Black hole entropy from Quantum Geometry

Jerzy Lewandowski (Uniwersytet Warszawski); Marcin Domagala

REVIEW

Black hole entropy from Quantum Geometry

Marcin Domagala, Jerzy Lewandowski (Uniwersytet Warszawski)

General Relativity and Quantum Cosmology · Submitted 14 Jul 2004 · 14 references

Pith's verdict: A genuine error in the ABCK counting, correctly identified; the exclusion of the old Barbero-Immirzi value is sound even though the lower bound is deferred to the companion paper.

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CitationJerzy Lewandowski (Uniwersytet Warszawski), Marcin Domagala. "Black hole entropy from Quantum Geometry." arXiv:gr-qc/0407051 (2004). https://pith.science/paper/3OHXXPKC

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Editor's note

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Bottom line: Domagala and Lewandowski have found a real error in the ABCK state counting — an ordering constraint on the spin labels that has no business being there — and once you drop it the old value γ = ln2/(π√3) is excluded. That is load-bearing for anyone who quotes the Immirzi parameter, and it undercuts the clean Dreyer quasi-normal-mode story (abstract:s5). What is solid: the reduction in Section III is clean. Eliminating the j's, deriving Σm_i = 0 and Σ|m_i| ≤ k/2 from the area bound, and getting a bijection between horizon labels b and half-integer sequences m is the right way to set the problem up. The upper bound N(a) ≤ 3^k follows from a recurrence (Eq. 41) I can check by hand, and it is rigorous. The identification of the ABCK mistake — imposing j1 ≤ ... ≤ jn as an admissibility condition — is specific and, as far as I can tell, correct. Soft spots, in order of size. The lower bound is what actually excludes γ_ABCK, and it rests on the random-walk statement N'_k = (C/√k)N^-_k, attributed to Meissner and proved in the companion paper [11], a preprint. Your reader flagged this; I think the worry is overstated. To rule out γ_ABCK you only need N(a) ≥ c^k with c > e^{πγ_ABCK} ≈ 1.49, and the heuristic gives c = 2. The zero-sum fraction would have to be exponentially small, roughly e^{-0.29k}, to break it. That is not plausible for a symmetric walk, so the gap is real but not load-bearing. Second, the paper says outright that it cannot verify S ∝ a itself and quotes the exact asymptotic from [11]. That is honest and the authors flag it, but it does mean the headline result here is a bound, not the entropy. Third, Section V's probability distribution inherits the same random-walk argument, so it stands or falls with it. Minor: using a_j ≤ a rather than an interval is fine, and they say so in a footnote. Who should read it: anyone working on LQG black hole entropy, and anyone who has quoted γ ≈ 0.127. It deserves referees — the error claim needs independent checking, and the dependence on [11] should be assessed alongside it.

Referee report

Significance

If correct, this is an important correction to a foundational result in loop quantum gravity. The combinatorial reduction in Eq. (36) is clean and quotable, the upper bound N(a) ≤ 3^k follows rigorously from the recurrence (41), and the resulting interval for γ excludes the value that had been used throughout the literature. The paper is candid about what it does not establish: it states explicitly that the lower bound rests on a random-walk argument due to Meissner with an exact proof in the companion paper [11], and that the sharp value γ_M is likewise taken from [11]. The derived spin distribution (Eq. 65) is a useful by-product, and the observation that the higher-spin contribution is not negligible has consequences for the conjectured relation to quasi-normal modes. The paper's honesty about the deferred proof is a strength, not a weakness.

