{"id":"59cc13ae-d578-4c70-8723-b02ee1a0acca","arxiv_id":"2607.20936","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The Green's function prescription for the in-out effective action omits endpoint vacuum wavefunctional contributions; including them cancels the boundary obstruction and recovers the Bogoliubov vacuum persistence probability.","lead":"This paper shows that the vacuum persistence probability, a basic quantity in quantum field theory, is not ambiguous: two standard calculation methods disagree only because one of them omits boundary terms. The authors derive the missing terms and show the complete result always agrees with the Bogoliubov formula.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Boundary two-point function (13) is an unproven prescription; the central cancellation in (17) depends on it.","rationale":"I read the paper as a careful formal attempt to show that the vacuum persistence probability is prescription independent once endpoint vacuum wavefunctionals are included. The algebraic steps in the bulk and endpoint variations are internally consistent, and the signs in the Supplemental Material check out; the cancellation in Eq. (17) would indeed remove the boundary obstruction (9) and leave Im W = 1/4 Tr log(αα†) if Eq. (13) is exactly the path-integral two-point function on the endpoint surface. The single most load-bearing assumption is therefore Eq. (13), exactly as the Reader identified. The paper's derivation of Eq. (13) consists of restricting the Feynman Green's function and using theta(x,x)=1/2, but for distinct boundary points this does not follow from the future/past step-function representation, and the path-integral boundary correlation is not manifestly the same as a time-ordered product. Because the endpoint Wronskian terms in (14)-(16) are the ones that cancel B_in and B_out, any additional boundary-local term in the two-point function would break the central conclusion. This is a genuine technical gap rather than a mere disagreement with convention, and it is testable in an exactly solvable Gaussian model. The lateral-boundary assumption and the sign convention in Eq. (2) are secondary; the former is explicit and may be valid in the intended compact cases, and the latter appears typographical. I therefore agree with the Reader's conditional verdict: the argument is coherent and plausible, but the unverified boundary two-point function is the soft spot on which the proof hinges.","tokens_in":17372,"tokens_out":27074,"duration_ms":253237,"concrete_test":"Use a solvable single-mode time-dependent harmonic oscillator, or a 1+1 dimensional scalar with known Bogoliubov coefficients, and compute the initial-boundary two-point function directly from the Gaussian path integral with the endpoint wavefunctionals (10), i.e. <phi(x)phi(y)> = Z^{-1} ∫ Dphi phi(x)phi(y) Ψ_out* Ψ_in e^{iS}, and compare it to Eq. (13) evaluated with the α^{-1} mode sum and theta=1/2. If the two differ by any local boundary term, such as c δ_Σ(x,y) or a term involving the imaginary part of K, recompute Eqs. (14)-(16) and check whether ∂_{m^2}WΨ still cancels B_in+B_out in Eq. (9). The Gaussian integral is exact analytically, so this settles whether the central cancellation holds as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central cancellation in Eq. (17) rests on the endpoint variation (14)-(16), whose Wronskian coefficients come from inserting the boundary two-point function (13) into the kernel variation i/2<phi·∂_{m^2}K·phi>. Equation (13) is not derived: it is obtained by restricting the in-out Feynman Green's function to the Cauchy surface and applying the coincident-limit prescription theta(x,x)=1/2. This prescription is not obviously valid for distinct boundary points x≠y: for two points on the same spacelike Cauchy surface neither lies in the future of the other, so the step functions in the mode representation (S25) vanish, and the boundary value of G_F is a distributional limit that can depend on the approach to the surface. The path-integral expectation <phi(x)phi(y)> on the endpoint is an unordered product, not a time-ordered one, so it need not equal the symmetrized Feynman restriction; a boundary-local contact term or a term from the normal discontinuity of G_F could shift the Wronskian coefficient in (14). Because the cancellation with B_in and B_out in (9) is exact only if the Wronskian coefficients in (14)-(16) match (9) with no extra terms, any such correction would leave a residual boundary term and the conclusion (19) would not follow. The paper supplies no derivation of (13) beyond the theta=1/2 convention.