REVIEW 4 major objections 4 minor 9 references
Some Comments on infinities on Quantum field Theory : A functional integral approach
T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper claims that one coupling rescaling, $g_{\mathrm{bare}} = g_{\mathrm{rem}}/((1-\alpha)^{1/2}v)$, makes a two-dimensional scalar field theory's generating functional finite at the ultraviolet limit $\alpha = 1$.
desk verdict A recycling of the author's own fractional-Laplacian regularization, with the finite-limit claim resting on an unproved determinant expansion and a circular coupling choice. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the analytically regularized covariance $L^{-1}_{\alpha,\Omega,m} = \chi_\Omega [(-\Delta)^\alpha + m^2]^{-1}\chi_\Omega$, which for $\alpha > D/2$ is trace-class and, by the Minlos–Bochner theorem, defines a probability measure supported on $L^2(\Omega)$ — ordinary square-integrable functions instead of distributions — so interactions become legitimate objects. The renormalization prescription $g_{\mathrm{bare}}(\alpha,v) = g_{\mathrm{rem}}/((1-\alpha)^{1/2} v)$ is engineered so that its $(1-\alpha)^{-1/2}$ blow-up matches, at order $N$, the $(1-\alpha)^{N/2}$ leading behavior supplied by the determinant expansion (22), $L_N(\alpha,v) = \int_{\Omega^N} d^2x_1\cdots d^2x_N\, \det^{-1/2}[L^{-1}_\alpha(x_i,x_j)] = (1-\alpha)^{N/2}C_N + \cdots$, with $C_N = v^N (4\pi)^{N/2}(\det A)^{-1/2}$ and $A$ the matrix that is zero on the diagonal and one off it. The determinant of $A$ contributes the factor $(N-1)^{-1/2}$ that suppresses higher orders, making the perturbative series summable; the author also claims the analogous kernel $(-\Delta^2)^{-\alpha}$ carries the same argument in four dimensions.
What would settle it
Take the two-point case on a torus, where the fractional Laplacian has an explicit eigenbasis, and compute $L_2(\alpha,v) = \int_{\Omega^2} d^2x_1\, d^2x_2\, \det^{-1/2}[L^{-1}_\alpha(x_i,x_j)]$; the cancellation (24) at second order requires $L_2(\alpha,v) \to (1-\alpha)\,4\pi v^2$ as $\alpha \to 1$, up to terms that vanish after multiplication by $(g_{\mathrm{bare}})^2$. Repeating the check at third order — where the predicted limit involves $(N-1)^{-1/2}$ with $N=3$ — would test whether the remainder terms genuinely drop out; any mismatch in the power of $(1-\alpha)$ or a surviving remainder falsifies the finiteness claim.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is the finiteness statement of Section 3: the functional path integral (14), interpreted through the Feynman perturbative definition (18) — limits nested so that $m^2 \to 0$ first, then $\alpha \to 1$, then $N \to \infty$, and finally $v \to \infty$ — has a finite limit at $\alpha = 1$ once the bare coupling is set to $g_{\mathrm{bare}}(\alpha,v) = g_{\mathrm{rem}}/((1-\alpha)^{1/2}v)$. The proof is an estimate: each $N$-th order term is bounded by $|g_{\mathrm{rem}}|^N (4\pi)^{N/2}/(N-1)^{1/2}$ times the norm of the interaction's Fourier transform, so the whole series is dominated by $C^2 e^C$ with $C = (4\pi)^{1/2}|g_{\mathrm{rem}}|\|\tilde{V}\|_{L^\infty}$; consequently the generating functional is finite and continuous in the source $j \in L^2(\Omega)$. The same construction is claimed to work in four dimensions with the square-Laplacian kernel $(-\Delta^2)^{-\alpha}$, while the competing propagator prescription (29), which converges to the ordinary massless two-dimensional Green function, is shown to fail because that Green function is not a tempered distribution.
Load-bearing premise
The finiteness claim rests on two unproved premises: the determinant expansion (22), which is asserted from the author's earlier work and must have leading term $(1-\alpha)^{N/2}C_N$ with vanishing remainders, and the specific nested order of limits in (18) that identifies the path integral with the $N \to \infty$ limit of its perturbative series before the volume diverges.
