Revisiting the Immirzi parameter: Landauer's principle and alternative entropy frameworks in Loop Quantum Gravity

Everton M. C. Abreu; Jorge Ananias Neto; Ronaldo Thibes

REVIEW 3 major objections 4 minor 55 references

Revisiting the Immirzi parameter: Landauer's principle and alternative entropy frameworks in Loop Quantum Gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper derives the Immirzi parameter of Loop Quantum Gravity from the Landauer principle alone, obtaining exactly $\gamma = \ln 2/(\pi\sqrt{3})$ without Boltzmann-Gibbs microstate counting.

desk verdict Landauer-based Immirzi derivation reproduces the standard value but the 'no Boltzmann-Gibbs' claim doesn't survive the equations; the new Barrow/Kaniadakis formulas are honest algebra with an effective-quantity caveat. read the letter →

arxiv 2412.14156 v2 pith:HBLNR5DI submitted 2024-12-18 gr-qc

Everton M. C. Abreu , Jorge Ananias Neto , Ronaldo Thibes This is my paper · ORCID

classification gr-qc

keywords ImmirziparameterLandauerprincipleLoopQuantumGravityBekenstein-HawkingentropyBarrowKaniadakisblackholeinformationareaquantization

topics Quantum Gravity

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

This paper tries to show that the Immirzi parameter of Loop Quantum Gravity, a free constant governing the area spectrum, can be fixed by an information-theoretic argument instead of the usual counting of gravitational microstates. The route is Landauer's principle: erasing one bit of black hole information costs entropy $\Delta S = k_B \ln 2$, and when that loss is written through the Bekenstein-Hawking area law it fixes the area change to $\Delta A = 4 l_p^2 \ln 2$. Equating that change with the smallest LQG area quantum, $a(1/2)=4\pi l_p^2 \gamma \sqrt{3}$, gives exactly $\gamma = \ln 2/(\pi\sqrt{3})$, the same value obtained from Boltzmann-Gibbs counting. The same procedure is then run inside Barrow and modified Kaniadakis entropy frameworks, producing new area-dependent expressions for $\gamma$ that reduce to the standard value in the appropriate limits. The result connects black hole information erasure to the quantum geometry of loop gravity and offers a thermodynamic route into alternative entropy models.

What carries the argument

The argument runs on three identities joined by one equality. The first is the saturated Landauer principle, $\Delta S = k_B \ln 2$, for erasing one bit. The second is the Bekenstein-Hawking area law, $S = k_B A/(4l_p^2)$, whose variation gives $\Delta S = (k_B/4l_p^2)\Delta A$. The third is the LQG area quantum for a puncture of spin $j$, $a(j) = 8\pi l_p^2 \gamma\sqrt{j(j+1)}$, evaluated at the minimum spin $j=1/2$ as $a(1/2)=4\pi l_p^2\gamma\sqrt{3}$. The load-bearing move is to identify the one-bit area change $\Delta A = 4l_p^2 \ln 2$ with $a(1/2)$, which turns the free parameter $\gamma$ into a fixed number. For Barrow and modified Kaniadakis entropies, the same move is repeated with their modified entropy variations, Eqs. (21) and (29), which is what makes $\gamma$ area-dependent in those frameworks.

What would settle it

Measure or compute the area change that accompanies a one-bit loss of information from a black hole horizon. If it is not $4l_p^2\ln2$, for instance if it is a multiple of that or depends on the horizon area, then equating it to the single spin-1/2 puncture quantum $4\pi l_p^2\gamma\sqrt3$ fails and the derived $\gamma$ cannot be sustained. Within LQG itself, a calculation showing that macroscopic black hole entropy is dominated by punctures with $j>1/2$ would equally falsify the identification used here.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Immirzi parameter does not need the Boltzmann-Gibbs microstate counting to be determined. Starting from the saturated Landauer equality $\Delta S = k_B \ln 2$ and the Bekenstein-Hawking law $S = k_B A/(4l_p^2)$, one obtains $\Delta A = 4l_p^2 \ln 2$ for the area decrease when the black hole loses one bit. Setting this equal to the area quantum of a single spin-$1/2$ puncture, $a(1/2) = 4\pi l_p^2 \gamma \sqrt{3}$, yields $\gamma_{\rm Land} = \ln 2/(\pi\sqrt{3})$, identical to the value from Eq. (7). Repeating the same identification for Barrow entropy gives $\gamma_B = \frac{\ln 2}{\pi\sqrt{3}} \frac{1}{1+\Delta/2}\left(\frac{4l_p^2}{A}\right)^{\Delta/2}$, and for modified Kaniadakis entropy gives $\gamma_{MKE} = \frac{\ln 2}{\pi\sqrt{3}}\sqrt{1+\kappa^2\left(\frac{A}{4l_p^2}\right)^2}$; both reduce to the standard $\gamma$ when $\Delta\to 0$ or $\kappa\to 0$. The paper interprets the resulting area dependence as an effective, model-dependent feature rather than a change in the true LQG parameter.

