arxiv: 2604.23215 · v1 · submitted 2026-04-25 · ✦ hep-th · gr-qc

Recognition: unknown

Fractions and Fakeons in Quantum Field Theory

Damiano Anselmi

Authors on Pith no claims yet

Pith reviewed 2026-05-08 07:36 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords fractional powersfakeonsd'Alembertianperturbative unitaritygauge theoriesgravity theoriesWard identitiesCutkosky identities

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The pith

Fractional powers of the d'Alembertian operator can be incorporated into quantum field theories using fakeon prescriptions while preserving perturbative unitarity and gauge identities.

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

The paper investigates formulations of quantum field theories whose kinetic terms contain fractional or continuous powers of the d'Alembert operator. It requires that any such theory maintain perturbative unitarity and possess a well-defined classical limit with only a finite number of initial conditions. One approach continues Euclidean correlation functions to Minkowski space by applying the fakeon prescription directly to the fractional part of the power. An alternative decomposes that fractional part into a continuum of ordinary fakeons, and infinitely many such decompositions exist. These options produce inequivalent Minkowski theories that share the same Euclidean counterpart, and the constructions extend to covariant d'Alembertians in gauge and gravity theories where the Ward and Cutkosky identities remain valid in every formulation.

Core claim

Formulations of quantum field theories with fractional or continuous powers of the d'Alembert operator can be constructed so that perturbative unitarity holds and the classical limit is well-defined with a finite number of initial conditions. This is achieved either by continuing from Euclidean space with the fakeon prescription applied to the fractional part or by representing the fractional part as a continuum of ordinary fakeons. The decompositions are infinite in number and give inequivalent Minkowskian theories with identical Euclidean versions. The method works at tree level and for bubble diagrams, and it extends to continuous powers of covariant d'Alembertians in fractional gauge and

What carries the argument

The fakeon prescription applied to the fractional part of the power during Euclidean-to-Minkowski continuation, together with its alternative representation as a continuum of ordinary fakeons.

If this is right

Infinitely many inequivalent Minkowskian theories can share the same Euclidean counterpart.
The constructions work at tree level and for bubble diagrams with unitarity preserved.
Ward identities hold in all formulations of fractional gauge theories.
Cutkosky identities remain valid across every decomposition in gravity theories.
Potential pitfalls in analytic continuation and decomposition calculations are identifiable and avoidable.

Where Pith is reading between the lines

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

The same Euclidean theory could be realized by different physical Minkowski versions whose scattering amplitudes differ at higher orders.
Extensions to full loop calculations would provide concrete tests of whether unitarity survives beyond the demonstrated cases.
The multiplicity of formulations suggests that additional physical criteria may be needed to select among the inequivalent theories.

Load-bearing premise

The fakeon prescription for the fractional part of the power ensures perturbative unitarity and a well-defined classical limit with finite initial conditions.

What would settle it

An explicit computation of a diagram beyond bubble order that violates unitarity or requires an infinite number of initial conditions in the classical limit.

Figures

Figures reproduced from arXiv: 2604.23215 by Damiano Anselmi.

**Figure 1.** Figure 1: Numerical comparison between −2iB2M(1)/λ2 (thick plot) and −2iB′ 2M(1)/λ2 (dashed plot) as functions of δ = γ − 1/2 for 0 < δ < 1/2. We do not have the explicit result for ∆B2M, but for p 2 > 0 we can compare B2M(p 2 ) numerically to (2.7) and another simple option, which is B ′ 2M(p 2 ) = ic′ view at source ↗

read the original abstract

We investigate formulations of quantum field theories whose kinetic terms involve fractional or continuous powers of the d'Alembert operator. The primary requirements are perturbative unitarity and a well-defined classical limit with a finite number of initial conditions. A direct approach consists of continuing the correlation functions from Euclidean space to Minkowski spacetime using the fakeon prescription for the fractional part of the power. Alternative formulations arise through decomposition, in which the fractional part is represented as a continuum of ordinary fakeons. These options are infinite in number and yield inequivalent Minkowskian theories with the same Euclidean counterpart. We demonstrate these features at tree level and for bubble diagrams. We also point out potential pitfalls in the calculations. Finally, we show how to treat continuous powers of covariant d'Alembertians in fractional gauge and gravity theories. The Ward and Cutkosky identities hold in all formulations.

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

Anselmi decomposes fractional d'Alembertian powers into continua of fakeons to get infinitely many inequivalent Minkowski theories with the same Euclidean version, but only checks the identities at tree level and bubbles.

read the letter

The new piece is the explicit decomposition of the fractional part of the power into a continuum of ordinary fakeons. This produces infinitely many different Minkowski formulations that all reduce to the same Euclidean theory. The paper shows that the Ward and Cutkosky identities survive in every version, at least for the cases it checks. It also flags some calculation pitfalls up front, which keeps the presentation straightforward. The tree-level and bubble-diagram calculations are concrete and support the claim that the fakeon continuation works for the fractional operator in those diagrams. The construction itself is a direct extension of the earlier fakeon work, and the multiple inequivalent decompositions add something that was not already in the literature. The limitation is exactly what the stress-test note says: the preservation of unitarity and the classical limit is verified only up to bubbles. Higher-loop diagrams with the fractional power inside loops are not checked, so the general claim rests on the assumption that no new branch-cut or measure problems appear. The paper does not claim to have done those checks, so the gap is stated rather than hidden. This is narrow work aimed at people already using fakeons or fractional operators in gauge and gravity theories. A reader who cares about non-local or fractional QFTs will find the decomposition and the low-order identities useful. It is solid enough on its own terms to go to a serious referee, even though the higher-order verification is missing and would need to be addressed in revision.

