Finite-difference zeta function regularisation and spectral weighting in effective actions

Keisuke Okamura

arxiv: 2604.11460 · v1 · submitted 2026-04-13 · 🧮 math-ph · math.MP

Finite-difference zeta function regularisation and spectral weighting in effective actions

Keisuke Okamura This is my paper

Pith reviewed 2026-05-10 15:05 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords zeta function regularisationeffective actionTsallis statisticsinformation geometryfinite differencespectral weightingnonextensive scaling

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The pith

Finite-difference replacement in zeta regularisation enables scale-dependent spectral weighting.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard zeta function regularisation fixes the relative weights of spectral contributions independently of scale. The paper relaxes this by replacing the derivative at s=0 with a finite-difference construction that uses the values of the zeta function at zero and at q-1. In finite systems this produces Tsallis-type quantities and a q-controlled information geometry once the macroscopic limit is taken. In infinite dimensions the same change defines an effective action whose variation is the trace of A to the power -q times the variation of A. The result is that zeta regularisation, effective actions, nonextensive scaling and information geometry all appear as different expressions of one finite-difference principle for aggregating spectra.

Core claim

Replacing the derivative at s=0 in the zeta function with a finite-difference expression built from ζ_A(0) and ζ_A(q-1) turns the regularisation prescription into a scale-dependent one. This yields an effective action Γ_q whose variation satisfies δΓ_q = Tr(A^{-q} δA), realises nonextensive scaling in finite systems, and produces a q-controlled information-geometric structure; zeta regularisation, effective actions, nonextensive scaling and information geometry thereby become different realisations of the same finite-difference spectral aggregation.

What carries the argument

Finite-difference construction based on ζ_A(0) and ζ_A(q-1) that replaces the derivative at s=0, thereby introducing a tunable parameter q that controls the relative weight of each spectral contribution.

If this is right

In finite systems the macroscopic limit produces Tsallis-type quantities.
A q-controlled information-geometric structure emerges.
In infinite dimensions the effective action satisfies the variation δΓ_q = Tr(A^{-q} δA).
Zeta function regularisation, effective actions, nonextensive scaling and information geometry become different views of finite-difference spectral aggregation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same finite-difference idea could be applied to other regularisation schemes to introduce tunable scale dependence.
Numerical checks on small lattices could verify whether the q-dependent weighting matches observed nonextensive behavior in finite quantum systems.
The construction may extend naturally to interacting theories or curved backgrounds where standard zeta regularisation is already used.

Load-bearing premise

The finite-difference replacement of the derivative at s=0 by a construction based on ζ_A(0) and ζ_A(q-1) preserves the essential regularization properties while legitimately producing the claimed scale-dependent weighting and macroscopic limits without introducing inconsistencies.

What would settle it

A direct computation of the one-loop effective action for a free scalar field on a finite lattice that shows the proposed variation δΓ_q = Tr(A^{-q} δA) either fails to reproduce the standard result at q=1 or produces divergences absent from ordinary zeta regularisation.

Figures

Figures reproduced from arXiv: 2604.11460 by Keisuke Okamura.

**Figure 2.** Figure 2: FIG. 2. Spectral weighting induced by the finite-difference effective [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

Standard zeta function regularisation enforces a scale-independent prescription for spectral aggregation, effectively fixing the relative weight of spectral contributions. We relax this constraint by replacing the derivative at $s=0$ with a finite-difference construction based on $\zeta_{A}(0)$ and $\zeta_{A}(q-1)$. In finite systems, it gives rise in the macroscopic limit to Tsallis-type quantities and a $q$-controlled information-geometric structure. In infinite dimensions, it yields an effective action whose variation $\delta\Gamma_{q}=\mathrm{Tr}(A^{-q}\delta A)$ realises scale-dependent spectral weighting. Within this framework, zeta function regularisation, effective action, nonextensive scaling, and information geometry emerge as manifestations of a common principle of finite-difference spectral aggregation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The finite-difference zeta idea links regularization to Tsallis weighting in a new way, but the claimed effective-action variation misses a term from the analytic continuation of zeta(0).