Load-bearing premise

The whole chain rests on identifying the erasure of one bit with the removal of exactly one spin-1/2 puncture, so that $\Delta S=k_B\ln2$ and $\Delta A=a(1/2)$; if a bit erases several punctures, or the dominant puncture spin is not 1/2, or the saturated Landauer equality fails inside non-extensive entropies, every derived value of $\gamma$ changes.

Editorial extensions

If this is right

The Immirzi parameter is fixed by information thermodynamics alone, giving exactly $\gamma = \ln 2/(\pi\sqrt{3})$, in agreement with the value from Boltzmann-Gibbs microstate counting.
If both routes agree, the Landauer principle provides independent thermodynamic support for the standard value of the Immirzi parameter in Loop Quantum Gravity.
Under Barrow entropy the effective $\gamma_B$ decreases with the fractal exponent $\Delta$; under modified Kaniadakis entropy $\gamma_{MKE}$ grows with $\kappa$, with both tending to the standard value in the limiting cases.
Because the Barrow and Kaniadakis derivations make $\gamma$ depend on the horizon area, those entropies should be regarded as effective thermodynamic descriptions; the underlying LQG parameter and geometric operators stay fixed.
Black hole evaporation that loses one bit at a time saturates Landauer's bound, so the information loss occurs with maximum thermodynamic efficiency.

Reading between the lines

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

If the per-bit/per-puncture identification is taken literally, a black hole radiating $N$ bits should lose area in integer multiples of $a(1/2)$; observing a different area quantum would force the spin labelling or the saturated-equality assumption to change.
The same Landauer-based construction could be applied to other entropy proposals (Tsallis, R\'enyi, loop-quantum-corrected logarithms), yielding a family of predicted effective $\gamma$ values that could be compared against horizon-temperature or evaporation-time constraints.
The equality between the Landauer and Boltzmann-Gibbs derivations is not an accident in the paper's own reasoning: both count $2^N$ states, suggesting that the spin-$1/2$ puncture is itself a bit, though the paper does not develop this reading into an independent justification.
A testable extension would be to compute $\gamma$ with a subleading spin $j>1/2$ or with a distribution of spins; if such corrections shift the predicted $\gamma$ away from $\ln 2/(\pi\sqrt3)$, the one-bit-one-puncture assumption would be falsifiable.

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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

Colleague,

Two things to know off the bat. First, the paper's central derivation – Eqs. (12)–(19) – does exactly what it says algebraically: using the saturated Landauer equality ΔS = k_B ln2 together with the LQG area quantum for j=1/2 gives the standard Immirzi value. Second, the advertised novelty over Boltzmann-Gibbs counting is not real. k_B ln2 is the BG entropy of a two-state system, and the microstate counting W = 2^N is sitting right in the paper's own Sec. 2. The Landauer route is a repackaging of Dreyer's count, not an independent derivation.

What is actually new are the two area-dependent expressions, Eq. (23) for Barrow entropy and Eq. (31) for modified Kaniadakis entropy. The algebra checks; the limits Δ→0 and κ→0 recover the standard γ. The paper also deserves credit for flagging in Sec. 4 that a running γ is inconsistent with LQG's canonical structure and should be treated as an effective quantity. That caveat is prominent enough that the abstract's unqualified claim about deriving "the Immirzi parameter" in those frameworks is the main overreach. The abstract should carry the same qualifier.

The soft spots are all interpretive. The stress-test note is right: Landauer is an inequality, and the paper uses saturation without independent justification. For the non-extensive entropies, the paper simply reuses ΔS=k_B ln2, but the Landauer cost per bit in Barrow or Kaniadakis statistics is not derived; it is assumed. So Eqs. (23) and (31) are conditional on a premise that may fail in those frameworks. Also, the paper does not sharply delineate what is new relative to ref. [20], which already applies Landauer to area quantization. The Sec. 2 result seems to be the same constraint.