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses the apparent discrepancy between the Bogoliubov and Green's function prescriptions for the imaginary part of the in-out effective action, which gives the vacuum persistence probability. The authors argue that the Green's function prescription computes only the bulk mass variation of the in-out path integral and omits the contribution from the endpoint vacuum wavefunctionals. They derive the endpoint wavefunctional contribution and show that it cancels the boundary obstruction terms that prevent the Green's function result from agreeing with the Bogoliubov expression. The final result is Im W = 1/4 Tr log(α α†), so P_vac = e^{-2 Im W_B}, implying that the vacuum persistence probability is not prescription dependent. The derivation is formal, with a supplemental material providing the mode representation of the in-out Green's function, the reduction of the bulk integral to endpoint Wronskians, and the evaluation of the Gaussian wavefunctional variation.","tokens_in":17621,"tokens_out":20404,"duration_ms":189356,"significance":"If the central claim holds, the paper resolves a known and actively discussed ambiguity in the computation of vacuum persistence probabilities in curved spacetime and external backgrounds, particularly the de Sitter mismatch reported in Refs. [23,24]. The bulk-obstruction calculation, Eqs. (7)-(9), is an independent and clean derivation that the Green's function prescription contains endpoint Wronskian terms in addition to the Bogoliubov variation. The paper also provides a concrete, falsifiable identity, Eq. (19), and the supplemental material contains explicit derivations of the Gaussian wavefunctional normalization and variation. The main weakness is that the boundary two-point function entering the endpoint variation is assumed rather than derived; this is the load-bearing step for the cancellation in Eq. (17).","major_comments":[{"comment":"The identification of the boundary two-point function ⟨φσ(x)φσ(y)⟩ with the symmetrized restriction of the in-out Feynman Green's function is not derived. For x and y on the same Cauchy surface, the step-function representation (S25) is ambiguous because the points are spacelike separated, and the path-integral expectation is an unordered product rather than a time-ordered one. A boundary-local contact term or a contribution from the normal discontinuity of G_F could modify the Wronskian coefficients in (14), and the cancellation in (17) is exact only if Eq. (13) holds with no extra terms. The authors should provide a derivation of (13) from the path integral with the endpoint wavefunctionals in (3), or at minimum a careful distributional definition of the boundary restriction.","section":"Boundary completion by vacuum wavefunctionals, Eq. (13)"},{"comment":"The mass derivatives ∂_{m^2} u_in*_k and ∂_{m^2} u_out_k' are not uniquely defined: one can add any solution of the homogeneous Klein-Gordon equation to ∂_{m^2} u, and the added homogeneous part changes the Wronskian boundary terms in (8) and (14)-(16). The paper does not specify a convention for differentiating the mode bases (for example, fixing the normalization and phase of each mode as a function of m^2), nor does it show explicitly that the final combination in (17) is invariant under such a redefinition. The cancellation of the B terms must be independent of this convention for Eq. (19) to be well defined.","section":"In-out amplitude with endpoint states, Eqs. (8), (14)-(16)"},{"comment":"The final expression Im W = 1/4 Tr log(α α†) is a trace over all modes and is generally ultraviolet divergent. The paper states that the regulator is removed after the calculation is completed, but it does not specify the regularization or demonstrate that the imaginary part of the renormalized trace is regulator-independent. Given that the cited de Sitter example showed sensitivity to the regularization of the Green's function prescription, the authors should explain why the completed expression does not inherit a similar regulator dependence, and how Im W_0=0 in the large-mass reference condition is established beyond the classical suppression of particle production.","section":"Conclusion, Eq. (19)"}],"minor_comments":[{"comment":"The phrase 'with the latter running up to the physical mass m^2' is confusing given the notation ∫_{m^2}^{+∞} d\\bar{m}^2; please clarify the direction of integration.","section":"Eq. (2)"},{"comment":"The text uses 'wavefunctional' and 'wave functional' interchangeably; please choose one spelling and use it consistently.","section":"Throughout"},{"comment":"The derivation of the normalization constant would be clearer if the Gaussian integral convention were stated explicitly; as written, the powers of π and 2 are easy to misread.","section":"Supplemental Material, Eq. (S56)"},{"comment":"The sentence 'the imaginary part of effective action reduce to the Bogoliubov expression' in the supplemental material contains a grammatical error; it should read 'reduces to'.","section":"Eq. (19)"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the unproven identification in Eq. (13). If the authors can supply a rigorous derivation of the boundary two-point function from the path integral with the endpoint wavefunctionals, the paper's central claim is likely correct and the result would be a valuable resolution of the prescription-dependence problem. The derivation is otherwise internally coherent, and the bulk-obstruction calculation is a genuine independent check. I would encourage the editor to solicit a revised version with the technical gaps filled."