{"id":"a2bc9027-33e5-4866-ad30-83208553bc04","arxiv_id":"2509.09257","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new plane-wave representation of anti-de Sitter scalar two-point functions is proposed, yielding new Feynman propagator and special-function identities.","lead":"This paper derives a new plane-wave expansion for quantum fields in anti-de Sitter space, a setting where such expansions have been missing for decades. If correct, it gives physicists a more practical way to compute Feynman diagrams in AdS, including a direct link between Euclidean and real-time calculations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved relative-cycle construction in Eq. (22) is the load-bearing weak point; for generic λ the needed branch/twisted-homology prescription is missing.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: the existence and properties of the relative homology cycle in Eq. (22). I agree with the CONDITIONAL verdict. The paper's main result depends on a one-sentence assertion about a topological cycle; for generic complex λ the integrand is multivalued, so the ordinary relative-homology statement is insufficient and a branch/twisted-homology prescription is needed. No proof of convergence, z1-independence, or AdS-invariance is given, and Eq. (23) is presented as the evaluation without derivation. The later Feynman-propagator representation and diagrammatic claims inherit this gap. However, I see no concrete evidence that the formula is false; the explicit Q-function form is plausible and could be verified independently. Therefore the conditional verdict should stand: the missing construction is a condition for full acceptance, not grounds for rejection.","tokens_in":7151,"tokens_out":23155,"duration_ms":284936,"concrete_test":"Supply the missing construction in the simplest nontrivial case: d=2, λ=-1/2. Choose explicit z1∈Z^-, z2∈Z^+ with z1·z2∉[-1,1], parametrize the relevant cone, and define γ(z1) by lifting a path with endpoints in {ζ·z1=0} to a covering where (z1·ζ)^λ(z2·ζ)^{1-d-λ} is single-valued. Numerically evaluate the integral for two different choice of lift and for an AdS-transported pair; compare with the RHS of Eq. (23). If the value depends on the lift or differs from the Q-function, the relative-cycle formulation as stated fails. Alternatively, re-derive Eq. (22) from the standard hypergeometric contour for Q_λ and check that the period reproduces Eq. (23) term by term without an extra boundary term.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim stands or falls on Eq. (22): an integral over a relative homology cycle γ(z1) in H_{d-1}(C^-, {ζ:ζ·z1=0}) is asserted to converge, be independent of the representative and of z1, be AdS-invariant, and equal the Legendre Q-function in Eq. (23). The only support is the sentence that for Re λ > -1 the cycle 'should belong' to such a class. This is not a proof. For generic λ the integrand is multivalued; the paper itself notes the e^{-4πiλ} monodromy of the full-cone integral around the (ξ0,ξd) cycle. An ordinary relative homology cycle does not by itself determine a branch of the integral or a well-defined lift. One needs a specified covering (or twisted relative homology with local coefficients), a proof that the boundary contribution at ζ·z1=0 is harmless, and a proof of z1-independence under the AdS group. None is supplied; Eq. (23) is simply announced as the value. Since Eq. (30) and the diagrammatic applications are derived from Eq. (22), any failure of this construction would invalidate the paper's main contribution. The formula may well be true, but at this precise point the argument is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a manifestly covariant plane-wave expansion for the Wightman function of a free scalar field in anti-de Sitter spacetime. The central formula, Eq. (22), represents the AdS two-point function as an integral over a relative homology cycle in a chiral null cone of products of the new AdS plane waves (z·ζ)^λ, and Eq. (23) evaluates this integral in terms of a Legendre Q-function. From this representation the author derives an integral representation of the Feynman propagator in a Poincaré patch, Eq. (30), discusses its relation to Euclidean AdS diagrams, and exhibits new identities for Legendre and Bessel functions. The paper is a letter and is largely organized around the claimed validity of Eq. (22).","tokens_in":7478,"tokens_out":2973,"duration_ms":36815,"significance":"If the central plane-wave expansion is established, it would be a substantive contribution: it would fill a long-standing gap in manifestly covariant momentum-space descriptions of AdS quantum fields, provide a new integral representation for the AdS Feynman propagator, and explain the relation between Euclidean and Lorentzian AdS diagrams in explicit examples. The paper also contains a number of concrete, checkable special-function identities. However, the central formula is not proved at the advertised