Major objections (3)

[§IV, Eq. (50) and Eq. (54)] The lower bound N'_k = (C/√k) N^-_k is the step that excludes γ_ABCK ≈ 0.127, since the rigorous upper bound alone gives only γ ≤ ln3/π ≈ 0.35. The argument is a random-walk heuristic attributed to Meissner, with the exact proof deferred to the companion preprint [11]; the authors acknowledge this, which is appropriate. What is missing is a statement of the margin. To exclude γ_ABCK one needs only N(a) ≥ c^k with c > exp(πγ_ABCK) ≈ 1.49, whereas the heuristic gives c = 2 (since N^-_k ≈ (2/3)2^k); a polynomial correction 1/√k is far above that threshold. Please state this explicitly, and state that the constant C in Eq. (50) is bounded below (ideally giving its value from the local central limit theorem), so that the exclusion does not appear to rest on an unquantified constant.
[§IV, final paragraph] The diagnosis of the ABCK error is the paper's stated contribution [abstract:s2], but it is given only as a paraphrase: 'an assumption, that one can order the j_i labels in (23) such that j_1 ≤ ... ≤ j_n, and impose this as a constraint on admissible states we count.' Please quote Eq. (44) of [8] and show explicitly how the ordering constraint reduces the count to the (±1/2) sequences, so that a reader can verify the diagnosis without consulting [8]. As it stands the characterization of the error must be taken on trust, and the paper's framing depends on it.
[Abstract, [abstract:s5]] The abstract states that 'the value of the Barbero-Immirzi parameter ... is different than previously derived.' What the paper actually establishes here is an interval (Eq. 54) together with the exclusion of γ_ABCK; the sharp value γ_M is taken from [11] (Eqs. 56–57). The abstract should say that the old value is excluded and that γ is constrained to the interval, rather than implying that a new sharp value is derived in this paper.

Minor objections (5)

[§III, Eq. (34) and the sentence following it] The uniqueness of the lift from a sequence b to a sequence m is asserted with the non-strict bound sum |m_i| ≤ k/2. As stated, the boundary case sum |m_i| = k/2 admits more than one lift: for n = 2, m = (k/4, −k/4) and m' = (−k/4, k/4) give the same b and both satisfy the bound. The strict inequality sum |m_i| < k/2, which follows from (31)–(33) together with sqrt(|m|(|m|+1)) > |m|, removes the ambiguity; please state the strict version.
[§IV, Eq. (52)] It would help to make [abstract:s4] quantitative. With a = 4πγℓ²k, the corrected entropy lies between k ln2 and k ln3, while Eq. (53) gives (ln2/√3)k ≈ 0.40k; the correction is therefore a factor of roughly 1.7 to 2.7. A sentence stating this ratio would make the size of the higher-spin contribution concrete.
[§V, Eqs. (64)–(65)] The derivation of the spin distribution again invokes the random-walk approximation that is proved in [11]. Please flag the deferral at Eq. (64) as you do at Eq. (50), so that the reader knows which steps are established here and which are quoted.
[Footnote 3] The claim that using the inequality ≤ a rather than an interval [a−δa, a+δa] gives the same result is asserted without support. A sentence explaining why the interval only changes O(δa) terms would close the gap.
[Typesetting and typographical errors] Several typographical errors should be corrected: 'greater then' in the abstract; 'labeles' and 'Bekenshtein' in §I; 'randon' in §V; 'Discretness' in the title of [6]; 'termodynamics' in the title of [9]; 'etropy' in the title of [11]; and the affiliation 'Ho˙ za 69', which should read 'Hoża 69'. In Eq. (7) the range of m_i is printed as {−j_1, ..., j_i} and should be {−j_i, ..., j_i}, and the symbol used for 'not equal' in Eq. (8) is malformed.

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The paper

The paper claims

A counting error in the quantum-geometry account of black-hole entropy: removing a spurious ordering of spin labels raises the number of horizon states and forces the Barbero-Immirzi parameter into a new range.

The reading

Quantum Geometry — the graph-and-spin-network version of Loop Quantum Gravity — is supposed to supply the microscopic states whose logarithm gives a black hole's entropy. The paper argues that the standard counting recipe in the Ashtekar-Baez-Corichi-Krasnov isolated-horizon framework quietly imposes an ordering on the spin labels at the horizon punctures, and that this ordering throws away most of the admissible states. Once the constraint is dropped, the entropy is bounded between $S_1 = (a/4)(1-1/e)$ and $S_2 = (a/4)(1-1/e^2)$ in units of $4G$, and the free parameter of the theory is pinned to a new window instead of the value published earlier. The consequence is that spins greater than $1/2$ carry a non-negligible share of the entropy, so the earlier picture in which only the simplest punctures matter — and the quasi-normal-mode coincidence built on it — has to be rethought.

