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A non-perturbative construction of the de Sitter late-time boundary

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Every bulk scalar in a Bunch-Davies de Sitter theory with a Källén-Lehmann representation has a well-defined late-time conformal boundary built from principal-series primary operators.

desk verdict A serious, genuinely new proposal for a continuous dS boundary, but its central completeness relation is imported from AdS and needs direct scrutiny. read the letter →

arxiv 2412.00183 v1 pith:3KUXLZVJ submitted 2024-11-29 hep-th

Kamran Salehi Vaziri This is my paper · ORCID
classification hep-th MSC 81T2083C47 PACS 04.62.+v
keywords deSitterlate-timeboundarybulk-to-boundaryexpansionprincipalseriesinversionformulaKällén-Lehmannspectraldensityboundary-to-bulkconnectorcosmologicalcorrelators
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

The paper tries to fix a gap in the non-perturbative understanding of de Sitter spacetime: the late-time boundary should be conformal, but the natural AdS-style expansion fails because it would produce boundary operators with dimensions off the unitary irreducible representations of the de Sitter group. The proposal is a continuous bulk-to-boundary expansion: instead of summing over discrete primaries, integrate over the principal series $\Delta=d/2+i\lambda$. The central technical result is an inversion formula that constructs the boundary primaries as integrals of the bulk field against a bulk-to-boundary propagator, with normalization fixed by the Källén-Lehmann spectral density. This gives a well-defined boundary operator content for interacting bulk theories and reproduces the known perturbative results in the free-field limit. A sympathetic reader should care because it turns the late-time de Sitter boundary into a concrete object that could support a conformal bootstrap for cosmological correlators.

What carries the argument

The central object is the boundary-to-bulk connector $\hat D_\Delta(\eta,y)=(-\eta)^\Delta\,{}_0F_1(\Delta-d/2+1,\tfrac14\eta^2\partial_y^2)$, a differential operator that lifts a boundary two-point function to a bulk-to-boundary propagator and a free bulk-to-bulk propagator; together with the inversion kernel $K^\pm_\Delta$ and the completeness relation $\int d\lambda\,(1/2\pi\xi^\pm_\Delta)\,\Pi^\pm_\Delta(\sigma)=\delta(Y_1,Y_2)$ obtained by Wick rotation from AdS. The connector packages descendants automatically and reduces de Sitter diagram identities (split representation, V diagram, broken leg) to algebraic steps. The inversion formula is the inverse of the expansion only insofar as this completeness relation holds.

What would settle it

Evaluate the left side of the de Sitter completeness relation, $\int d\lambda\,(1/2\pi\xi^\pm_\Delta)\,\Pi^\pm_\Delta(\sigma)$, against a smooth compactly supported test function of two bulk points in planar coordinates, without Wick rotation; if the result is not $\delta(Y_1,Y_2)$ on the full manifold, the inversion formula fails.

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Extended reading notes

Core claim

On its own terms, the central discovery is equation (4.10): the boundary operator $O_\Delta(y)$, with $\Delta=d/2+i\lambda$ on the principal series, is obtained from a bulk field $\phi$ by $O_\Delta(y)=\frac{1}{N_\Delta}\int_{\text{bulk}} K^\pm_\Delta(\eta',y';y)\,\phi(\eta',y')$, where $K^\pm_\Delta$ is the bulk-to-boundary propagator and $N_\Delta=2\pi\xi^\pm_\Delta a_\Delta$ is fixed by the Källén-Lehmann spectral density. This inverts the bulk-to-boundary expansion $\phi(\eta,y)=\int d\lambda\,a_\Delta\,\hat D_\Delta(\eta,y)O_\Delta(y)$, so the boundary theory is constructible directly from bulk data. The resulting boundary operators satisfy conformal Ward identities, have two-point functions of the form $\delta_{\lambda_1\lambda_2}/y_{12}^{2\Delta_1}+\gamma_{\Delta_1}\delta_{\lambda_1,-\lambda_2}\delta(y_1-y_2)$, and reproduce perturbative cosmological correlators through the broken-leg identity.

Load-bearing premise

Everything rests on a distributional completeness relation in de Sitter that the paper derives only by analytic continuation from anti-de Sitter harmonic analysis; if that identity does not hold as a genuine de Sitter distribution, the inversion formula does not invert the expansion.