level of rigor: the construction of the relative homology cycle is asserted rather than demonstrated, and the analyticity and branch issues for generic complex λ are not resolved. The strength of the paper's conclusions therefore currently rests on an unverified topological/analytic assumption.","major_comments":[{"comment":"The central identity (22)-(23) is load-bearing and is not established. The cycle γ(z1) is introduced only by the statement that for Re λ > -1 it 'should belong' to a relative homology class in H_{d-1}(C−, {ζ:ζ·z1=0}). No proof is given that such a cycle exists, that the integral converges, that the result is independent of the representative and of z1, or that it is AdS-invariant. Moreover, for generic complex λ the integrand (z1·ζ)^λ(z2·ζ)^{1-d-λ} is multivalued; an ordinary relative homology cycle does not select a branch. A twisted relative homology construction or an explicit covering must be specified, and the boundary contribution at ζ·z1=0 must be analyzed. Since Eq. (23), Eq. (25), and all subsequent applications depend on Eq. (22), this missing proof is a major obstacle.","section":"Eqs. (21)-(23)"},{"comment":"The odd-dimensional formula (25) is announced without derivation. It cannot be obtained by simply substituting d=2n+1 into Eq. (22) because the prefactor in (22) contains 1/cos(πd/2), which vanishes for odd d. The author does not explain how the logarithmic term arises as a limit or regulated version of the relative-cycle integral, nor which branch of the logarithm is used. Since AdS3 and AdS5 are the physically most relevant odd-dimensional cases, this is not a minor omission.","section":"Eq. (25)"},{"comment":"The Feynman propagator representation (30) is deduced from Eq. (29), but the deduction is only described as 'after some pain'. The paper does not show the Fourier transform calculation or the analytic continuation that turns the Wightman function into the Feynman propagator; in particular, the iε prescription and the contour deformation are not specified. This matters because the advertised applications to Wick rotation of Euclidean diagrams rely on (30). A complete proof or a reference with the full calculation is needed.","section":"Eqs. (29)-(30)"},{"comment":"The two-line diagram identity below Eq. (33) is asserted to be 'shown at first for p²>0' by using Eq. (23) and the Euclidean version of Eq. (30), but the actual verification is not presented. Since the coincidence of Euclidean and Lorentzian banana diagrams is one of the main claimed applications, the reader cannot check the argument from the letter as written. The one-line integral (32) is explicit and helpful, but the two-line case needs a detailed derivation or a clear reference.","section":"Diagrammatic applications, after Eq. (33)"}],"minor_comments":[{"comment":"There are numerous typographical and formatting issues: missing spaces ('dimensiond', 'AdSd'), inconsistent accents ('Poincar´ e' vs 'Poincaré'), and 'Bunch-Davis' should probably be 'Bunch-Davies'. These should be corrected in a revision.","section":"Throughout"},{"comment":"The shadow representation (24) is said to follow from (22) by letting z2 tend to the boundary. This limiting procedure is not explained; either provide the derivation or state it as a conjecture.","section":"Eq. (24)"},{"comment":"The multiplication theorem (27) is called 'perhaps unknown'. The author should either provide a proof or a precise reference; the domain of validity in x and λ should also be stated.","section":"Eq. (27)"},{"comment":"The final conjecture about Witten diagrams is clearly labelled as open, which is appropriate. However, the wording could be tightened to distinguish proved statements from conjectural ones in the diagrammatic section.","section":"End of paper"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's earlier work [1,2,5], which is reasonable given the technical continuity. The main concern is that the central identity is presented as a theorem but the proof is only a sketch of a relative homology construction. This is not a matter of external disagreement; it is an internal gap. The paper would be suitable for a letter only after the missing proof of (22) (and the odd-dimensional version (25)) is supplied, or the paper is recast as a conjecture plus supporting evidence. I would also ask the editor to consider whether the journal's letter format accommodates the required technical detail; if not, a longer article may be more appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read on arXiv:2509.09257. The headline: this paper has a plausible and potentially important new result — a single-integral plane-wave expansion for AdS scalar Wightman functions, Eqs. (22)–(23), with a manifestly covariant kernel and a Legendre Q-function value. If it holds, it supplies the missing momentum-space tool for AdS QFT and a bridge between Euclidean and Lorentzian Feynman diagrams. The novelty is real: the earlier iϵ attempt by Gary–Giddings–Penedones is different and didn't get the single integral. The Legendre multiplication theorem (27) is a nice byproduct. The Fourier representation (29) and the Feynman propagator representation (30) are substantive, and the banana-diagram Wick rotation is a concrete payoff.