What carries the argument

The load-bearing device is the reduction of horizon state counting to a one-dimensional combinatorial problem. Each horizon state is labelled by a sequence of non-zero half-integers $m_i$ that must sum to zero and whose area weights $sqrt{|m_i|(|m_i|+1)}$ must fit under the fixed horizon area. Dropping the sum-to-zero condition and keeping only the area budget gives an over-counting set with recurrence $N^+_k = 2 N^+_{k-1} + ... + 2 N^+_0 + 1$, solved by $N^+_k = 3^k$, hence the upper bound. Replacing $sqrt{|m|(|m|+1)}$ by $|m| + 1/2$ gives a second recurrence $N^-_k = 2 N^-_{k-2} + ... + 2 N^-_0 + 1$ with solution $N^-_k = (2/3) 2^k + (-1)^k / 3$, and a random-walk argument supplies the fraction $C/sqrt{k}$ of those sequences whose entries sum to zero, giving the lower bound. The same random-walk estimate yields the limiting spin distribution $P(m) = e^{-2 pi gamma_M sqrt{|m|(|m|+1)}}$, which is the exponential of minus the entropy of a single puncture of that spin, and which sums to one by the defining equation for $gamma_M$.

Extended reading notes

Core claim

The paper's central claim is that the Ashtekar-Baez-Corichi-Krasnov state count undercounts horizon states because it assumes the spin labels $j_1, ..., j_n$ at the punctures can be sorted, $j_1 <= ... <= j_n$, and treats that as a constraint on admissible states. The authors replace the recipe with a clean combinatorial problem: count finite sequences $(m_1, ..., m_n)$ of non-zero half-integers with $m_1 + ... + m_n = 0$ and $8 pi gamma ell_{Pl}^2 sum_i sqrt(|m_i|(|m_i|+1)) <= a$, plus the empty sequence. From this they derive the bounds $(ln 2 / 4 pi gamma ell_{Pl}^2) a <= ln N(a) <= (ln 3 / 4 pi gamma ell_{Pl}^2) a$, which translate into a permitted window $ln 2 / pi <= gamma <= ln 3 / pi$ for the Barbero-Immirzi parameter. The previously derived value $gamma = ln 2 / (pi sqrt{3})$ falls outside that window, so it cannot be right. The authors state plainly that they could not themselves prove the entropy is proportional to the area, and they point to a companion paper for the exact asymptotics $ln N(a) = (gamma_M / gamma) a / (4 ell_{Pl}^2) - (1/2) ln a + O(1)$, where $gamma_M$ solves $1/2 - sum_{j != 0} e^{-2 pi gamma_M sqrt{j(j+1)}} = 0$.

Load-bearing premise

The lower bound, which is what excludes the old value of the Barbero-Immirzi parameter, rests on the claim that a fraction of order $1/sqrt{k}$ of the counted sequences have entries summing to zero; here that is argued by a random-walk heuristic and delegated to a companion paper for an exact proof.

Editorial extensions

If this is right

The Barbero-Immirzi parameter, which multiplies the spectra of every geometric operator in the theory, is confined to $ln 2 / pi <= gamma <= ln 3 / pi$, and the previously published value $ln 2 / (pi sqrt{3})$ is excluded.
Punctures with spin greater than $1/2$ contribute to the leading term of the entropy, so the horizon is not well described by a model in which each puncture carries a single bit.
The conjectured match between Quantum Geometry and the black-hole quasi-normal mode spectrum, which assumed a minimum spin of 1 rather than $1/2$, loses its original basis and must be re-derived.
Taking the companion paper's exact result as given, the entropy is proportional to the horizon area in the leading order with a coefficient $gamma_M / gamma$, and carries a sub-leading $- (1/2) ln a$ correction.
The probability that a given puncture carries label $m$ falls off as $e^{-2 pi gamma_M sqrt{|m|(|m|+1)}}$, i.e. as the exponential of minus the entropy of that puncture, independent of how $gamma$ is fixed.

