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REVIEW 4 major objections 5 minor 33 references

The (No) Boundary Proposal and excited states in de Sitter holography

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that adding a second Euclidean boundary with arbitrary Dirichlet data to the no-boundary path integral defines excited states in de Sitter holography, and it computes their correlators.

desk verdict A carefully computed proposal for excited dS states via an extra Euclidean boundary; the central map is posited and gravity is left for later, but it deserves a real referee. read the letter →

arxiv 2506.16943 v2 pith:TB2PRH2L submitted 2025-06-20 hep-th

Marcelo Botta-Cantcheff , Facundo Lorenzo Cruz , Pedro J. Martinez This is my paper · ORCID
classification hep-th
keywords deSitterholographyno-boundaryproposalexcitedstatesdS/CFTcorrespondenceholographiccorrelatorsEuclideanpathintegrallate-timealpha-vacua
open problems Quantum Gravity
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 aims to establish that excited states in de Sitter holography (the dS/CFT correspondence) can be built by relaxing the no-boundary proposal: instead of a Euclidean geometry whose only boundary is the Cauchy surface, one considers a Euclidean region with a second compact boundary $\Sigma_b$ and fixes arbitrary Dirichlet data $\phi_b$ there. The central claim is that the resulting wave functional $\Psi'_{\phi_b}$ equals the CFT generating function with a nonzero source $J_b$, so the dual operator acquires a nonvanishing one-point function and late-time cosmological correlators are modified. This matters because it supplies a concrete holographic dictionary for states other than the Euclidean vacuum in de Sitter holography, a gap highlighted by the paper. The construction is worked out explicitly for a free massive scalar in $dS_{1+1}$, yielding corrected one- and two-point functions and a wave-function renormalization controlled by $\phi_b$.

What carries the argument

The load-bearing object is the generalized Euclidean path integral with two boundaries, equation (2.5), which prepares the state by evolving from an arbitrary field configuration $\phi_b$ at Euclidean time $\tau_b$ to the Cauchy surface. The identity that carries the argument is the generalized holographic prescription $\Psi'_{\phi_b} = Z_{\mathrm{CFT}}(J_b)$, checked by functional differentiation: the linear kernel $K_b(x,y)$ in the on-shell action couples the asymptotic source to the boundary data and produces the one-point function. The explicit computation in $dS_{1+1}$ relies on mode solutions written in terms of associated Legendre functions $P^{|n|}_{\Delta-1}$ and on splitting the renormalized on-shell action into an asymptotic piece, a $\Sigma_b$ piece, and a cross term; the asymptotic piece yields the deformed two-point function while the cross term yields the one-point kernel.

What would settle it

Include back-reaction in the $dS_{1+1}$ computation, solve the junction conditions and the Hamiltonian constraint on the two-boundary geometry, and check whether a finite-norm wave functional exists for finite $\tau_b$ with generic $\phi_b$; if only $\phi_b = 0$ or $\tau_b \to -\infty$ survive, the construction fails. A lighter test is to compute the corrected two-point function (3.18)-(3.19) near $\tau_b \to 0$, where the small-hole expansion breaks down; a divergence or a failure of the CFT-source interpretation there would signal that the holographic identification does not hold.

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Extended reading notes

Core claim

The paper's central claim is that the identity $\Psi'_{\phi_b}(\phi,T) = Z_{\mathrm{CFT}}(J)$ holds when the initial state is prepared by a Euclidean path integral on a compact manifold with two boundaries, $\Sigma_0$ and $\Sigma_b$, with arbitrary Dirichlet data $\phi_b$ on $\Sigma_b$. Functional differentiation then identifies wave-functional coefficients with CFT $n$-point functions in the source $J_b$; in particular, the linear term gives $\langle O(x)\rangle_{J_b} = \int_{\Sigma_b} dy\, K_b(x,y)\, \phi_b(y)$, which is nonzero for generic $\phi_b$. In the explicit $dS_{1+1}$ example, the late-time two-point function becomes the vacuum correlator times a correction depending only on the boundary position $\tau_b$, the one-point function is governed by the kernel $K_b$, and the boundary data renormalize the wave function even at zero field. The paper also argues that these states are not $\alpha$-vacua, since the corrected correlators exhibit no antipodal singularities.

Load-bearing premise

The whole construction rests on the premise that, once gravity is turned on, a Euclidean geometry with an extra boundary at finite Euclidean time is still a legitimate configuration of the path integral; the paper assumes this by neglecting back-reaction, so if the constraints of quantum gravity forbid such a boundary, the proposed excited states would not exist.