\n\nThe soft spot is exactly where the stress-test pointed. The load-bearing item is the relative homology cycle γ(z1) in Eq. (22). The paper says that for Re λ > −1 the cycle 'should belong' to H_{d−1}(C^−, {ζ·z1=0}) — that's a hope, not a proof. For generic complex λ the integrand is multivalued; an ordinary relative cycle doesn't fix a branch without a covering or a twisted local-coefficients prescription. The paper itself notes the e^{−4πiλ} monodromy on the full cone, so the issue is real. Without a construction of the cycle and a check that the boundary term at ζ·z1=0 is harmless, Eq. (22) is an unproven central claim. Eqs. (29) and (30) are also announced with 'after some pain' and 'it may be deduced'; the two-line identity is 'can be shown'. For a letter that's compressed, but a referee should ask for the details. I don't think the paper is wrong — the normalization is checked against the known Wightman function and the Klein–Gordon check is explicit — but the current write-up leaves the central machinery at the level of an assertion.\n\nCitation pattern is fine. The paper builds on the author's own dS plane-wave work and AdS analyticity results, which are independent published support. The self-citations are appropriate here.\n\nWho it's for: AdS/CFT practitioners and anyone working on momentum-space representations in curved spacetime. Worth a serious referee: yes. The result is important enough that the gap in the cycle construction should be either fixed or formally treated in a companion paper. I'd recommend sending to peer review, with the request that the author supply the homological details or at minimum a concrete contour prescription for the integral and a proof of the propagator representation. I would not cite it yet in my own work until the cycle issue is closed.","headline":"A genuinely new plane-wave representation for AdS scalar Wightman functions, worth taking seriously, but the central integration cycle is asserted rather than proved.","tokens_in":7915,"tokens_out":3016,"would_cite":false,"duration_ms":33577,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The AdS scalar two-point function can be written as an integral of plane waves over a relative homology cycle, giving a covariant momentum-space picture that was previously missing.","keywords":["AdS plane waves","Wightman function","relative homology cycle","Feynman propagator","Legendre functions","Bessel functions","Wick rotation","AdS quantum field theory"],"falsifier":"Evaluate the right-hand side of Eq. (22) for a concrete case, say d=2 and λ=0 (or λ=1/2), using an explicit parametrization of a relative cycle γ(z1), and compare with the closed form in Eq. (23): a mismatch for any non-coincident pair z1,z2 would disprove the claimed plane-wave representation. Alternatively, check whether the integral's value is independent of continuous deformations of γ(z1) that keep the endpoints on the zero set; if not, the relative-homology construction fails.","tokens_in":7011,"feed_emoji":"🌊","tokens_out":4757,"duration_ms":53166,"temperature":0.7,"pith_summary":"The paper aims to give anti-de Sitter scalar quantum fields the same tool that plane waves provide in Minkowski and de Sitter spaces: a manifestly covariant integral representation of the two-point Wightman function as a superposition of simple exponentials (z·ζ)^λ. The representation is claimed to hold for all complex mass parameters and in all spacetime dimensions, and it is built on a relative homology cycle that replaces the ordinary momentum-space contour and encodes the topology of the AdS covering. From it the paper derives a new integral representation of the AdS Feynman propagator in Poincaré coordinates, which allows Euclidean and Lorentzian AdS Feynman diagrams to be linked by Wick rotation in concrete examples (banana diagrams). The same expansion yields new identities for Bessel and Legendre functions, including a multiplication formula for Legendre functions of the second kind. If correct, this closes a long-standing gap in the covariant formulation of AdS quantum field theory and opens a direct route to computing loop diagrams in real space.","feed_headline":"Plane-wave expansion found for AdS quantum fields","feed_subtitle":"A cycle integral represents AdS two-point functions for all masses and dimensions, and rotates Euclidean diagrams into Lorentzian ones.","key_machinery":"The central objects are the AdS plane waves φ_λ^±(z,ζ)=(z·ζ)^λ, defined as globally univalued holomorphic functions on the product of chiral tuboids and chiral cones, together with the relative homology cycle γ(z1) ∈ H_{d-1}(C^-, {ζ: ζ·z1=0}) that makes the integral AdS-invariant and convergent