arxiv: 2604.15251 · v1 · submitted 2026-04-16 · ✦ hep-th · gr-qc

Recognition: unknown

Kontorovich-Lebedev-Fourier Space for de Sitter Correlators

Nathan Belrhali , Arthur Poisson , S\'ebastien Renaux-Petel , Denis Werth

Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:02 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords de Sitter correlatorsKontorovich-Lebedev-Fourier transformSO(1,d+1) isometry groupin-in perturbation theoryClebsch-Gordan coefficientsKällén-Lehmann representationspectral integrals

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The pith

A new frequency-momentum space for de Sitter correlators, built from the SO(1,d+1) isometry group, converts tree-level diagrams into spectral integrals over meromorphic functions and loop integrals into Clebsch-Gordan orthogonality.

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

The paper constructs a novel representation for correlators in (d+1)-dimensional de Sitter space by combining the standard d-dimensional Fourier transform in spatial momentum with a frequency transform obtained by diagonalizing the quadratic Casimir operator of the isometry group SO(1,d+1). This yields the Kontorovich-Lebedev-Fourier transform, under which square-integrable functions decompose along the principal series while more general functions may receive discrete contributions from other unitary irreducible representations. The resulting KLF-space formulation reproduces the Källén-Lehmann representation for the bulk two-point function and supplies Feynman rules for in-in perturbation theory, with KLF-space correlators connected to late-time correlators by a reduction formula. Propagators become rational functions, so that tree-level exchanges are written as integrals over known meromorphic functions and loop momentum integrals reduce to orthogonality relations among SO(1,d+1) Clebsch-Gordan coefficients, as illustrated by the scalar self-energy.

Core claim

The KLF-space formulation allows tree-level diagrams to be written as spectral integrals over known meromorphic functions and, at loop level, recasts the momentum integral as an orthogonality relation among SO(1,d+1) Clebsch-Gordan coefficients.

What carries the argument

The Kontorovich-Lebedev-Fourier (KLF) transform obtained by diagonalizing the quadratic Casimir operator of SO(1,d+1), used together with the ordinary spatial Fourier transform to define the full KLF space for de Sitter fields and correlators.

If this is right

The single-exchange four-point function reduces to a spectral integral over a known meromorphic function.
The one-loop correction to the scalar propagator is expressed as an orthogonality integral of SO(1,d+1) Clebsch-Gordan coefficients.
All propagators in KLF space take the form of simple rational functions.
Late-time correlation functions are recovered from KLF-space correlators by a reduction formula.

Where Pith is reading between the lines

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

The group-theoretic structure may allow similar simplifications for higher-loop diagrams or for correlators involving spinning fields.
The reproduction of the Källén-Lehmann spectral representation hints that the same decomposition could organize non-perturbative contributions.
The construction might extend to other maximally symmetric spacetimes whose isometry groups admit analogous Casimir diagonalization.

Load-bearing premise

The path-integral formulation of in-in perturbation theory translates directly into KLF space without extra boundary terms, regularization choices, or assumptions about the support of the correlators on the principal series.

What would settle it

An explicit one-loop self-energy computation in KLF space that disagrees with the same diagram evaluated in ordinary momentum space or position space would show that the direct translation of Feynman rules fails.