Editorial extensions

If this is right

  • Any scalar bulk two-point function admitting a Källén-Lehmann representation defines a family of boundary primaries with principal-series dimensions, so the late-time boundary exists without assuming a discrete spectrum.
  • Boundary two-point functions necessarily contain local contact terms in addition to the CFT power law, and the boundary operators fail to commute; these are direct consequences of bulk unitarity and canonical commutation relations.
  • Perturbative de Sitter in-in correlators are recovered from the inversion formula, and the boundary-to-bulk connector reorganizes standard diagrammatic identities (split representation, V diagram, broken leg), simplifying actual calculations.
  • The bulk-to-boundary coefficient decays exponentially at large scaling dimension for theories that flow to a CFT in the UV, making the continuous expansion well-behaved inside correlation functions.

Reading between the lines

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

  • If the completeness relation can be proven directly in de Sitter without the AdS continuation, the same inversion logic should extend to complementary and exceptional series, turning the suspected discrete sums into a theorem.
  • The non-commutativity of the boundary operators suggests the late-time boundary is not an ordinary Euclidean CFT; any bootstrap based on these operators will need to treat contact terms and operator ordering as part of the crossing equations.
  • The exponential suppression of $a_\Delta$ at large $\Delta$ implies the principal-series integral can be truncated in numerical studies of de Sitter correlators, offering a concrete path to finite-coupling checks in models such as the $O(N)$ model.
  • A spinning generalization of the inversion formula, once exceptional-series representations are included, would provide the de Sitter analogue of a boundary stress tensor and a handle on massless spin-2 physics in de Sitter.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a non-perturbative description of the late-time conformal boundary of QFT in de Sitter spacetime. It introduces a continuous family of boundary primary operators O_Δ(y) living on the principal series, together with a bulk-to-boundary expansion in which a bulk scalar is written as an integral over these operators with kernel determined by the Källén-Lehmann spectral density. The main result is an inversion formula, Eq. (4.10), that constructs the boundary operators from the bulk field by integrating against the bulk-to-boundary propagator. The paper also defines a boundary-to-bulk connector operator, studies the large-Δ behaviour of the spectral density, and verifies the inversion formula by recovering boundary two-point functions and by reproducing perturbative correlators, including an explicit three-point example. Several appendices provide supporting identities, including a completeness relation imported from AdS harmonic analysis.

Significance. If the construction is valid, it gives a concrete, symmetry-based definition of boundary operators for de Sitter that is adapted to the continuous principal-series spectrum, going beyond the formal discrete AdS-like expansion whose operators generically lie off the dS unitary representations. The proposed inversion formula and the boundary-to-bulk connector are potentially useful technical tools for cosmological bootstrap computations. The paper is also commendable for its explicit consistency checks, including the two-point function with contact terms, the V-diagram and broken-leg identities, and a three-point example. However, the central inversion step depends on a completeness relation that is obtained only by analytic continuation from AdS, and the convergence of the expansion inside correlation functions is explicitly left open; these are load-bearing gaps that need to be addressed before the construction can be regarded as fully established.