read the letter

The core move is replacing the s=0 derivative in zeta regularization with a finite difference built from zeta_A(0) and zeta_A(q-1). This produces a tunable q that weights the spectrum differently at different scales. In finite systems the construction recovers Tsallis-type quantities and an information-geometric structure in the macroscopic limit, which is a clean and explicit connection not standard in the usual zeta literature. That part reads as a genuine addition within the subfield of spectral regularization methods. The paper also states that in infinite dimensions the resulting effective action Gamma_q satisfies delta Gamma_q = Tr(A^{-q} delta A), which would realize the scale-dependent weighting directly. If the derivations hold, this would be useful for people who want a one-parameter family of regularized actions without introducing an explicit cutoff scale by hand. The main soft spot is exactly there. The finite-difference formula for the variation assumes delta zeta(0) = 0 so that only the zeta(q-1) term survives. But after analytic continuation, zeta(0) receives contributions from the Seeley-DeWitt coefficients in the heat-kernel expansion, so its variation with A is generally nonzero. That leaves an extra term delta zeta(0)/(1-q) in delta Gamma_q that the claimed expression omits. The abstract does not indicate how this is handled, and without an explicit check against the standard trace formulas in the convergence region versus the continued values, the central claim for infinite-dimensional cases looks incomplete. The work is aimed at specialists in zeta-function methods for QFT and statistical mechanics who already know the usual regularization pitfalls. It deserves a serious referee to verify whether the variation formula can be made consistent or whether the extra term can be absorbed without breaking the Tsallis and information-geometry limits. I would send it out for review rather than desk-reject, but the authors would need to address the continuation issue head-on.

Referee Report

2 major / 2 minor

Summary. The paper introduces a finite-difference regularization of the zeta function, replacing the derivative d/ds ζ(s) at s=0 with a difference quotient constructed from ζ_A(0) and ζ_A(q-1) for a parameter q. In finite-dimensional systems this produces Tsallis-type quantities and q-dependent information geometry in the macroscopic limit. In infinite dimensions it defines an effective action Γ_q whose variation is stated to be δΓ_q = Tr(A^{-q} δA), thereby implementing scale-dependent spectral weighting. The framework is presented as unifying standard zeta regularization, effective actions, nonextensive scaling, and information geometry under a common principle of finite-difference spectral aggregation.

Significance. If the variation formula and the preservation of regularization properties under the finite-difference replacement can be established rigorously, the construction would supply a tunable spectral weight q that interpolates between conventional zeta regularization (q→1) and other regimes, with concrete links to Tsallis statistics. The unification claim is ambitious but would be strengthened by explicit checks against known heat-kernel coefficients and by reproducible derivations of the claimed variation.

major comments (2)

[abstract and the section deriving δΓ_q] The derivation of the central variation δΓ_q = Tr(A^{-q} δA) (stated in the abstract and presumably detailed in the section on infinite-dimensional effective actions) inserts the formal trace formula δζ(s) = −s Tr(A^{-s−1} δA) into the finite-difference quotient. This formula holds only for Re(s) sufficiently large that the trace converges absolutely. After analytic continuation, ζ(0) is determined by the Seeley-DeWitt coefficient a_{d/2} and therefore varies non-trivially with A, so δζ(0) ≠ 0 for generic δA. The resulting extra term δζ(0)/(1−q) is omitted from the claimed variation, undermining the assertion that Γ_q realises scale-dependent weighting.
[sections on finite and infinite systems] The weakest assumption—that the finite-difference replacement based on ζ_A(0) and ζ_A(q-1) preserves the essential regularization properties of the analytic continuation while legitimately producing the macroscopic limits—is not verified against standard heat-kernel expansions or known counter-examples where ζ(0) changes under deformations of A.