Editorial extensions
If this is right
- The full generating functional (14), under the limit order (18), is finite and continuous in the source $j \in L^2(\Omega)$, so the model yields a genuine Euclidean field-theory object rather than a formal divergent series.
- Every truncation of the perturbative series obeys the uniform bound $|I_N| \le C^2 e^C$ with $C = (4\pi)^{1/2}|g_{\mathrm{rem}}|\|\tilde{V}\|_{L^\infty}$, so convergence is dominated by an ordinary exponential series.
- For $\alpha > 1$ and $\delta > 0$, the exponential-cutoff interactions have finite $N$-point functions (eq 31), so the framework supports non-Gaussian measures before any cutoff is removed.
- The same proof is claimed to extend to a four-dimensional model whose kinetic operator is the square Laplacian $(-\Delta^2)^\alpha$ on a finite volume in $\mathbb{R}^4$.
- The naive propagator prescription (29) is shown to fail because the massless two-dimensional Green function is not a tempered distribution, so the finite-limit claim is tied to the specific determinant-cancellation route rather than to any propagator that approaches the standard Green function.
Reading between the lines
- If expansion (22) can be proved with explicit control of its remainder — e.g., by diagonalizing the fractional Laplacian on a torus — the same 'matching powers' rule, pairing a $(1-\alpha)^{-1/2}$ coupling divergence against a $(1-\alpha)^{N/2}$ determinant factor, should generalize to other super-renormalizable models with power-like ultraviolet singularities.
- The designated order of limits in (18) is load-bearing: the paper sends $m^2 \to 0$, then $\alpha \to 1$, then $N \to \infty$, then $v \to \infty$, and a numerical check of the low-order coefficients under a reversed order — say $N \to \infty$ before $\alpha \to 1$ — would show whether the divergence the construction is designed to cancel reappears.
- The claimed four-dimensional extension with kernel $(-\Delta^2)^{-\alpha}$ is the natural place to look for a counterexample, since the kernel's power in $|x-y|$ differs; computing $L_2$ and $L_3$ there would show whether the determinant cancellation survives in higher dimension.
- The contrast between the successful determinant route and the failed naive propagator suggests a general moral the paper leaves implicit: in a measure-based formulation, renormalizability is a property of the pairing between the coupling's regulator dependence and the covariance's singular behavior, not of the bare action alone.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a finite-volume analytic regularization of Euclidean scalar field theory based on fractional powers of the Laplacian. For α > D/2 the Gaussian measure with covariance χ_Ω(−Δ)^{−α}χ_Ω is constructed via the Minlos–Bochner theorem (Section 2). In Section 3 the author defines a bare coupling g_bare(α,v) = g_rem/((1−α)^{1/2} v) in Eq. (17) and claims that the perturbative generating functional Eq. (14), understood through the nested limit in Eq. (18), has a finite limit at α = 1 and v → ∞. The remaining sections review the Wiener measure construction and propose a Feynman–Wiener geometrodynamical propagator, with appendices on distributional Fourier transforms and functional calculus.
Significance. If the central claim were established, the paper would provide a concrete finite-volume regularization in which a 2D scalar interaction has a finite perturbative generating functional, and it would clarify the nature of ultraviolet infinities in a mathematically explicit setting. The first two theorems of Section 2 are standard and correctly identify the range α > D/2 in which the fractional Laplacian inverse is trace class; the bound in Eq. (19) is a real estimate given the determinant expansion. However, the central removal step is not proved: the determinant expansion Eq. (22) is asserted and cited to the author's own previous work, the covariance used to compute its leading coefficient appears inconsistent with the covariance of Eq. (15), and the higher-order terms are discarded without uniform control. No empirical prediction or machine-checked proof is supplied. The result therefore remains an unsupported claim rather than an established theorem.
major comments (4)
- [§3, Eq. (22)] The entire cancellation that converts the divergent prefactors (1−α)^{−N/2} into the finite expression in Eq. (24) relies on the expansion L_N(α,v) = (1−α)^{N/2} C_N + ⋯, but this expansion is not proved in the manuscript; it is cited to the author's earlier papers [2,3]. Since this expansion is the only mechanism that neutralizes the singular bare coupling of Eq. (17), the central conclusion 'has a finite limit for α = 1' is unsupported.