Reading between the lines

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

Because the limiting spin distribution is exactly a Boltzmann weight in the puncture area, the horizon state count reads like a partition function over independent punctures; that suggests the $- (1/2) ln a$ term is the standard logarithmic correction from a gas of such punctures rather than an accident of the recurrence.
If the exact count is accepted, matching to the Bekenstein-Hawking coefficient fixes $gamma = gamma_M$, turning a free parameter of the theory into a computable pure number and giving a sharp numerical target for anyone solving the defining equation.
The paper asserts in a footnote that counting with an area window $[a - delta a, a + delta a]$ instead of the strict inequality gives the same answer, but does not show it; a reader who wants the entropy claim to be robust should check that the window does not change the coefficient of the leading term.
Any revival of the quasi-normal-mode link now needs a mechanism that suppresses higher spins, since the correction here shows they dominate rather than decouple.

Original abstract

Quantum Geometry (the modern Loop Quantum Gravity using graphs and spin-networks instead of the loops) provides microscopic degrees of freedom that account for the black-hole entropy. However, the procedure for state counting used in the literature contains an error and the number of the relevant horizon states is underestimated. In our paper a correct method of counting is presented. Our results lead to a revision of the literature of the subject. It turns out that the contribution of spins greater then 1/2 to the entropy is not negligible. Hence, the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived. Also, the conjectured relation between Quantum Geometry and the black hole quasi-normal modes should be understood again.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,418 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Quantum Geometry (the modern Loop Quantum Gravity using graphs and spin-networks instead of the loops) provides microscopic degrees of freedom that account for the black-hole entropy. However, the procedure for state counting used in the literature contains an error and the number of the relevant horizon states is underestimated. In our paper a correct method of counting is presented. Our results lead to a revision of the literature of the subject. It turns out that the contribution of spins greater then 1/2 to the entropy is not negligible. Hence, the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived. Also, the conjectured relation between Quantum Geometry and the black hole quasi-normal modes should be understood again.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Quantum Geometry (the modern Loop Quantum Gravity using graphs and spin-networks instead of the loops) provides microscopic degrees of freedom that account for the black-hole entropy.	background 1.00	0.11	0.12	asserted 1.00
s1	However, the procedure for state counting used in the literature contains an error and the number of the relevant horizon states is underestimated.	gap 0.85	0.21	0.14	asserted 1.00
s2	In our paper a correct method of counting is presented.	contribution 0.73	0.93	0.12	asserted 1.00
s3	Our results lead to a revision of the literature of the subject.	interpretation 0.85	0.32	0.13	asserted 0.99
s4	It turns out that the contribution of spins greater then 1/2 to the entropy is not negligible.	result 0.99	0.88	0.23	asserted 1.00
s5	Hence, the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived.	result 0.75	0.85	0.28	asserted 1.00
s6	Also, the conjectured relation between Quantum Geometry and the black hole quasi-normal modes should be understood again.	interpretation 0.89	0.19	0.06	asserted 0.43

What kind of paper the classifier reports this is: theoretical result 0.77 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
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levels run

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S sections
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (14)

[11]

Meissner KA 2004 Black Hole etropy in Loop Quantum Gravi ty gr-qc/0407052

View on Pith Pith review arXiv 2004
[8]

Ashtekar A, Corichi A and Krasnov K 1999 Isolated Horizon s: The Classical Phase Space Adv. Theo. Math. Phys. 3 419 Ashtekar A, Baez J C, Krasnov K 2000 Quantum geometry of isola ted horizons and black hole entropy Adv. Theo. Math. Phys. 4 1-95

View on Pith 1999
[1]

Thiemann T 2003 Lectures on Loop Quantum Gravity in Quantum Gravity: From The- ory to Experimental Search Lecture Notes in Physics Preprint gr-qc0112038 Thiemann T 2004 Modern Canonical Quantum General Relativity (Cambridge: Cam- bridge University Press) at press gr-qc/0110034

