Remarks on "The Modification of Feynman Diagrams in Curved Space-Time"

Noah M. MacKay

REVIEW 4 major objections 4 minor 9 references

Remarks on "The Modification of Feynman Diagrams in Curved Space-Time"

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A curved-space Feynman modification predicts zero gravitational effect on massless particle probabilities

desk verdict A clear, honest comment whose central massless limit is a coordinate artifact; the physical redshift factor survives. read the letter →

arxiv 2501.15672 v1 pith:5BFZWHFQ submitted 2025-01-26 physics.gen-ph

Noah M. MacKay This is my paper · ORCID

classification physics.gen-ph PACS 04.62.-v11.10.-z

keywords FeynmandiagramscurvedspacetimeSchwarzschildmetricmasslessparticlesgeometricprobabilitymodifierweak-fieldapproximationeffectivemassgravitationallensing

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 review argues that in the Feynman-diagram modification of the cited preprint, the geometric probability modifier for massless particles becomes exactly 1 as the mass goes to zero. Taking that modifier at face value, photons and gluons would feel no gravitational modification of their interaction probabilities on any spacetime. The author treats this as a concerning result because null geodesics are known to be bent and redshifted by curvature, so the unitary limit seems to erase known gravitational effects. The review also flags the limits of the weak-field approximation and proposes a modified massless dispersion relation with a small effective mass as a possible remedy.

What carries the argument

The geometric probability modifier $B(k)$, defined in Eq. (1) as the fourth root of the ratio $\sqrt[4]{(E^2 - m^2(1 - r_S/r_0))/(E^2 - m^2(1 - r_S/r))}$, with $r_S$ the Schwarzschild radius. Taking the massless limit drives both numerator and denominator to $E^2$, forcing $B(k)=1$; this limit is the lever that produces the unitary-probability conclusion.

What would settle it

Compute the full weak-Schwarzschild propagator for a massless scalar field and evaluate the interaction probability including coordinate phase factors; if the probability gains any curvature dependence beyond the factor B(k), the unitary conclusion fails. A direct measurement of photon-photon or photon-gluon scattering near a massive body that exhibits a gravitational correction would also contradict the claim.

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Extended reading notes

Core claim

The central claim is the identity $\lim_{m\to 0} B(k) = 1$, where $B(k)$ is the fourth root of the ratio of Schwarzschild-modified dispersion relations at the observer and emission radii. Interpreting $B(k)$ as the complete geometric probability modifier, this gives unitary probability for massless particles in any spacetime, meaning the cited formalism would leave the probability of distant photons and gluons unmodified by gravity. The author argues this contradicts the intuitive picture of gravitational lensing and redshift of null geodesics and therefore calls the result into question.

Load-bearing premise

The load-bearing premise is that B(k) is the complete physical probability modifier for a particle interaction in curved spacetime; if lensing and redshift enter through the phase or other parts of the propagator, then B(k)=1 does not imply unmodified massless probabilities.

Editorial extensions

If this is right

If $B(k)\to 1$ holds for massless particles, then the cited formalism predicts no gravitational correction to photon or gluon interaction probabilities, conflicting with lensing and redshift intuition.
The weak-field approximation ($r_0, r \gg r_S$) may break down for black-hole geometries; the review suggests a transition point at $r_1 = 2r_{\mathrm{ISCO}}$ for where the weak-to-strong transition could be marked.
A modified massless dispersion of the form $k^2 = E^2 - A k^\alpha f(r)$ with small $|A|$ and $\alpha = 0$ preserves Lorentz invariance while assigning an effective mass, which could screen singularities in Feynman exchange channels.
Vector and tensor particles obey generalized Laplace-Beltrami wave equations, so the curvature corrections could extend to photons, gluons, and gravitational waves in weak Schwarzschild geometry.

Reading between the lines

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

The unitary limit may be an artifact of treating $B(k)$ as a standalone probability modifier; a full propagator calculation would introduce phase and focusing factors that depend on curvature even at zero mass, potentially restoring gravitational effects.
One testable extension is to compute the gravitational redshift of photon energies in the effective-mass dispersion; a nonzero effective mass would produce a measurable deviation from general relativity's prediction.
The proposed weak-to-strong transition at $2r_{\mathrm{ISCO}}$ is heuristic; a more principled criterion could compare the size of curvature corrections to the characteristic interaction scale of the quantum process.
If the effective-mass prescription is applied to gluons, the quark-gluon plasma in a curved background could exhibit modified energy loss or screening patterns, a consequence not developed in the review.