major comments (4)
  1. [§4.2, Eq. (4.14), and Appendix D] The claim that Eq. (4.10) is the inverse of the bulk-to-boundary expansion (3.1) relies on the de Sitter completeness relation (4.14). The derivation in Appendix D is exclusively by Wick rotation from the AdS harmonic-analysis completeness relation (D.7), and the paper itself states that (4.14) must be understood through analytic continuation to AdS. The later consistency checks in §4.3.1 and §4.3.2 use the split representation, the V-diagram and the broken-leg identity, all of which are derived from the same analytic continuation, so they do not provide an independent test of (4.14). I ask for a direct de Sitter derivation or a coordinate-space distributional check of (4.14) — for example, by acting with both sides on the mode-function basis of §2.1 — because if (4.14) fails as an identity, Eq. (4.10) does not invert Eq. (3.1).
  2. [§3.3 and concluding bullet in §5] The convergence of the bulk-to-boundary expansion inside correlation functions is not established. Equation (3.31) controls only the large-Δ behaviour of the bulk-to-boundary coefficient a_Δ; the matrix elements ⟨ψ_i|O_Δ|ψ_j⟩ in Eq. (3.13) also enter the convergence analysis and their large-Δ asymptotics are not determined. The paper's own conclusion states this explicitly. Since the non-perturbative construction is intended to justify inserting (3.1) into correlators, this missing bound is a load-bearing gap. Please either supply the needed estimate or state clearly in the abstract and introduction that the expansion is formal and that only the large-Δ behaviour of a_Δ is analysed.
  3. [§3.1, Eq. (3.1), and §5] The construction covers only the principal-series part of the de Sitter spectrum. For a generic scalar whose Källén-Lehmann density has complementary-series or exceptional-series contributions, Eq. (3.1) is not an equality and Eq. (4.10) defines only a principal-series component of the boundary theory. The title and abstract claim a construction of the de Sitter late-time boundary without this restriction. Please qualify the claims and state explicitly that the paper constructs the principal-series sector, with other representations left to future work.
  4. [§4.3.1, Eqs. (4.20)–(4.21)] The recovery of the power-law coefficient of the boundary two-point function is not an independent check. With the normalization α_Δ=1, Eq. (3.10) fixes a_Δ = sqrt(2 g_Δ ρ(Δ)), and substituting this into Eq. (4.21) makes α_Δ equal to 1 by construction. The independent content of the two-point check is the contact-term coefficient β_Δ, which is not fixed by the normalization convention. The paper should present the two-point check in this way and should highlight that γ_Δ in Eq. (3.12) is independent of the spectral density, since this is a non-trivial prediction of the construction.
minor comments (4)
  1. [Eq. (3.31)] Combining (3.14) and (3.30) appears to give the exponent δ - d/4 - 1/2 rather than the displayed δ - (d-2)/4; please check the exponent and the surrounding constants.
  2. [Throughout] The notation for the continuous operators O_Δ and the discrete operators O_Δ is easy to confuse, especially since both appear in the same equations; please use a more distinct typesetting consistently from the first occurrence.
  3. [Acknowledgements] There is a truncated grant reference in the acknowledgements ('European Union funding (ERC, , 101118787)'); the missing grant number should be filled in.
  4. [§4.4, Eqs. (4.39)–(4.44)] The coefficients ζ_± are constrained by hermiticity only up to their real part; the paper should state explicitly that the imaginary part of ζ_± is left undetermined by the present arguments.

Circularity Check

1 steps flagged · score 2.0 of 10

Mild by-construction normalization in one consistency check; the inversion formula itself is a genuine harmonic-analysis derivation.

  1. self definitional [Section 4.3.1, after Eq. (4.21)]
    "These match exactly with what we found by inserting the Källén-Lehmann decomposition into equations (3.10) and (3.11). As before, one may also normalize the boundary operators by setting αΔ = 1, thereby fixing aΔ in terms of the Källén-Lehmann spectral density to obtain (3.12)."

    The CFT coefficient in Eq. (4.20) is αΔ = Γ(Δ)Γ(d/2−Δ)/(2π^{d/2+1}) ρ(Δ)/aΔ². The same paragraph then sets αΔ = 1, which fixes aΔ = sqrt(2gΔρ(Δ)). Because aΔ (and hence the inversion normalization NΔ = 2πξ±Δ aΔ) is fixed by demanding the very coefficient later said to be 'recovered,' the unit power-law term in the boundary two-point function is an input normalization rather than an independent output of the inversion formula. The contact coefficient βΔ and the perturbative matching in §4.3.2 remain non-trivial checks, so the circularity is mild.

full rationale

The central object is the inversion formula (4.10), OΔ(y) = (1/NΔ)∫ K±Δ φ, with NΔ = 2πξ±Δ aΔ. The direct derivation in §4.1 starts from the proposed expansion (3.1) and uses Kontorovich–Lebedev orthogonality (4.8) to obtain (4.5), giving a genuine inversion of that expansion. The reverse direction in §4.2 uses the completeness relation (4.14), imported from AdS harmonic analysis by Wick rotation in Appendix D; the paper explicitly cautions that the distribution must be understood through analytic continuation to AdS and offers no independent de Sitter proof. That is a load-bearing mathematical assumption, but it is not circular re-use of the target result. The one partly by-construction element is the consistency check in §4.3.1: after computing αΔ ∝ ρ(Δ)/aΔ², the paper sets αΔ = 1 and thereby fixes aΔ = sqrt(2gΔρ(Δ)). Hence the standard power-law term in the recovered two-point function is normalized into existence rather than predicted. The contact-term coefficient βΔ is not fixed by that normalization and does provide an independent check, as do the perturbative correlators in §4.3.2 via the broken-leg identity. Self-citations [4,22] supply background results such as the Källén–Lehmann representation, the Lorentzian inversion formula, and UIR resolutions, but the core inversion formula is derived in-paper from standard Bessel-function orthogonality, so those self-citations are not load-bearing in a circular sense. Overall there is no foundational circularity; the minor by-construction normalization in one verification step keeps the score low but not zero.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The construction depends on an assumed spectral density rho(Delta), the principal-series resolution of identity, and a completeness relation imported from AdS via analytic continuation. No numerical fits are performed, but the spectral density itself is a free theory-dependent function. The continuous boundary operators are the central new objects; they are defined by the inversion formula rather than independently observed.