Editorial extensions

If this is right

  • If the generalized prescription is correct, late-time de Sitter correlators in an excited state are not vacuum correlators: one-point functions become nonzero and the two-point function picks up the explicit correction of equations (3.18)-(3.19).
  • Taking $\phi_b \to 0$ does not restore the undeformed vacuum; the presence of the boundary at $\tau_b$ leaves a residual correction, so the geometry itself deforms the state.
  • The states defined by $(\Sigma_b, \phi_b)$ form a family distinct from $\alpha$-vacua, because their correlators show no antipodal singularities order by order in the small-hole expansion.
  • The wave functional is renormalized by the boundary data, as in equation (3.22), so normalization of the de Sitter wave function encodes information about the state preparation.
  • Each excited state has a semiclassical bulk geometry by construction, so the dictionary maps dual CFT excitations to geometries with an extra Euclidean boundary.

Reading between the lines

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

  • Not stated in the paper: the residual $\tau_b$ dependence at $\phi_b = 0$ suggests the boundary position acts like a state label interpolating between the vacuum at $\tau_b \to -\infty$ and a maximally deformed state at $\tau_b \to 0$; this could be probed by searching for the predicted correction in primordial non-Gaussianity data.
  • Not stated in the paper: the same two-boundary construction should generalize to higher-dimensional de Sitter and to weakly interacting fields, where the leading $\psi_1$ correction to the three-point function gives a concrete bispectrum shape; measuring this shape in the CMB would test the proposal.
  • Not stated in the paper: because no $\alpha$-vacuum trace appears, the boundary data behave like a single-trace source rather than a double-trace deformation; checking whether a double-trace deformation reproduces the same corrections would clarify the dictionary.
  • Not stated in the paper: including back-reaction through the junction conditions and imposing the Hamiltonian constraint of quantum gravity may select a discrete family of allowed $\phi_b$ configurations, which would quantize the space of excited states.
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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 / 5 minor

Summary. This paper proposes a generalization of the Hartle-Hawking no-boundary proposal in de Sitter space. Instead of preparing the vacuum by a Euclidean path integral over a compact geometry with a single boundary, the authors introduce an additional compact boundary Σ_b in the Euclidean region and impose arbitrary Dirichlet data φ_b on it. The resulting wavefunctional Ψ'_φ_b(φ,T) is then identified, via Eq. (2.8), with the CFT generating function Z_CFT(J) on the future boundary, with φ_b playing the role of a source for a dual operator. In Section 2 the authors argue heuristically that this describes excited states, and in Section 3 they carry out an explicit free massive scalar computation in dS_{1+1} in the complementary series, obtaining a nonvanishing one-point function (3.20) and modified two-point function (3.18)-(3.19) in the small-hole limit τ_b → -∞. They also argue in Appendix B that these states are not α-vacua. Gravity is excluded from the analysis, and the gravitational constraints are left for future work.

Significance. If the proposal is correct, it would provide a concrete holographic dictionary for excited states in de Sitter space, extending the dS/CFT correspondence beyond the Euclidean vacuum and giving explicit, computable modifications to late-time cosmological correlators. The free-field computation in Section 3 is careful and self-contained, and the comparison with α-vacua in Appendix B is a useful falsifiable distinction. The main strength is that the paper gives explicit formulas rather than a purely formal proposal. However, the central equality (2.8) is assumed rather than derived, gravity is excluded from the outset, and the physical interpretation relies on a CFT dictionary that is not established independently. These issues are acknowledged by the authors but they are load-bearing for the claim that the construction defines physical excited states of de Sitter quantum gravity.