View on Pith Pith review arXiv 2003
[2]

Rovelli C 1998 Loop quantum gravity Living Rev. Rel 1 1

View on Pith 1998
[3]

Ashtekar A and Lewandowski J 2004 Background Independen t Quantum Gravity: A Status Report Class. Quantum Grav. 21 R1-R100 gr-qc/0404018

View on Pith Pith review arXiv 2004
[4]

Krasnov K 1998 On Quantum Statistical Mechanics of a Schw arzschild Black Hole Gen.Rel.Grav. 30 53-68

View on Pith 1998
[5]

Rovelli C 1996 Black Hole Entropy from Loop Quantum Gravi ty Phys.Rev.Lett. 77 3288-3291

View on Pith 1996
[6]

Rovelli C and Smolin L 1995 Discretness of area and volume in quantum gravity Nucl. Phys. B442 593-622; Erratum: Nucl. Phys. B456 753

View on Pith 1995
[7]

Ashtekar A and Lewandowski J 1997 Quantum theory of geome try I: Area operators Class. Quant. Grav. 14 A55-A81

View on Pith 1997
[9]

Bekenstein J D 1973 Black holes entropy Phys. Rev. D7 2333 Bekenstein J D 1974 Generalized second law of termodynamics in black hole physics Phys. Rev. D9 3292 Bardeen J W, Carter B and Hawking S W 1973 The four laws of black hole mechanics Commun. Math. Phys. 31 161

View on Pith 1973
[10]

Dreyer O 2003 Quasinormal Modes, the Area Spectrum, and Black Hole Entropy Phys.Rev.Lett. 90 081301

View on Pith 2003
[12]

Ashtekar A, Domagala M, Lewandowski J, Meissner KA 2004 in preparation

View on Pith 2004
[13]

Ashtekar A and Krishnan B 2004 Isolated and Dynamical Ho rizons and Their Applica- tions gr-qc/0407042 Ashtekar A, Beetle C, Dreyer O, Fairhurst S, Krishnan B, Lewa ndowski J, Wisniewski J 2000 Generic Isolated Horizons and their Applications Phys.Rev.Lett. 85 3564-3567

View on Pith Pith review arXiv 2004
[14]

Domagala M 2004 Kwantowy operator pola powierzchni ora z entropia czarnej dziury Diploma Thesis, Uniwersytet Warszawski, Warsaw, Poland 14

View on Pith 2004

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How later papers used it

86 later papers cite it

Evidence: substantial. 85 of the citing papers share no author with it, and 25 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 150 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool.

Uses its method or tool 2 of 3 papers with this label

Gave a complete computation of the black hole degeneracy spectrum and entropy for the first eigenvalues of the area spectrum according to this paper's DL prescription.

Detailed black hole state counting in loop quantum gravity (2011) · 0.84 · paraphrased: the licence does not allow quoting
Took from this paper the idea that the counting problem can be simplified by implementing the two conditions of another work in separate steps.

Toward explaining black hole entropy quantization in loop quantum gravity (2007) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

84 arXiv papers cite this in 148 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 101 sentences without a confident label

Contests it 4 papers

The concept of `quantum hair' $N$ introduced in [ref] has been adored by several authors resulting in a series of papers where it has been naively overlooked that as a consequence of the introduction of `quantum hair' $N$, $$ can neither be uniquely determined as in [ref] , nor it is a free parameter as opposed to the claim of [ref] .

The Microcanonical Entropy of Quantum Isolated Horizon, `quantum hair' $N$ and the Barbero-Immirzi parameter fixation · Introduction · contest 0.77
Another significant attempt was to fix the Immirzi parameter in the framework of Loop Quantum Gravity [ref] but it was unsuccessful [ref] .

Quantum Tunneling in Black Holes · Overview · contest 0.84
An error in the state counting for the derivation of black hole entropy in [ref] was corrected by Domagala and Lewandowski in [ref] , showing that the dominance of spin $1/2$ representations was incorrect.