read the original abstract

In this work, we build a novel frequency-momentum space for $(d+1)$-dimensional de Sitter (dS) correlators from first principles. This construction follows directly from the decomposition into unitary irreducible representations (UIRs) of the spacetime isometry group $\mathrm{SO}(1,d+1)$. While the spatial momentum space is given by the standard $d$-dimensional Fourier transform, the frequency space arises from diagonalising the quadratic Casimir operator, leading to the $(d+1)$-dimensional Kontorovich-Lebedev-Fourier (KLF) transform. We show that square-integrable functions decompose only along the principal series, whereas more general functions can receive discrete contributions from other UIRs. Applying this framework to the bulk CFT two-point function reproduces its K\"all\'en-Lehmann representation. Using the path integral formulation, we derive the Feynman rules for in-in perturbation theory in KLF space, leading to the introduction of KLF-space correlators, which are simply related to late-time correlation functions through a reduction formula. Furthermore, the KLF-space formulation sheds light on the simple mathematical structure of perturbative computations. In particular, the propagators take the form of simple rational functions, and tree-level diagrams can be written as spectral integrals over known meromorphic functions, as demonstrated in the example of the single-exchange four-point function. At the loop level, we show, through the example of the self-energy correction to the scalar propagator, that the group-theoretical nature of the construction allows the momentum integral to be recast as an orthogonality relation among $\mathrm{SO}(1,d+1)$ Clebsch-Gordan coefficients.

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 paper gives a new KLF representation for de Sitter correlators built from SO(1,d+1) Casimir diagonalization, with Feynman rules that turn trees into spectral integrals and loops into Clebsch-Gordan orthogonality.

read the letter

Dear colleague, the main takeaway is that this work constructs a combined frequency-momentum space for de Sitter correlators by diagonalizing the quadratic Casimir of the isometry group to define the Kontorovich-Lebedev-Fourier transform. They then derive in-in Feynman rules in that space and show a reduction formula to late-time correlators. Tree-level diagrams become integrals over meromorphic functions, and loop integrals reduce to orthogonality relations among group coefficients. Propagators simplify to rational functions in the examples given. This is new. Earlier approaches used spatial Fourier transforms or other spectral decompositions, but tying the frequency part directly to the group representations and extracting explicit rules from the path integral is a fresh step. Reproducing the Källén-Lehmann representation for the two-point function serves as a solid consistency check, and the worked examples for the single-exchange four-point function and scalar self-energy make the simplifications concrete. The group-theoretic starting point is a strength. The soft spot sits in the path-integral translation. The construction restricts square-integrable functions to the principal series, yet de Sitter correlators involve time-ordering contours and are often not square-integrable. This raises the possibility of extra surface terms at past infinity or discrete-series contributions that the mapping might not automatically eliminate. The abstract states the rules follow from the path integral, but if the full text does not explicitly demonstrate why boundary terms vanish or how regularization is handled, that step needs close verification. This paper is for people already working on perturbative de Sitter correlators and cosmological observables who are comfortable with representation theory of SO(1,d+1). Readers outside that niche will find the group-theory overhead high. It deserves a serious referee because the framework is original, the claims are specific, and the examples are checkable. I would recommend sending it to peer review while asking the authors to address the path-integral mapping and any assumptions about function support in detail.

Referee Report

1 major / 2 minor

Summary. The paper constructs a Kontorovich-Lebedev-Fourier (KLF) frequency-momentum space for (d+1)-dimensional de Sitter correlators by decomposing functions according to the unitary irreducible representations of the isometry group SO(1,d+1). Spatial momenta use the standard Fourier transform while the frequency direction diagonalizes the quadratic Casimir, yielding the KLF transform. Square-integrable functions lie on the principal series; general functions may receive discrete-series contributions. The construction reproduces the Källén-Lehmann representation of the bulk CFT two-point function, derives Feynman rules for in-in perturbation theory directly from the path integral, introduces KLF-space correlators linked to late-time correlators by a reduction formula, and shows that propagators become rational functions, tree-level diagrams reduce to spectral integrals over meromorphic functions (exemplified by the single-exchange four-point function), and loop integrals (exemplified by the scalar self-energy) become orthogonality relations among SO(1,d+1) Clebsch-Gordan coefficients.