major comments (4)
  1. [Section 2.1, Eqs. (2.5)-(2.8)] The central identification Ψ'_φ_b(φ,T) = Z_CFT(J) is posited rather than derived. The consistency check in Section 2.2 is structural: Z_CFT(J) is introduced through the same bulk wavefunctional, so the nonvanishing one-point function (2.14) and the shift J = J_b + δJ restate the ansatz rather than provide independent evidence. To make the dictionary claim load-bearing, the paper should derive this equality from a concrete microscopic construction (such as an explicit CFT computation on S^1 with a source) or at least formulate it as a precise conjecture with a specified domain of validity, rather than presenting it as an immediate consequence of the path integral.
  2. [Section 2 (before Section 2.1) and Section 4] Gravity is excluded and the de Sitter background metric is fixed, so the additional boundary Σ_b is imposed by hand. The existence of the proposed states depends on Σ_b being a legitimate Euclidean saddle of the full gravitational path integral, but the paper never checks compatibility with the gravitational constraints; the authors themselves list the Wheeler-DeWitt equation and the gravitational Gauss law as open questions in Section 4. This is a load-bearing gap: if linearized gravitational constraints around de Sitter admit only the Bunch-Davies vacuum (as suggested by the cited references [9,10]), then arbitrary boundary data φ_b could be forbidden. A minimal consistency check in minisuperspace or in the linearized theory is needed before the states can be claimed to be states of quantum gravity rather than states of a scalar field on a fixed background.
  3. [Section 3, Eqs. (3.18)-(3.21)] All explicit results are obtained in the small-hole expansion τ_b → -∞. The two-point correction (3.19) is proportional to e^{2τ_b}, and the one-point kernel (3.21) is also given at leading order in this expansion. The paper motivates this regime as the one closest to the Euclidean vacuum, but the physically interesting excited states are those with finite τ_b, and the current computation does not control that regime. It is also not shown that the expansion is uniform in the mode number n or that the resulting states are normalizable. Without such control, the leading-order results cannot be safely extrapolated to the regime where the excitation is significant.
  4. [Section 2.2 and Section 3, Eqs. (2.14), (3.20)] The bulk boundary data φ_b on Σ_b are identified with the CFT source J_b only through the formal equality (2.8). The explicit one-point function (3.20) is a convolution of φ_b with the kernel K_b on S^1, but it is not shown that this response matches a CFT perturbed by a local source J_b(φ) inserted on the future boundary I^+. Without this identification, the phrase 'excited states' describes a bulk construction whose dual CFT interpretation is asserted rather than demonstrated. The paper should clarify the precise map between φ_b and J_b, including whether J_b is an arbitrary local function or is constrained by the bulk dynamics on the Euclidean interval.
minor comments (5)
  1. [Throughout] There are several typographical errors: 'Schwinger-Kelldysh' (Sections 2 and 2.1), 'pertubately' (Section 3), 'Riemmanian' (Introduction), 'ans' (Appendix A), and 'We must us verify' (Section 2.1). These should be corrected before publication.
  2. [Section 2.3, Eq. (2.23)] In the displayed formula for ⟨ϕ1ϕ2ϕ3⟩_{J_b=0}, the product of the three factors (−2Re ψ̂2(k_i)) appears without an explicit division symbol; the intended expression is presumably 2Re[ψ̂3] δ(...) divided by that product. Please make the notation unambiguous.
  3. [Section 2.2, after Eq. (2.13)] The phrase 'terms that are bilinear in ϕ and ϕb respectively' is unclear; it presumably means terms quadratic in ϕ and terms quadratic in ϕ_b, but the sentence should state this explicitly.
  4. [Section 3] The computation is restricted to the complementary series ∆ ∈ [0,1/2], while the proposal in Section 2 is phrased generally. A brief comment on whether the principal series or the massless case would change the construction would be useful.
  5. [Figure 2] The relationship between the left Schwinger-Keldysh contour and the right saddle geometry could be clarified, especially the location of Σ_b and the direction of the Euclidean interval (τ_b, 0).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bulk path integral defines a state, (2.8) is an openly proposed dictionary, and the correlator modifications in Sec. 3 are independent free-field computations.

full rationale

The paper does not fit parameters and then rename them as predictions. Equation (2.5) independently defines a bulk wave functional from a Euclidean path integral on M^b_E with an added boundary Sigma_b and Dirichlet data phi_b. Equation (2.8) is explicitly presented as a claim or proposal ('Our claim is that... So we can write the more general holographic prescription...'), i.e., a dictionary postulate rather than a derived consequence. The CFT-side notion of an excited state is the standard source insertion J_b in (2.9), and the identification of phi_b with J_b is part of the proposed dictionary, not a hidden input reused as an output. The subsequent one-point and higher-point relations (2.13)-(2.17) follow by functional differentiation of the postulated equality, and the substantive content is the explicit solution of the Klein-Gordon equation on the two-boundary geometry and the resulting on-shell action (3.3)-(3.16), which yields concrete formulas (3.18)-(3.21). The admitted exclusion of gravity and the open status of the Wheeler-DeWitt/Gauss-law constraints are limitations on whether these states exist in quantum gravity, not circular reasoning: they are explicitly flagged in Section 4. The citations to prior AdS constructions ([17,18,19]) are heuristic support; the one self-citation [19] is not the load-bearing justification, since the dS computation in Section 3 stands on its own and the central proposal is openly postulated rather than derived from [19]. The comparison with alpha-vacua in Appendix B provides an independent external benchmark. Therefore no step reduces by construction to its own input.