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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

Short version: an honest, clearly written comment on Li's curved-space Feynman diagram preprint. The author spots that B(k) → 1 as m → 0 and reads that as saying gravity leaves massless-particle probabilities untouched. That reading doesn't survive: Eq. (1) is built from conserved coordinate energy, so the E² terms cancel identically for m=0. A static observer measures local momentum with a redshift factor; the physically relevant ratio is [f(r)/f(r0)]^{1/4}, not 1. The stress-test note is right, and it's a bigger problem than the reader's 'B(k) might be incomplete' worry: even as a purely kinematic factor, B(k) in Eq. (1) is not the local quantity that would enter a probability.

What's good: the paper flags a genuine feature of Li's formula, and the writing is transparent. Guesses are labeled as guesses — r1 = 2rISCO is called a 'safe bet', the mass term is an 'ansatz'. The gauge counting for photons, gluons, and gravitons in Section IV is correct textbook material, and the Laplace-Beltrami generalization is a sensible direction to raise, even though the execution omits connection terms for vector/tensor fields.

Where it falls down: the central Section II inference is unsupported, and because the coordinate/local distinction is the whole issue, it's not a minor fix. Section III's transition marker is purely heuristic, with no derivation or physical argument. Section IV is a list of open questions rather than an analysis. The paper also cites LIGO's graviton mass bound correctly, but the proposed effective-mass modification is speculative and doesn't engage with the massive-gravity literature beyond a citation. No data or code, which is fine for a comment, but it means there's no independent check.

Who it's for: someone teaching from Li's preprint or thinking about whether B(k) is the right probability modifier. It could serve as a discussion note. But as a research contribution it doesn't establish anything new. The main claim is likely wrong; the rest is either heuristic or standard.

Recommendation: not for peer review as is. The author should first confront the coordinate-observer issue. If the critique is reworked around local momentum, there might be a viable short comment for a pedagogical venue. As it stands, I'd let it be a blog post.

Referee Report

4 major / 4 minor

Summary. The manuscript is a short critical review of Li's preprint on Feynman diagrams in curved spacetime. It examines the massless limit of the geometric probability modifier B(k), proposes a heuristic marker for the weak-field validity domain, and attempts to extend the scalar d'Alembertian generalization to vector and tensor fields. The central conclusion is that for massless particles B(k) tends to 1 and therefore gravity leaves massless-particle probabilities unmodified, a result the author presents as a defect in Li's framework.

Significance. If the central conclusion were sound, it would identify a serious problem in Li's curved-space Feynman-diagram formalism and would warrant attention. The paper does correctly compute the formal limit lim_{m→0} B(k)=1 from Eq. (2), and it raises a legitimate question about the applicability of the weak-field approximation near black holes. However, the massless limit is an artifact of using coordinate momentum rather than locally measured momentum, and the vector/tensor equations proposed in Section IV are not the standard curved-space field equations. The paper therefore does not currently make a reliable contribution beyond a cautionary remark.

major comments (4)

[Section II, Eqs. (1)-(2)] The inference that gravity leaves massless-particle probabilities unmodified is not supported. In Eq. (1), B(k) is a ratio of coordinate 3-momenta built from the same conserved coordinate energy E in numerator and denominator. For a static observer at radius r, the locally measured energy is E/sqrt(1-r_s/r), so the physically relevant massless momentum ratio is (1-r_s/r)^{1/4} / (1-r_s/r0)^{1/4}, not 1. Thus lim_{m→0} B(k)=1 reflects the cancellation of coordinate energies, not an absence of gravitational effects on physical probabilities. The paragraph following Eq. (2) overinterprets a coordinate artifact as a physical statement.
[Section II, modified dispersion relation] The proposed modification k^2 = E^2 - A k^alpha f(r) is introduced without derivation or justification; for |A| << 1 and alpha = 0 it is simply a mass term with an arbitrary coefficient. No connection between this dispersion relation and B(k) or a physical mechanism is provided, so it does not address the coordinate-artifact problem identified in the previous comment.
[Section III, weak-to-strong transition marker] The marker r1 = 2 rISCO is asserted on the basis of a marble-downhill analogy and is not derived from any quantitative criterion, such as the size of neglected terms in the weak-field expansion. The manuscript itself labels the choice as 'naive' and a 'safe bet', so it cannot support a substantive claim about the validity domain of Ref. [1].
[Section IV, Eq. (7)] The scalar Laplace-Beltrami operator cannot be applied directly to vector and tensor fields. For a vector field, the covariant wave equation in Lorenz gauge is nabla^nu nabla_nu A^mu - R^mu_nu A^nu = 0 (in appropriate sign conventions), which includes curvature coupling terms absent from Eq. (7). Similarly, for the metric perturbation h_mu nu, the linearized Einstein equation involves the Lichnerowicz operator with curvature terms. Consequently, Eq. (7) is not a valid generalization, and the discussion of photons, gluons, and gravitons in that section rests on incorrect equations.