New Variables for Classical and Quantum Gravity in all Dimensions V. Isolated Horizon Boundary Degrees of Freedom · Introduction · contest 0.79

Uses its method or tool 3 papers

By using an interval of the form $[0,a_κ]$, Domagala and Lewandowski [ref] proved that the black hole entropy can be obtained according to the following prescription involving only the $m_I$ bulk labels:

Detailed black hole state counting in loop quantum gravity · The ABCK quantum isolated horizon and the DL counting · 0.83
It will also be useful to introduce the shorthand [equation] Finally we take from [ref] the idea that the counting problem can be simplified by implementing the two conditions of [ref] in separate steps.

Toward explaining black hole entropy quantization in loop quantum gravity · Counting states by counting paths · 1.00
We want to mention here that we restrict our analysis in the main body of the paper to the prescription given by Domagala and Lewandowski in [ref] to compute the entropy.

Statistical description of the black hole degeneracy spectrum · Introduction · 0.89

Uses its data or values 10 papers

Some studies such as [ref] suggest that $γ≈0.2375$, therefore $α≈1.1663$.

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For the area gap $Δ$ we use the value $Δ= 5.17$ that comes from the black hole entropy calculations [ref] .

Quantum Gravity in the Sky: Interplay between fundamental theory and observations · Introduction · 1.00
Logarithmic corrections due to the counting of the microstates of BH in LQG give as values $α̃=-1/2$ [ref] and $α̃=-3/2$ [ref] .

Entropy in Loop Quantum Cosmology · Generalized Second Law in LQC. · 0.93

Compares with it 7 papers

For example, exactly in the framework of the standard model, we insert the following numbers of fundamental modes: $N_0=4$ (Higgs), $N_1/2=45$, $N_1=12$, $N_l=3$ (neutrinos), yielding $γ=1/9≈0.11$, which is quite close to the (a bit obsolete) Ashtekar–Baez–Corichi–Krasnov value, $γ_abck=ln2/π√(3)≈0.13$ [ref] , [ref] (see [ref] , for a better estimation).

A relation between the Barbero-Immirzi parameter and the standard model · Introduction · 0.97
The coefficient $-1/2$ of the subleading logarithmic correction term matches the results of $U(1)$ Chern-Simons theory approaches [ref] .

Entropy of black holes with arbitrary shapes in loop quantum gravity · Entropy of Isolated Horizons with Arbitrary Shapes · 0.97
Let us now return to the problem if and how our expression for the entropy ([ref]) can be reconciled with the Loop Quantum Gravity calculation [ref] , [ref] , [ref] , [ref] according to which the black hole entropy computed by counting the black hole horizon microstates equals [equation] where $γ_M$ is a parameter, whose numerical value is between $0.2$ and $0.3$, accompanied by higher order corrections; see [ref] for detailed discussion.

Gravity as a constrained BF theory: Noether charges and Immirzi parameter · Discussion: what about the Immirzi parameter? · 0.99

Cites it as support 2 papers

Computations of black hole entropy via state counting [ref] , as well as entanglement entropy [ref] , are consistent with this expectation.

Path integral renormalisation in loop quantum cosmology · Introduction · 1.00
They must be, to a certain extent, distinguishable to account for a linear area-entropy relation [ref] .

Black hole horizons from within loop quantum gravity · Gauge invariant solutions · 0.96

Cites it as background 14 papers

The theory [ref] predicts the area-dependence of the entropy as [equation] $γ$ is the Barbero-Immirzi-parameter, and $γ_c$ a numerical constant[$γ_c$ depends on the precise definition of which states are counted as surface states.

Toward explaining black hole entropy quantization in loop quantum gravity · Introduction · 0.86
These calculations have a long and continuing history (see for example [ref] and for reviews [ref] ), including some controversy over the correct evaluation of the number of states.

Renormalization and black hole entropy in Loop Quantum Gravity · Introduction · 1.00
The main approaches to the problem of counting the configurations giving the statistical entropy in the microcanonical area ensemble for a back hole are those of DL [ref] and ENP [ref] .

The thermodynamic limit and black hole entropy in the area ensemble · Black hole area ensembles · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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