for Re λ > -1. The cycle carries the topological information that distinguishes AdS from dS: a full real-cone cycle overcounts momentum directions and breaks invariance for generic λ. Working in the Poincaré foliation and using the Hankel transform then converts the plane-wave integral into the Feynman-propagator representation (30).","core_discovery":"The paper establishes that the AdS scalar Wightman function W_λ^{AdS}(z1,z2), holomorphic in the chiral tuboid domain Z^- × Z^+, admits the plane-wave decomposition W_λ^{AdS}(z1,z2) = c_d(λ) ∫_{γ(z1)} (z1·ζ)^λ (z2·ζ)^{1-d-λ} dμ_γ(ζ), where γ(z1) is a relative homology cycle in the punctured chiral cone, and that this integral equals the explicit Legendre-function expression (23). This is the AdS counterpart of the dS plane-wave expansion, with the essential difference being topological: the integration cycle must be relative to the zero set ζ·z1=0 to achieve AdS invariance for generic complex λ. The paper further derives a new integral representation of the Feynman propagator in Poincaré coo","pith_inferences":["If the relative-cycle construction is robust, it supplies a natural topological characterization of AdS momentum space: the space of momentum directions is a covering of the complex null cone, and the homology cycle encodes the covering, which may clarify why integer and half-integer mass parameters behave differently.","The propagator representation (30) could be used as a numerical tool: it reduces AdS loop integrals to Minkowski loop integrals with an additional Hankel transform, potentially making higher-loop AdS calculations tractable by computing ordinary massive Minkowski integrals.","The multiplication theorem (27) may be a special case of a larger family of identities connecting Legendre functions on different sheets of the cut plane; the plane-wave representation suggests such identities follow from the invariance of the relative cycle under deformation.","Because the λ -> (1-d-λ) symmetry is broken in AdS (unlike dS), the plane-wave expansion may help identify which values of λ correspond to stable or unitary representations, tying the topological cycle to the admissible mass spectrum."],"forward_implications":["The AdS scalar two-point function now has a momentum-space form as covariant as the Minkowski one, so amplitudes can be written directly in real-space AdS without first going to the Euclidean continuation.","Euclidean banana diagrams in AdS can be Wick-rotated to the Lorentzian Poincaré patch and give identical results, as shown for the one-loop and two-line examples.","The representation yields concrete new identities relating Legendre functions of the second kind to series of associated Legendre functions (Eq. 27), with analogous formulas in general dimension.","Tensorial and spinorial AdS correlation functions can be obtained by applying differential operators to the scalar plane-wave formula.","The paper conjectures, but leaves open, that Witten diagrams integrated over the Poincaré patch are AdS invariant; the mechanism shown for simple diagrams suggests a general proof may be within reach."],"fun_headline_variants":["Plane-wave expansion cracks AdS scalar fields","New cycle integral for AdS propagators","AdS plane waves link Euclidean and Lorentzian","Topological cycle simplifies AdS Feynman diagrams","AdS Wightman functions as plane-wave sums"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole construction rests on the claim that for every z1 and Re λ > -1 there exists a relative homology cycle γ(z1) in the punctured chiral cone, with the integral convergent, independent of the choice of representative, and AdS-invariant; the paper states this but does not give a proof.","fun_headline_variants_meta":{"raw":{"variants":["Plane-wave expansion cracks AdS scalar fields","New cycle integral for AdS propagators","AdS plane waves link Euclidean and Lorentzian","Topological cycle simplifies AdS Feynman diagrams","AdS Wightman functions as plane-wave sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000235,"raw_usage":{"total_tokens":1289,"prompt_tokens":647,"completion_tokens":642,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":391,"completion_tokens_details":{"reasoning_tokens":570}},"tokens_in":391,"tokens_out":642,"duration_ms":7548,"temperature":1.0,"reasoning_tokens":570,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T19:24:03.238116+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the right-hand side of Eq. (22) for a concrete case, say d=2 and λ=0 (or λ=1/2), using an explicit parametrization of a relative cycle γ(z1), and compare with the closed form in Eq. (23): a mismatch for any non-coincident pair z1,z2 would disprove the claimed plane-wave representation. Alternatively, check whether the integral's value is independent of continuous deformations of γ(z1) that keep the endpoints on the zero set; if not, the relative-homology construction fails.","supporting_citations":[],"review_version":1}