiv 2020

[5] [5]

Marolf and H

D. Marolf and H. Maxfield, JHEP 04, 272 (2021), arXiv:2010.06602 [hep-th]

arXiv 2021

[6] [6]

Chandrasekaran, N

V. Chandrasekaran, N. Engelhardt, S. Fischetti, and S. Hern´ andez-Cuenca, JHEP 11, 110 (2022), arXiv:2207.09472 [hep-th]

arXiv 2022

[7] [7]

Marolf and H

D. Marolf and H. Maxfield, Int. J. Mod. Phys. D 30, 2142027 (2021), arXiv:2105.12211 [hep-th]

arXiv 2021

[8] [8]

Rocek and R

M. Rocek and R. M. Williams, Phys. Lett. B 104, 31 (1981)

1981