None of this is fatal. The paper is short, readable, and honest about its limitations. The formulas (23) and (31) are new enough to be of use to people working on corrections to black-hole entropy. I would take it as a modest but legitimate contribution in a crowded field: the derivation of γ is not new in substance, but the Barrow/Kaniadakis applications are.

Send it to a referee, yes. The referee's main job is to pin down what is actually new versus [20] and to make the authors add the effective qualifier to the abstract. It deserves a serious referee, even if the expected outcome is a revised, more carefully framed paper.

Referee Report

3 major / 4 minor

Summary. The paper applies Landauer's principle to derive the Immirzi parameter in Loop Quantum Gravity, obtaining γ = ln 2/(π√3) in Sec. 2, and then extends the same procedure to Barrow entropy and a modified Kaniadakis entropy, obtaining area-dependent expressions γ_B and γ_MKE in Eqs. (23) and (31). The stated novelty is that the derivation avoids the usual Boltzmann-Gibbs microstate counting. The paper also notes in Sec. 4 that the area-dependent expressions should be interpreted as effective quantities because a varying Immirzi parameter is inconsistent with the canonical structure of LQG.

Significance. If the derivation were an independent route to the Immirzi parameter, it would be a notable conceptual result connecting information thermodynamics and quantum geometry. The paper has clear strengths: the algebraic steps in Eqs. (22)-(23) and (29)-(31) are straightforward and correct, the limits Δ→0 and κ→0 recover the standard value, and the authors explicitly acknowledge in Sec. 4 that area-dependent γ conflicts with LQG and should be read as effective. However, as I argue below, the central claim of independence from Boltzmann-Gibbs entropy is not supported, and the non-extensive extensions assume rather than derive the relevant Landauer bound. The paper is best read as a consistency check or reformulation of the standard Dreyer calculation, not as a parameter-free derivation.

major comments (3)

[Sec. 2, Eqs. (12)-(19)] The claim that the derivation avoids the Boltzmann-Gibbs entropy is not supported. The saturated Landauer equality ΔS = k_B ln 2 in Eq. (12) is exactly the BG entropy change of a two-state system, and for a spin-1/2 puncture the LQG entropy contribution is also k_B ln 2 under the standard counting W = (2j_min+1)^N. Equating ΔA = 4 l_p^2 ln 2 from the Bekenstein-Hawking law with the area quantum a(1/2) = 4π l_p^2 γ√3 is algebraically identical to setting S_BH = N k_B ln 2 with N = A/a(1/2), which is the Dreyer/BG calculation. Thus Eq. (19) is a reformulation of the standard counting, not an independent derivation, and the abstract's phrase 'without using the typical procedure that involves the Boltzmann-Gibbs entropy' should be substantially tempered.
[Secs. 3 and 4, Eqs. (22)-(23) and (30)-(31)] The paper applies the saturated Landauer equality ΔS = k_B ln 2 to Barrow entropy and modified Kaniadakis entropy without deriving or justifying the Landauer bound for those non-extensive frameworks. For Kaniadakis entropy in particular, Eqs. (24)-(25) show that S_κ is not proportional to ln W, so erasing one bit of information does not automatically correspond to an entropy change of k_B ln 2; the entropy change per bit is generally model-dependent. Unless the Landauer bound for these entropies is established, Eqs. (23) and (31) rest on an additional assumption and are not derived consequences of the stated principles. This is load-bearing for the paper's main new results.
[Abstract and Sec. 4, final paragraph] The abstract and the main text (e.g., Eq. (23) and Fig. 1) present γ_B and γ_MKE as 'Immirzi parameter' values, while Sec. 4 concedes that an area-dependent γ is inconsistent with LQG and should be interpreted as an effective quantity. This qualifier is essential and should appear wherever the symbols are introduced and in the abstract; currently the abstract presents the results without that caveat, which is misleading about the physical status of the derived expressions.

minor comments (4)

[Sec. 2, around Eq. (12)] Eq. (14) is described as Landauer's principle 'when it is saturated'; it would be helpful to state explicitly that this follows from combining the Bekenstein-Hawking relation with the assumption that erasing one bit removes exactly one unit of entropy k_B ln 2, since that assumption is the operative input.
[Sec. 4, Eq. (26)] The notation S_BH is used in Eq. (26) before it is defined in the following line; reorder or define it explicitly at first use.
[Sec. 4, Eq. (28)] The paper should clarify whether S_κ^* is intended as a new black-hole entropy formula or as an effective parametrization; the text says both things at different points, and a single consistent statement would avoid confusion.
[Figs. 1 and 2] The y-axis labels 'Immirzi parameter' should be changed to 'effective Immirzi parameter' to match the caveat in Sec. 4, and the captions should state that the plotted quantities are area-dependent effective parameters.