Referee Report

2 major / 0 minor

Summary. The paper investigates QFT formulations whose kinetic terms involve fractional or continuous powers of the d'Alembertian. It proposes analytic continuation from Euclidean to Minkowski space via the fakeon prescription applied to the fractional part of the power, together with alternative decompositions of the fractional part into a continuum of ordinary fakeons. These yield infinitely many inequivalent Minkowskian theories sharing the same Euclidean counterpart. Explicit verifications of the prescription, the optical theorem, and the Ward/Cutkosky identities are given at tree level and for bubble diagrams; potential pitfalls are noted; and the construction is extended to continuous powers of covariant d'Alembertians in gauge and gravity theories.

Significance. If the results generalize, the work supplies a concrete route to consistent perturbative treatments of fractional powers in gauge and gravity theories while preserving unitarity and a finite number of initial conditions. The explicit low-order demonstrations and the observation that multiple inequivalent Minkowskian theories arise from one Euclidean theory constitute genuine additions to the fakeon framework. The discussion of pitfalls is also a positive feature.

major comments (2)

[Abstract] Abstract: the assertion that the Ward and Cutkosky identities hold in all formulations rests on explicit checks performed only at tree level and for bubble diagrams. The central claim of perturbative unitarity for arbitrary loop order therefore depends on the unverified assumption that the fakeon continuation (or its continuum decomposition) introduces no new branch-cut or integration-measure obstructions when the fractional power acts on covariant operators inside higher-loop diagrams. This assumption is load-bearing for the unitarity requirement.
[Abstract] The well-defined classical limit with a finite number of initial conditions for the continuum-of-fakeons formulation is stated as a primary requirement but is not constructed explicitly beyond the tree-level and bubble cases already checked; the extension is therefore not yet demonstrated to be free of the very initial-condition pathologies the fakeon prescription is meant to avoid.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive major comments. We respond to each point below, agreeing with the need for greater precision regarding the scope of our explicit checks, and we will make corresponding revisions to the abstract and discussion sections.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that the Ward and Cutkosky identities hold in all formulations rests on explicit checks performed only at tree level and for bubble diagrams. The central claim of perturbative unitarity for arbitrary loop order therefore depends on the unverified assumption that the fakeon continuation (or its continuum decomposition) introduces no new branch-cut or integration-measure obstructions when the fractional power acts on covariant operators inside higher-loop diagrams. This assumption is load-bearing for the unitarity requirement.

Authors: We agree that the explicit verifications of the Ward and Cutkosky identities are limited to tree level and bubble diagrams, and that the general claim for perturbative unitarity at arbitrary loop order relies on the assumption that the fakeon prescription (or continuum decomposition) introduces no new obstructions in higher-order diagrams. In the revised version we will qualify the abstract to state that the identities are verified at tree level and for bubble diagrams, with the expectation of general validity based on the analytic structure observed in these cases. We will add a note in the conclusions acknowledging that a complete demonstration at all loop orders is left for future work. revision: yes
Referee: [Abstract] The well-defined classical limit with a finite number of initial conditions for the continuum-of-fakeons formulation is stated as a primary requirement but is not constructed explicitly beyond the tree-level and bubble cases already checked; the extension is therefore not yet demonstrated to be free of the very initial-condition pathologies the fakeon prescription is meant to avoid.

Authors: We acknowledge that the explicit construction of the well-defined classical limit with a finite number of initial conditions for the continuum-of-fakeons formulation is demonstrated only at tree level and for bubble diagrams. While the general fakeon framework motivates this property, we agree that the manuscript should not overstate the explicit verification. We will revise the abstract and relevant sections to clarify that the classical limit is established in the orders considered, and we will outline the general argument based on the decomposition into ordinary fakeons without introducing new pathologies. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies the established fakeon prescription to fractional powers of the d'Alembertian, explicitly demonstrating the resulting formulations, decompositions, and identities at tree level and for bubble diagrams. The central claims (Ward/Cutkosky identities holding across formulations, inequivalent Minkowskian theories from different decompositions) are shown by direct construction and verification in those cases rather than reducing by definition or statistical fit to the inputs. Reliance on prior fakeon results is present but not load-bearing for the new fractional extension, as the manuscript performs independent checks and points out pitfalls without smuggling an ansatz or renaming a known result as a derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on extending the fakeon prescription (from prior work) to fractional powers and assuming that different decompositions produce valid, unitary theories with matching Euclidean limits and finite classical initial conditions.

axioms (1)

domain assumption The fakeon prescription provides a consistent analytic continuation from Euclidean to Minkowski space for fractional powers while preserving perturbative unitarity.
Invoked throughout to define the physical theory and ensure the classical limit has a finite number of initial conditions.

invented entities (1)

Continuum of ordinary fakeons no independent evidence
purpose: To represent the fractional part of the power as an alternative to the direct fakeon prescription.
Introduced as one of infinitely many inequivalent formulations; no external falsifiable prediction is provided in the abstract.

pith-pipeline@v0.9.0 · 5433 in / 1435 out tokens · 60448 ms · 2026-05-08T07:36:46.605843+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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Reference graph

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