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper gives a concrete mechanism for resolving a known ambiguity in the vacuum persistence probability. The new piece is the explicit boundary completion—endpoint vacuum wavefunctionals contributing Wronskian terms that cancel the boundary obstruction in the Green's function prescription. Prior work (refs 23, 27) had flagged endpoint states as relevant; this paper actually carries out the cancellation and lands on the Bogoliubov answer. I think that's a genuine step forward, even if the final answer is familiar.\n\nWhat it does well: the bulk reduction to boundary Wronskians (Eqs. 7–9) is clean and doesn't rely on the endpoint argument; the Green–Lagrange trick is a nice way to isolate the obstruction. The wavefunctional variation is also laid out in enough detail in the supplemental material that a patient reader can check the algebra. The de Sitter discussion at the end is sensible: the cutoff sensitivity is diagnosed as boundary sensitivity that cancels when both bulk and endpoint pieces are included in the same scheme.\n\nWhere I'd push back: the load-bearing step is Eq. (13), the claim that the boundary two-point function equals the symmetrized restriction of the in-out Feynman Green's function. For distinct points on the same Cauchy surface, the step functions in (S25) are not defined, and the coincident-limit convention θ=1/2 doesn't obviously extend. The endpoint expectation in the path integral is an unordered product; it could in principle carry a boundary-local contact term from the Gaussian wavefunctional. The cancellation in (17) is exact only if those Wronskian coefficients match (9) with no extra terms. The paper asserts (13) rather than derives it from the path integral with the explicit boundary terms. That's the gap I'd want closed before I trust the proof.\n\nThere's also a mild circularity worry: the wavefunctionals are built from the same in/out mode bases whose overlap defines α, so the completion is partially guaranteed by construction. The bulk obstruction calculation is independent, but the final equality is less of a surprise. That said, a formal clarification that shows the two prescriptions agree after including endpoint terms is still worth having.\n\nWho's this for? People computing vacuum persistence in curved space or strong fields, especially the de Sitter puzzle. The paper is a formal clarification, not new phenomenology, but it should kill a lingering confusion.\n\nMy recommendation: send it to referees, but the referee should ask for a proper derivation of (13), either from the Gaussian path integral or from a boundary-to-boundary propagator calculation. If that gap is filled, the paper is solid. It's not a desk-reject.","headline":"A clean formal mechanism that resolves the vacuum-persistence prescription ambiguity by cancelling the boundary obstruction with endpoint vacuum wavefunctionals, but the central boundary two-point function is asserted rather than derived.","tokens_in":18147,"tokens_out":4444,"would_cite":true,"duration_ms":41118,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T20","83C47"],"pacs":["04.62.+v"],"model":"deepseek-v4-flash","headline":"Once endpoint vacuum wavefunctionals are included, the Green's function prescription agrees with the Bogoliubov formula, so vacuum persistence is prescription independent.","keywords":["vacuum persistence probability","in-out effective action","Bogoliubov coefficients","Green's function prescription","endpoint vacuum wavefunctionals","Schrödinger wavefunctional","Wronskian boundary terms","particle production in curved spacetime"],"falsifier":"Take a free scalar on a compact (1+1)-dimensional globally hyperbolic spacetime with a time-dependent mass profile, choose explicit in- and out-mode bases, and compute both sides of equation (18) within one regulator, including the endpoint Gaussian prefactor and kernel variations; if the imaginary part of the left-hand sum differs from $\\tfrac{1}{4}\\operatorname{Tr}\\log(\\alpha\\alpha^\\dagger)$ for any allowed endpoint phase or boundary basis, the boundary completion fails.","tokens_in":17145,"feed_emoji":"⚫️","tokens_out":8395,"duration_ms":79345,"temperature":0.7,"pith_summary":"The paper addresses a long-standing ambiguity: the vacuum persistence probability computed from Bogoliubov coefficients and from the coincident Feynman Green's function can look different, for instance in de Sitter spacetime. It argues that this is not an ambiguity of the physical probability but a boundary incompleteness of the Green's function prescription, which records only the bulk-action response to a change in mass squared. Including the mass variation of the initial and final vacuum wavefunctionals produces boundary Wronskian terms that exactly cancel the mismatch. The completed amplitude yields $\\operatorname{Im} W = \\tfrac{1}{4}\\operatorname{Tr}\\log(\\alpha\\alpha^\\dagger)$ and $P_{\\rm vac}=e^{-2\\operatorname{Im} W_B}$, so the vacuum persistence probability has one value independent of which prescription was used.","feed_headline":"Vacuum