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

The central claim rests on the proposed generalized holographic formula, a semiclassical treatment, imported renormalization, and the neglect of gravity. The only hand-chosen numerical parameter is the boundary location tau_b, which is not fitted to data.

free parameters (1)
  • location of the additional boundary tau_b = not fitted; small-hole limit tau_b -> -infinity used
    Position of the extra Euclidean boundary introduced by hand in eq. (2.5); it controls the deformation away from the no-boundary vacuum, and the paper studies the limit tau_b -> -infinity.
assumptions (6)
  • domain assumption The standard dS/CFT dictionary, eq. (1.1): Psi_dS[phi] = Z_CFT[J] with delta phi = delta J.
    The paper assumes the existing holographic relation is valid for the vacuum and extends it; this remains a conjecture in de Sitter holography.
  • ad hoc to paper The generalized formula (2.8): Psi'_phi_b(phi,T) = Z_CFT(J) with the extra boundary data acting as a source.
    This is the central proposal of the paper, stated as a definition and checked only for consistency in a free-field example.
  • domain assumption Saddle-point approximation and gluing conditions (2.3) determine the semiclassical state.
    The path integral is evaluated on shell and junction conditions are imposed, following standard holographic practice.
  • domain assumption AdS/CFT counterterm renormalization (Lambda ~ 1/epsilon, source rescaling ~ epsilon^Delta) applies to dS.
    The renormalization prescription is imported from AdS/CFT (Appendix A); de Sitter renormalization is not independently established.
  • ad hoc to paper Gravity is excluded and backreaction of Sigma_b is negligible.
    The paper says 'we exclude gravity from the current study and assume low-energy field fluctuations', so the boundary is not a gravitational saddle.
  • domain assumption The scalar field is in the complementary series with Delta in [0,1/2] and dual operator dimension 1 - Delta.
    This choice simplifies the holographic map; the paper notes principal series is more subtle.
invented entities (1)
  • additional Euclidean boundary Sigma_b
    purpose: Defines a family of excited states in the no-boundary de Sitter path integral by carrying arbitrary Dirichlet data phi_b.
    The boundary is introduced by hand rather than derived from gravitational dynamics; its effects are computed only within the model, so there is no independent falsifiable handle outside the paper.

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Pith. "Pith review of The (No) Boundary Proposal and excited states in de Sitter holography." pith.science (2026). https://pith.science/paper/TB2PRH2L

@misc{pith2026250616943,
  author       = {Pith},
  title        = {Pith review of: The (No) Boundary Proposal and excited states in de Sitter holography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TB2PRH2L}},
  note         = {Machine review of arXiv:2506.16943}
}
abstract

In the AdS/CFT framework, vacuum and excited states are systematically described by imposing arbitrary Dirichlet boundary conditions at the AdS boundary. Furthermore, there are explicit relations connecting the quantum states to their corresponding dual Euclidean AdS geometries, in line with the Hartle-Hawking (HH) construction. The ground state therefore corresponds to the dominant saddle point under trivial conditions on the asymptotic boundary, which is the exact Euclidean AdS geometry. In contrast, the situation in de Sitter spacetime differs significantly, as there is no natural region analogous to the AdS boundary. Thus, the Hartle Hawking approach precisely defines the ground state as a path integral over smooth (Euclidean) geometries ending on a spatial Cauchy surface, with \textit{no} additional boundary or past singularity, known as the no boundary proposal. In this work, we revisit the no boundary proposal to describe excited states within the framework of de Sitter Holography. Specifically, we investigate the possibility of defining a family of excited states by introducing an additional boundary in the Euclidean region and imposing arbitrary Dirichlet boundary conditions on it. As a result, we demonstrate that the computation of $n$ point correlation functions is consistent with the presence of excited states, and furthermore, show that cosmological late-time observables in these states undergo non-trivial modifications. This study may have significant implications for the development of the holographic dictionary for de Sitter spacetimes.

Figures

Figures reproduced from arXiv: 2506.16943 by the authors.

Figure 1
Figure 1. de Sitter spacetime is represented as the real-time evolution from the vacuum state at τ = t = 0. This is described as the lower half of Euclidean dS space.The corresponding Schwinger-Keldysh diagram in the complex plane is shown on the left. of the CFT defined on the Riemmanian d-sphere, whose points x can be identified with I +, and thus J(x) is an arbitrary source for the primary operator O, considered dual to th… view at source ↗
Figure 2
Figure 2. The left figure is the Schwinger-Keldysh complex diagram, where the initial state is described as a path integral on a finite imaginary time interval (τb, 0). The right figure represents the corresponding saddle point geometry. We must us verify that in fact this new proposal in fact fits into the expected feature of a holographic prescription for excited states, as discussed in the forthcoming subsection. In what f… view at source ↗
Figure 3
Figure 3. In (a) we show ⟨O(φ ′ )O(φ)⟩0 in blue and ⟨O(φ ′ )O(φ)⟩corr ⟨O(φ′)O(φ)⟩0 real and imaginary pieces in orange and green. In (b) we are showing a sample form for the real and imaginary piece of Kb(φ) 1  e iπ∆P |n| ∆−1 (− tanh(τb)) − e iπ|n|P |n| ∆−1 (+ tanh(τb)) ∼ e |n|τb πeiπ∆ csc(π∆) Γ(|n|) + . . . (3.15) Before computing correlation functions from these contributions, we compute Ib. This piece of the on shell act… view at source ↗

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