free parameters (2)
  • Bulk-to-boundary coefficient a_Delta
    Theory-dependent function fixed by normalization alpha_Delta = 1 to sqrt(2 g_Delta rho(Delta)) in Eq. (3.12); the spectral density rho(Delta) is an input from the bulk QFT, not derived here.
  • Coefficients zeta_plus and zeta_minus in the hermitian linear combination O_Delta = zeta_+ O_Delta^+ + zeta_- O_Delta^- = Re zeta_± = 1/2, Im zeta_± unconstrained
    Hermiticity of the bulk field fixes the real part to 1/2 in Section 4.4; the imaginary part is left open.
assumptions (6)
  • domain assumption Bunch-Davies (Euclidean) vacuum with Wightman functions defined by analytic continuation from the sphere.
    Stated in Section 2.1; all correlators and the Källén-Lehmann representation rely on this vacuum choice.
  • domain assumption Resolution of identity over principal series UIRs, Eqs. (2.14) and (2.18).
    Taken from [4]; the completeness of principal series for scalar states in dS is assumed.
  • domain assumption Completeness relation (4.14) for de Sitter Pi propagators, continued from AdS harmonic analysis.
    Derived in Appendix D by Wick rotation from AdS; no independent dS derivation is provided.
  • domain assumption Large-Delta limit of the spectral density is controlled by the UV CFT behavior of the two-point function, Eq. (3.28).
    Used in Section 3.3 to derive exponential decay of a_Delta; assumes the bulk theory flows to a CFT in the ultraviolet.
  • standard math Kontorovich-Lebedev orthogonality relation, Eq. (4.8).
    Quoted from [47,48] and used to derive the inversion formula; no proof is given in the paper.
  • domain assumption Spectral density shadow symmetry rho(Delta) = rho(d - Delta).
    Used throughout to simplify the Källén-Lehmann representation; a property of principal series representations assumed to hold for the bulk theory.
invented entities (1)
  • Boundary primary operators O_Delta(y) with continuous dimension Delta = d/2 + i lambda on principal series
    purpose: Serve as the late-time boundary theory in the bulk-to-boundary expansion (3.1) and inversion formula (4.10).
    These objects are defined by the inversion formula and their two-point functions contain contact terms; the paper checks consistency with free theory and perturbation theory, but no independent falsifiable handle outside the paper is given.

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Pith. "Pith review of A non-perturbative construction of the de Sitter late-time boundary." pith.science (2026). https://pith.science/paper/3KUXLZVJ

@misc{pith2026241200183,
  author       = {Pith},
  title        = {Pith review of: A non-perturbative construction of the de Sitter late-time boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KUXLZVJ}},
  note         = {Machine review of arXiv:2412.00183}
}
read the original abstract

We propose a new approach for constructing the late-time conformal boundary of quantum field theory in de Sitter spacetime. A boundary theory which consists of a continuous family of primary operators residing on unitary irreducible representations, the principal series. These boundary operators exhibit two-point functions that include contact terms alongside standard CFT two-point functions. We introduce a bulk-to-boundary expansion in which a bulk operator, when pushed to the boundary, is represented as an integral over boundary operators. The kernel of this integral is related to the K\"all\'en-Lehmann spectral density, and we examine the convergence of the expansion by deriving the spectral density's large dimension limit. Additionally, we derive an inversion formula for the bulk-to-boundary expansion, where, given a bulk theory, the boundary operator content is constructed as an integral of the bulk operator times the bulk-to-boundary propagator. We verify the inversion formula by recovering the boundary two-point function and reproducing perturbation theory. Along the way, we define an operator that generates both the bulk-to-boundary and free bulk-to-bulk propagators from the boundary two-point function, proving to be a powerful tool for simplifying de Sitter diagrams.

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