Circularity Check

1 steps flagged · score 5.0 of 10

Main 'Landauer without BG' derivation re-derives the known BG result from an equivalent input; alternative entropy results are model-dependent but not circular.

renaming known result [Abstract and Sec. 2, Eqs. (12)-(19)]
"By leveraging the Landauer principle in conjunction with the Bekenstein-Hawking entropy law, we derive the usual value for the Immirzi parameter precisely, γ = ln2/(π√3), without using the typical procedure that involves the Boltzmann-Gibbs entropy. [...] the matching value obtained from the application of the BG statistic and the Landauer principle likely arises because the number of configurations (microstates) on a punctured surface, as expressed in Eq."

The Landauer input is ΔS = k_B ln2 (Eq. 12), which is precisely the Boltzmann-Gibbs entropy for a two-state system (S=k_B ln W with W=2). Substituting this into the Bekenstein-Hawking variation (Eq. 16) gives ΔA = 4 l_p^2 ln2 (Eq. 17), and equating with the spin-1/2 area quantum (Eq. 18) yields γ = ln2/(π√3) (Eq. 19). This is the same algebraic identity as the BG derivation (Eqs. 3-7); the 'Landauer' route is the BG microstate counting W=2^N dressed as information erasure. The paper itself concedes this by attributing the agreement to W=2^N = Ω=2^N. Thus the abstract's 'without BG' claim is not a first-principles derivation of γ; it is the known Dreyer/BG result re-derived from an equivalent input.

full rationale

The core calculation in Sec. 2 is a self-contained algebraic derivation from three stated inputs (Landauer equality, Bekenstein-Hawking area law, LQG area spectrum); if those inputs are granted, Eq. (19) follows. However, the advertised independence from BG statistics is not real: ΔS=k_B ln2 is the two-state BG entropy, and the paper itself identifies the microstate count W=2^N with the bit count Ω=2^N. So the standard γ result is a renaming/repackaging rather than a new independent constraint. The Barrow and modified-Kaniadakis sections are model-dependent extensions whose starting entropy functions are explicitly imported (for Kaniadakis, from the authors' prior work [42]); no fitted parameter is called a prediction, and no uniqueness theorem is invoked. The reliance on self-cited [42] is transparent and the paper labels it phenomenological, so it does not constitute load-bearing circularity. Overall, the central novelty claim is overstated by one step, but the paper is not systematically circular.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The derivation leans on the LQG area spectrum, the Bekenstein-Hawking law, the choice j_min = 1/2, and the identification of a spin-1/2 puncture with one bit. The Barrow and Kaniadakis branches add two entropy models and reuse the same saturated Landauer equality without independent justification. No new entities are posited.

free parameters (4)

minimum spin j_min = 1/2
Standard Dreyer choice; the dominant configurations are assumed to saturate at the lowest spin. This directly sets the area quantum that fixes γ.
horizon area A = not fixed; 16π l_p² and 64π l_p² used in plots
Appears in γ_B and γ_MKE (Eqs. 23, 31); the paper introduces an area dependence that is later declared effective.
Barrow exponent Δ = varies 0 to 1
Input from Barrow's model quantifying horizon fractal deformation; the derived γ_B depends on it.
Kaniadakis parameter κ = varies 0 to 1
Input from modified Kaniadakis entropy; the derived γ_MKE depends on it.

assumptions (5)

domain assumption The LQG area quantum is a(j) = 8π l_p² γ sqrt(j(j+1)) for a puncture with spin j (Eq. 1).
Standard LQG result taken from cited literature; central to linking ΔA to γ.
domain assumption The Bekenstein-Hawking entropy-area law S = k_B A/(4 l_p²) holds (Eq. 15).
Input benchmark used to fix γ.
ad hoc to paper Each spin-1/2 puncture carries exactly one bit of information, so its entropy is k_B ln2 and erasing it obeys the saturated Landauer equality ΔS = k_B ln2 (Eq. 12).
This is the key identification that makes the derivation work; it is equivalent to the BG microstate count W = 2^N.
domain assumption Barrow entropy S_B = k_B (A/(4l_p²))^{1+Δ/2} (Eq. 20) and modified Kaniadakis entropy S*_κ (Eq. 28) correctly describe black hole entropy.
Taken from cited prior literature; the paper's derivation is conditional on these models.
ad hoc to paper The saturated Landauer equality ΔS = k_B ln2 holds inside the Barrow and Kaniadakis entropy frameworks.
The paper applies the same ΔS = k_B ln2 input to modified entropies without further justification; for non-extensive entropies this is not automatic.