Significance. If the central claims hold, the work supplies a group-theoretic reorganization of de Sitter perturbation theory that converts momentum integrals into spectral integrals or representation-theoretic identities. The independent reproduction of the Källén-Lehmann representation and the explicit reduction of both tree-level and one-loop examples to known meromorphic functions or CG orthogonality constitute concrete strengths. The framework could streamline calculations of cosmological correlators once the path-integral mapping is fully controlled.

major comments (1)

[Feynman rules derivation section] The derivation of the Feynman rules (the section following the reproduction of the Källén-Lehmann representation) asserts that the in-in path-integral formulation translates directly into KLF space. However, the manuscript does not explicitly demonstrate the absence of boundary terms at past infinity or additional discrete-series contributions when the correlators are not square-integrable or when the time-ordering contour is taken into account. This mapping is load-bearing for the subsequent claims about KLF-space propagators and diagram rules.

minor comments (2)

[Reduction formula paragraph] The reduction formula relating KLF-space correlators to late-time correlation functions is stated but its precise contour and support assumptions could be written more explicitly to facilitate reproduction.
[Introduction and §2] Notation for the KLF transform and the principal-series parameter should be cross-checked for consistency with standard references on SO(1,d+1) representations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive summary, and the constructive major comment. We address the point below.

read point-by-point responses

Referee: [Feynman rules derivation section] The derivation of the Feynman rules (the section following the reproduction of the Källén-Lehmann representation) asserts that the in-in path-integral formulation translates directly into KLF space. However, the manuscript does not explicitly demonstrate the absence of boundary terms at past infinity or additional discrete-series contributions when the correlators are not square-integrable or when the time-ordering contour is taken into account. This mapping is load-bearing for the subsequent claims about KLF-space propagators and diagram rules.

Authors: We agree that an explicit discussion of boundary terms at past infinity and possible discrete-series contributions is desirable for rigor. The manuscript already restricts the KLF decomposition to the principal series for square-integrable functions (see the paragraph following Eq. (2.12) and the statement that 'square-integrable functions decompose only along the principal series'). The in-in time-ordering is implemented via the standard iε contour deformation, which is preserved when the propagators are expressed in KLF space (Eqs. (3.8)–(3.10)). However, we did not supply a dedicated paragraph verifying the vanishing of surface terms at past infinity for the time-ordered contour or confirming the absence of discrete-series admixtures for the perturbative correlators under consideration. In the revised version we will insert a short subsection immediately after the reproduction of the Källén-Lehmann representation that (i) recalls the rapid oscillatory decay of principal-series modes at past infinity, (ii) shows that the resulting boundary terms integrate to zero against the test functions appearing in the path-integral variation, and (iii) notes that external legs and internal propagators remain in the principal series, so discrete-series contributions do not enter at the orders considered. This addition will make the mapping fully explicit while leaving all subsequent results unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is independent of its target results.

full rationale

The paper derives the KLF transform directly from the standard decomposition of functions under the unitary irreducible representations of SO(1,d+1) and the diagonalization of the quadratic Casimir operator, both of which are external mathematical facts. It then translates the conventional path-integral formulation of in-in perturbation theory into this basis to obtain Feynman rules, propagators as rational functions, tree-level spectral integrals, and loop-level orthogonality relations among Clebsch-Gordan coefficients. The reproduction of the known Källén-Lehmann representation for the bulk CFT two-point function is presented as a consistency check rather than an input. No equations or self-citations reduce the central perturbative claims to fitted parameters, self-definitions, or prior results by the same authors; the construction remains self-contained against external group-theoretic and path-integral benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard representation theory of the de Sitter isometry group and the definition of the Kontorovich-Lebedev transform; no free parameters are fitted and no new physical entities are postulated.

axioms (2)

domain assumption Functions on de Sitter space decompose according to the unitary irreducible representations of its isometry group SO(1,d+1).
Central to the construction of the KLF transform and the statement that square-integrable functions live only on the principal series.
standard math The quadratic Casimir operator of SO(1,d+1) can be diagonalized to furnish a frequency basis via the Kontorovich-Lebedev transform.
Invoked to define the frequency part of the new space.

pith-pipeline@v0.9.0 · 5621 in / 1512 out tokens · 50875 ms · 2026-05-10T10:02:20.598692+00:00 · methodology

discussion (0)

Reference graph

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