- [§3, Eqs. (15) and (23)] The leading coefficient C_N in Eq. (23) is computed from the matrix A_ii = 0, A_ij = 1, which corresponds to a translation-invariant kernel whose diagonal vanishes for α > 1. But the actual covariance of the measure in Eq. (15) has a positive diagonal: its trace, computed in Eq. (9), diverges as α → 1, and a finite-volume spectral covariance has G_α(x,x) ∼ c/(α−1). A zero-diagonal determinant and a positive-diagonal determinant with similar off-diagonal behavior need not have the same leading coefficient, and the v-scaling of the remainder can differ. Therefore Eq. (22) is not justified as stated.
- [§3, Eqs. (18) and (22)] The higher-order terms in Eq. (22) are discarded without a uniform estimate in N and v. The limit order in Eq. (18) sends v → ∞ after α → 1, so a remainder of the form (1−α)^m v^p would survive the limit unless p = 0 or m grows with p. No such control is provided. Thus even if the leading term of Eq. (22) were correct, the claimed finite limit is not established.
- [§3, Eq. (18)] The nested limit in Eq. (18) is presented as the 'R.P. Feynman sense', but the proof of Theorem 1 bounds lim_{N→∞} lim_{α→1} |I_N| at fixed v. The order in Eq. (18) instead takes α → 1 before N → ∞ and before v → ∞. No dominated-convergence or uniformity argument is supplied to justify this interchange or even to show that the stated order is well defined. This ambiguity affects the meaning of the central claim.
minor comments (4)
- [References [4] and [5]] The text attributes statements to 'S. Coleman ([4])' and 'G. Hoft ([5])', but the listed references [4] and [5] are Green–Schwarz–Witten and Klaiber; the citations should be corrected.
- [Throughout] The manuscript contains many typographical and grammatical errors ('Districutions', 'quntum', 'theories by theirs turn', 'analitically', 'Feynman-Dhyson'), and the equations contain garbled symbols and inconsistent notation; a careful proofread is needed.
- [Appendix C] The inversion formula in Eq. (C-9) rests on an explicitly non-proved hypothesis and an 'open problem' in infinite-dimensional analysis; since this appendix is not used in the main argument, it should be labeled as formal or speculative.
- [§3, Eq. (24)] The determinant factor for the matrix A is written ambiguously as '|(−1)(N − 1)(−1)^N|'; the sign and absolute value should be stated in a single unambiguous expression, such as |det A| = N − 1 for N ≥ 2.
Circularity Check
The Section 3 finite-limit theorem is not derived from the path integral: the counterterm Eq. (17) is chosen to cancel the self-cited leading divergence Eq. (22), and the higher-order terms are never controlled.
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self definitional
[Section 3, Eqs. (17), (22), and (24)]
"Let us show that by defining the bare coupling constant by the renormalization prescriptions (v = vol(Ω)) gbare(α, v) = grem/((1−α)1/2v) (17) ... We conclude this, that the functional path integral eq(14) under the renormalization coupling constant eq(17) and rigorous Feynman perturbative definition eq(18) has a finite limit for α = 1."
Eq. (17) is a prescription, not a derived quantity: the exponent (1−α)^{−1/2} is chosen to be the exact reciprocal of the leading (1−α)^{N/2} factor that Eq. (22) assigns to L_N. Inserting the two into Eq. (24) makes the cutoff and volume factors cancel identically at every order N. The claimed conclusion 'has a finite limit' is therefore the definition of g_bare plus the assumed exponent, not an output of an independent calculation. What would make the result nontrivial — proving Eq. (22) and showing the discarded terms vanish under the Eq. (18) limits — is not supplied.