minor comments (4)

[Eq. (1)] The symbol rS in Eq. (1) is inconsistent with r_s used elsewhere in the manuscript; the notation should be unified.
[Last paragraph of Section IV] The text refers to 'LVK observatories' as 'L VK'; this should be written as LIGO-Virgo-KAGRA or an equivalent standard abbreviation.
[References] Reference [2] is cited for both the graviton mass bound and the statement that Lorentz invariance is not violated for |A| << 1 and alpha = 0; the latter claim would benefit from a dedicated reference on Lorentz-violation constraints.
[Figure 1] The figure has no caption in the manuscript text; the caption should define the black boundary, the purple lines, and the meaning of the axes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the m→0 limit is a direct algebraic consequence of the cited Eq. (1), not a fitted or independently constructed prediction.

full rationale

The paper's central computation is the limit lim_{m→0} B(k) = 1 in Section II. This follows immediately by substituting m = 0 into Eq. (1), where the E^2 terms in numerator and denominator cancel identically. That is an algebraic consequence of the formula taken from Ref. [1], not a parameter fit dressed up as a prediction. The physical interpretation of this result may be contested—for example, because B(k) is built from coordinate momenta rather than local static-observer momenta—but that is a correctness or validity objection, not circularity. The paper does not fit any parameter to data and then predict that same data; the only external numerical input is the LIGO/VIRGO graviton mass bound, which is independently cited. The author's own prior works (Refs. [7] and [8]) appear only as supporting examples in Section IV, concerning quark-gluon plasma and LIGO-like Compton scattering; they are not used to justify the load-bearing claim about massless B(k). No self-citation chain forces the conclusion, and no result is equivalent to its input by construction in the sense of the circularity rubric. The speculative proposals in Sections II and III are explicitly flagged as possibilities or 'safe bets' by the author, not derived predictions. Overall, the derivation chain is transparent: Eq. (1) → Eq. (2) → interpretive remark, with no hidden fitted input or definitional identity underlying the conclusion.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The paper introduces two ad hoc constructs, a transition marker and an effective-mass dispersion, without fitting or independent evidence. No numerical data are analyzed. It relies on background GR and QFT assumptions from Ref. [1] and standard gauge theory, plus a graviton mass bound imported from LIGO.

free parameters (3)

r1 (weak-to-strong transition marker) = r1 = 2 rISCO = 6 r_s (suggested, not derived)
Introduced by hand in Section III as a 'safe bet'. No calculation or data selects it.
A and alpha (modified dispersion parameters) = A: not set (suggested |A| << 1); alpha: 0 for effective-mass case, otherwise unspecified
Section II proposes k^2 = E^2 - A k^alpha f(r). The parameters are free and not fixed by any measurement.
effective mass mu = unspecified
Invoked in Section II to give photons or gluons a mass-like dispersion. No value assigned or predicted.

assumptions (4)

domain assumption The ratio B(k) defined from the dispersion relation is a valid geometric probability modifier that tracks gravitational modification of particle interactions.
Eq. (1) from Ref. [1]. The paper's central critique, that B(k)=1 means unmodified, depends on this interpretation.
domain assumption The weak-field and local-Minkowski approximation used in Ref. [1] is applicable to particle interactions in a Schwarzschild well except where the paper questions it.
Section I accepts Ref. [1]'s framework, and this frames the boundary of the critique.
ad hoc to paper The Laplace-Beltrami operator is the complete curved-space generalization of the d'Alembertian for vector and tensor fields, with no curvature coupling terms.
Section IV Eq. (7) uses this ansatz without derivation. Standard treatment includes connection terms, so this is an unproved assumption.
ad hoc to paper The potential f(r) = 1 - r_s/r can be used to locate a weak-to-strong transition point analogously to a marble rolling downhill.
Section III Figure 1 analogy. Used to justify r1 = 2 rISCO without quantitative support.

invented entities (1)

Effective mass mu for massless fields (massive photon or gluon)
purpose: Makes B(k) deviate from 1 for m->0 and screens Feynman exchange-channel singularities at ultra-relativistic kinematics.
Section II proposes assigning an effective mass to massless fields via a modified dispersion relation. No experimental observable is calculated and the mass value is unspecified. The paper cites external screening literature, but not for this specific entity.