minor comments (2)

[introduction] Notation for the finite-difference operator and the precise definition of Γ_q should be introduced with an explicit equation number early in the manuscript to avoid ambiguity when the variation is later stated.
[finite-system section] The manuscript would benefit from a short table comparing the q→1 limit with the standard zeta-regularized effective action for a concrete operator (e.g., the Laplacian on a circle) to make the claimed continuity explicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism of our manuscript. The points raised regarding the variation formula and the need for verification against heat-kernel methods are well taken. We provide detailed responses below and have updated the manuscript with corrections and additional material to address these concerns.

read point-by-point responses

Referee: The derivation of the central variation δΓ_q = Tr(A^{-q} δA) inserts the formal trace formula δζ(s) = −s Tr(A^{-s−1} δA) into the finite-difference quotient. This formula holds only for Re(s) sufficiently large. After analytic continuation, ζ(0) varies non-trivially with A, so δζ(0) ≠ 0. The resulting extra term δζ(0)/(1−q) is omitted from the claimed variation.

Authors: We acknowledge that the original derivation overlooked the contribution arising from the analytic continuation of ζ(s). The finite-difference quotient for the effective action Γ_q must account for δζ(0) in addition to the trace term from δζ(q-1). In the revised manuscript, we have updated the derivation in the relevant section and the abstract to include the complete expression δΓ_q = Tr(A^{-q} δA) + δζ(0)/(1-q). We further analyze the physical implications of this term, noting that it provides an additional scale-dependent correction consistent with the nonextensive framework. This revision ensures the regularization properties are rigorously preserved. revision: yes
Referee: The weakest assumption—that the finite-difference replacement based on ζ_A(0) and ζ_A(q-1) preserves the essential regularization properties of the analytic continuation while legitimately producing the macroscopic limits—is not verified against standard heat-kernel expansions or known counter-examples where ζ(0) changes under deformations of A.

Authors: We agree that direct verification is essential. We have added to the revised manuscript explicit computations in an appendix, employing the known heat-kernel coefficients for the scalar Laplacian on the circle S^1 and on the 2-sphere S^2. These examples demonstrate that the finite-difference construction yields the same regularized values as standard zeta regularization when q approaches 1, and that the macroscopic limits to Tsallis quantities remain valid even when ζ(0) is deformation-dependent. While a general theorem for arbitrary operators is beyond the scope of this work, the concrete checks support the assumption in physically relevant cases. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation remains self-contained

full rationale

The abstract presents the finite-difference replacement of the derivative at s=0 as a definitional relaxation of standard zeta regularisation, leading to the stated variation δΓ_q = Tr(A^{-q} δA) as a direct consequence of inserting the difference quotient into the effective action. No load-bearing step reduces to a self-citation, fitted parameter renamed as prediction, or ansatz smuggled from prior work by the same authors. The claimed emergence of nonextensive scaling and information geometry follows from the macroscopic limit of the same construction without invoking external uniqueness theorems or renaming known results. The derivation is therefore independent of its inputs once the finite-difference ansatz is adopted.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Ledger populated from abstract only; q appears as an introduced control parameter whose value is not derived from prior literature.

free parameters (1)

q
Parameter controlling the finite-difference step and the resulting scale-dependent weighting.

axioms (2)

standard math The zeta function ζ_A(s) admits analytic continuation and its derivative at s=0 defines standard regularization.
Invoked when the abstract contrasts the new finite-difference construction with the conventional derivative prescription.
domain assumption Finite systems possess a well-defined macroscopic limit that produces Tsallis-type quantities.
Stated as a direct consequence of the finite-difference construction in the abstract.

pith-pipeline@v0.9.0 · 5421 in / 1390 out tokens · 99478 ms · 2026-05-10T15:05:21.659134+00:00 · methodology

discussion (0)

Reference graph

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