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ansatz smuggled in via citation
[Section 3, Eq. (22), citing refs. [2] and [3]]
"At this point we note the Taylor expansion of the below written object L_N(α, v) = ∫ d2x1 · · · d2xn det^{−1/2}_{N×N}[L^{-1}_α(x_i,x_j)] = (1−α)^{N/2}C_n + (1−α)^{N/2+m_1}C_{N+1} + . . . (22) ... It yield thus ([2], [3]) for N > 1"
This expansion is the only step that converts the singular prefactors of Eq. (17) into the finite combination displayed in Eq. (24). It is not proved in the paper; the words 'It yield thus ([2], [3])' refer to two earlier papers by the same author. No independent proof, machine-check, or uniform estimate of the remainder terms C_{N+1}, ... is given, and those higher-order terms are dropped before α→1 and v→∞. Under the order of limits in Eq. (18), a remainder of the form (1−α)^m v^p would survive unless p=0, and no estimate rules this out. Thus the central finite-limit claim rests on an asymptotic ansatz imported from the author's own prior work; the derivation chain in this paper reduces to algebra on that self-cited expansion.
full rationale
The Section 3 derivation is circular in two connected ways. First, the bare coupling Eq. (17) is defined with exactly the (1−α)^{-1/2} v^{-1} scaling needed to cancel the leading (1−α)^{N/2} v^N behavior of L_N asserted in Eq. (22); after substitution, Eq. (24) is an identity, so the advertised 'finite limit' is built into the renormalization choice rather than deduced from the path integral. Second, the load-bearing expansion Eq. (22) is taken from the author's own references [2], [3] and is not proved here, and its omitted higher-order terms are not estimated in the N, v limits of Eq. (18). The paper itself later says 'we conjecture that the ultra-violet limit α→1 on the usual correlations functions ... should expected to be finite,' indicating that only the perturbative generating functional is claimed and that the corresponding correlation-function statement is left open. The surrounding paper does contain non-circular, independent measure-theoretic material (e.g., Theorem 1 and the construction of the Wiener measure), so the paper is not wholly circular; however, its central advertised result, the finite limit at α=1, reduces by construction to the chosen counterterm plus a self-cited asymptotic expansion. No empirical constants are fitted, so this is logical/definitional circularity rather than statistical circularity. Score 7 reflects that the central claim is close to being forced by self-citation and definition, though not quite score 8 because the paper also contains independent mathematical constructions and the expansion, if proved elsewhere, could in principle carry the result.
Assumptions & free parameters
free parameters (4)
- alpha (fractional Laplacian power) =
alpha > D/2, final limit alpha -> 1
- Omega and volume v = vol(Omega) =
finite box, limit v -> infinity
- delta (exponential cutoff in interaction) =
delta > 0, limit delta -> 0 conjectured elsewhere
- g_bare(alpha,v) scaling relation =
g_rem / ((1-alpha)^{1/2} v)
assumptions (4)
- standard math Minlos-Bochner theorem validates Gaussian measures from characteristic functionals.
- standard math The operator L^{-1}_{alpha,Omega,m} with kernel chi_Omega [(-Delta)^alpha + m^2]^{-1} chi_Omega is trace class for alpha > D/2.
- ad hoc to paper The determinant expansion eq(22) holds with leading term (1-alpha)^{N/2} C_N and vanishing higher-order contributions.
- domain assumption The path integral is interpreted as the N -> infinity limit of its perturbative expansion in g_rem, with the limit order of eq(18).
Cite this review
Pith. "Pith review of Some Comments on infinities on Quantum field Theory : A functional integral approach." pith.science (2026). https://pith.science/paper/4CVVMSUP
@misc{pith2026190901081,
author = {Pith},
title = {Pith review of: Some Comments on infinities on Quantum field Theory : A functional integral approach},
year = {2026},
howpublished = {\url{https://pith.science/paper/4CVVMSUP}},
note = {Machine review of arXiv:1909.01081}
}
read the original abstract
We analyze on the formalism of probability measures -functional integrals on function spaces , the problem of infinities on Euclidean field theories
Reference graph
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2015 doi
Reviewed August 14, 2026 · model on record in the stance chip above.
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