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Cite this review

Pith. "Pith review of Remarks on "The Modification of Feynman Diagrams in Curved Space-Time"." pith.science (2026). https://pith.science/paper/5BFZWHFQ

@misc{pith2026250115672,
  author       = {Pith},
  title        = {Pith review of: Remarks on "The Modification of Feynman Diagrams in Curved Space-Time"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5BFZWHFQ}},
  note         = {Machine review of arXiv:2501.15672}
}

read the original abstract

The arXiv preprint 2411.15164v1 \cite{Li:2024ltx} discusses how a weak Schwarzschild spacetime modifies Feynman diagram calculations and the governing equations of scalar and spinor particles. This short review offers an extended analysis on concerning points involving massless particles and limitations of weak-field approximation. Written for the Universit\"at Potsdam lecture \textit{New Developments in Astrophysics, Winter Semester 2024/25}.

Figures

Figures reproduced from arXiv: 2501.15672 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 5 canonical work pages

[1]

Li, [arXiv:2411.15164 [physics.gen-ph]]

B. Li, [arXiv:2411.15164 [physics.gen-ph]]

arXiv
[2]

B. P. Abbott et al. [LIGO Scientific and VIRGO], Phys. Rev. Lett. 118, no.22 (2017) 221101 [erra- tum: Phys. Rev. Lett. 121, no.12 (2018) 129901] doi:10.1103/PhysRevLett.118.221101 [arXiv:1706.01812 [gr-qc]]

arXiv 2017
[3]

Zhang, Comput

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work page arXiv 1998
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arXiv 2003
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Griffiths

D. Griffiths. Introduction to Elementary Particle Physics (John Wiley & Sons, Weinheim, 1987) ISBN: 978-3-527- 40601-2

work page 1987
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R. P. Feynman, F. B. Morinigo, W. G. Wagner, B. Hat- field, D. Pines. Feynman Lectures on Gravitation (West- view Press Inc., 2002) ISBN: 978-0-8133-4038-8

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for LIGO-like Compton scattering), which may be in- sightful for gravitational effects in the quantum scale. 3

work page

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arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Li, [arXiv:2411.15164 [physics.gen-ph]]

B. Li, [arXiv:2411.15164 [physics.gen-ph]]

arXiv

[2] [2]

B. P. Abbott et al. [LIGO Scientific and VIRGO], Phys. Rev. Lett. 118, no.22 (2017) 221101 [erra- tum: Phys. Rev. Lett. 121, no.12 (2018) 129901] doi:10.1103/PhysRevLett.118.221101 [arXiv:1706.01812 [gr-qc]]

arXiv 2017

[3] [3]

Zhang, Comput

B. Zhang, Comput. Phys. Commun. 109, 193-206 (1998) doi:10.1016/S0010-4655(98)00010-1 [arXiv:nucl- th/9709009 [nucl-th]]

work page arXiv 1998

[4] [4]

P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP 05, 051 (2003) doi:10.1088/1126-6708/2003/05/051 [arXiv:hep- ph/0302165 [hep-ph]]

arXiv 2003

[5] [5]

Griffiths

D. Griffiths. Introduction to Elementary Particle Physics (John Wiley & Sons, Weinheim, 1987) ISBN: 978-3-527- 40601-2

work page 1987

[6] [6]

R. P. Feynman, F. B. Morinigo, W. G. Wagner, B. Hat- field, D. Pines. Feynman Lectures on Gravitation (West- view Press Inc., 2002) ISBN: 978-0-8133-4038-8

work page 2002

[7] [7]

N. M. MacKay and Z. W. Lin, Eur. Phys. J. C 82, no.10, 918 (2022) doi:10.1140/epjc/s10052-022-10892-y [arXiv:2208.06027 [nucl-th]]

work page Pith review arXiv 2022

[8] [8]

for LIGO-like Compton scattering), which may be in- sightful for gravitational effects in the quantum scale. 3

work page