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Pith. "Pith review of Revisiting the Immirzi parameter: Landauer's principle and alternative entropy frameworks in Loop Quantum Gravity." pith.science (2026). https://pith.science/paper/HBLNR5DI

@misc{pith2026241214156,
  author       = {Pith},
  title        = {Pith review of: Revisiting the Immirzi parameter: Landauer's principle and alternative entropy frameworks in Loop Quantum Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBLNR5DI}},
  note         = {Machine review of arXiv:2412.14156}
}

abstract

This paper investigates the implications from area quantization in Loop Quantum Gravity, particularly focusing on the application of the Landauer principle -- a fundamental thermodynamic concept establishing a connection between information theory and thermodynamics. By leveraging the Landauer principle in conjunction with the Bekenstein-Hawking entropy law, we derive the usual value for the Immirzi parameter precisely, $\gamma = \ln2/(\pi \sqrt{3})$, without using the typical procedure that involves the Boltzmann-Gibbs entropy. Furthermore, following an analogous procedure, we derive a modified expression for the Immirzi parameter aligned with Barrow's entropy formulation. Our analysis also yields a new expression for the Immirzi parameter consistent with a corresponding modified Kaniadakis entropy for black hole entropy further illustrating, along with Barrow's entropy, the applicability of Landauer's principle in alternative statistical contexts within black hole physics.

Figures

Figures reproduced from arXiv: 2412.14156 by the authors.

**Figure 2.** FIG. 2: Values of the Immirzi parameter [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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C. Promsiri, E. Hirunsirisawat and W. Liewrian, Phys. Rev. D 102 (2020) 064014. 12

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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Y. Jun, M. Gavrilov and J. Bechhoefer, Phys. Rev. Lett. 113 (2014) 190601

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INTRODUCTION The quantum operator for area in Loop Quantum Gravity (LQG) has a discrete spec- trum [1]. This result plays a key role in the overall framework of LQG and suggests that geometric quantities such as area and volume, unlike in classical general relativity, are quantized at the Planck scale. In classical treatments of gravity, spacetime is cons...

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LANDAUER PRINCIPLE The Landauer principle [8–10], introduced by Rolf Landauer in 1961, is a fundamental concept that integrates ideas from information theory and thermodynamics. It states that erasing a single bit of information requires a minimum energy expenditure, mathematically described by the inequality ∆E≥k BTln 2,(8) wherek B is the Boltzmann cons...

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BARROW ENTROPY In a seminal recent contribution [21], Barrow investigated the possibility that quantum gravitational effects might lead to a complex fractal structure on the surface of a black hole. Such structure alters the actual horizon area, resulting in a new relation for black hole entropy, namely, SB =k B A 4l2 p 1+ ∆ 2 ,(20) whereAis the usual hor...

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MODIFIED KANIADAKIS ENTROPY The Kaniadakis statistics, also known asκ-statistics, provides a non-extensive gener- alization of the classical Boltzmann-Gibbs (BG) framework, introducing a deformation parameterκthat modifies the original entropy formulation [31–34]. Theκ-entropy is given by Sκ =−k B WX i=1 p1+κ i −p 1−κ i 2κ ,(24) wherek B is the Boltzmann ...

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Generally, the Immirzi parameter is arbitrary, but we propose a procedure to determine it by employing the Landauer principle

CONCLUSIONS In this paper, we examined the effect of the Landauer principle within the framework of LQG, focusing particularly on the Immirzi parameter with the lowest spin possible which is jmin = 1/2. Generally, the Immirzi parameter is arbitrary, but we propose a procedure to determine it by employing the Landauer principle. Next, from an analytical pe...

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ACKNOWLEDGMENTS We would like to thank the anonymous Referee for the important and constructive suggestions, which helped us to improve the clarity and scope of our manuscript. Jorge Ananias Neto would like to acknowledge CNPq (Conselho Nacional de Desenvolvimento Cient ´ ıfico e Tecnol´ ogico), Brazilian scientific support federal agency, for partial fin...

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O. Dreyer, Phys. Rev. Lett. 90(8) (2003) 081301-1

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