persistence probability has one value, not two","feed_subtitle":"Endpoint vacuum wavefunctionals cancel the boundary mismatch, fixing one value for vacuum persistence.","key_machinery":"The central objects are the endpoint Gaussian Schrödinger vacuum wavefunctionals $\\Psi_\\sigma[\\varphi_\\sigma] = N_\\sigma \\exp(\\tfrac{i}{2}\\varphi_\\sigma \\cdot K_\\sigma \\cdot \\varphi_\\sigma)$, with kernel $K_\\sigma$ built from the boundary data $q_k^\\sigma$, $p_k^\\sigma$ of the positive-frequency mode functions. Their mass variation is not negligible: it produces Wronskians such as $W_{\\Sigma_{\\rm in}}(u^{\\rm out}_{k'}, \\partial_{m^2} u^{\\rm in*}_k)$ that live on the Cauchy surfaces. The paper's key mechanism is the exact cancellation of those endpoint Wronskians with the boundary terms that arise when the bulk coincident Green's function is converted into a boundary integral. The identity $\\langle \\varphi_\\sigma(x)\\varphi_\\sigma(y)\\rangle$, set equal to the restricted in-out Feynman Green's function, is what allows the endpoint calculation to be expressed through the same Bogoliubov matrix $\\alpha$.","core_discovery":"The paper establishes that, for a free real scalar in a globally hyperbolic region bounded by initial and final Cauchy surfaces, the full in-out amplitude contains endpoint Gaussian vacuum wavefunctionals whose mass variation cannot be omitted. Varying with respect to $m^2$ separates the complete effective action into a bulk part and an endpoint part: $\\partial_{m^2}W = \\partial_{m^2}W_G + \\partial_{m^2}W_\\Psi$. The bulk Green's function term reduces, via the Green-Lagrange identity and Gauss' theorem, to the Bogoliubov mass variation minus two endpoint Wronskian terms, $\\partial_{m^2}W_G = \\partial_{m^2}W_B - B_{\\rm in} - B_{\\rm out}$. The endpoint vacuum wavefunctionals contribute exactly $+B_{\\rm in}+B_{\\rm out}$ together with a real phase, so the complete variation is $\\partial_{m^2}W = \\partial_{m^2}W_B + \\partial_{m^2}\\Theta$. Integrating from the large-mass reference point, where particle production is suppressed, gives $\\operatorname{Im} W = \\tfrac{1}{4}\\operatorname{Tr}\\log(\\alpha\\alpha^\\dagger)$, the Bogoliubov expression, and hence $P_{\\rm vac}=e^{-2\\operatorname{Im} W_B}$.","pith_inferences":["A natural extension the paper does not spell out is to Dirac or gauge fields, where the endpoint vacuum wavefunctional is not a simple scalar Gaussian; checking whether the same Wronskian cancellation survives would directly test the mechanism.","The argument implies that vacuum-energy or Casimir-type conclusions drawn from $\\operatorname{Re} W$ need an additional physical prescription for the endpoint phase, since only the imaginary part is protected.","In in-in or closed-time-path formulations the same endpoint states appear twice, so the boundary obstruction may reappear with a doubled structure; verifying an analogue of equation (18) there would be a concrete next step."],"forward_implications":["The vacuum persistence probability is fixed to $P_{\\rm vac}=e^{-2\\operatorname{Im} W_B}$, independent of whether one computes it from Bogoliubov coefficients or from Green's functions.","The apparent de Sitter discrepancy is not a physical ambiguity: cutoff sensitivity in the Green's function calculation is a boundary sensitivity that the endpoint wavefunctionals cancel within the same regularization scheme.","The real part of the in-out effective action is not uniquely fixed by bulk data alone; it depends on the normalization phases of the endpoint vacuum wavefunctionals.","A complete in-out path integral must treat the endpoint vacuum wavefunctionals as part of the amplitude, not as an optional normalization factor."],"supporting_citations":[{"why":"Supplies the definition of the in-out effective action and the Bogoliubov expression $W_B = \\tfrac{i}{2}\\operatorname{Tr}\\log\\alpha$ that the paper takes as the target result.","marker":"[6, 7]"},{"why":"Supplies the Green's function prescription formula (2) reconstructing the effective action from the coincident Feynman Green's function.","marker":"[8, 9]"},{"why":"States the previously assumed equivalence of the two prescriptions, which the paper shows requires the endpoint boundary completion.","marker":"[11]"},{"why":"Documents the disagreement between different methods for the imaginary part of the effective action and points to endpoint vacuum wavefunctionals as the missing piece.","marker":"[23]"},{"why":"Previous cutoff analysis by the same authors that traced the de Sitter mismatch to regularization dependence; the present paper reinterprets that sensitivity as a boundary effect.","marker":"[24]"},{"why":"Provides the Gaussian vacuum wavefunctional fixed by the annihilation conditions, used to construct $\\Psi_{\\rm in}$ and $\\Psi_{\\rm out}$.","marker":"[28]"},{"why":"Supplies the Green-Lagrange identity used to turn the bulk mode-product integral into boundary Wronskians.","marker":"[31]"},{"why":"Gives the Schrödinger wavefunctional vacuum-state construction in curved spacetime, including the Gaussian kernel and its normalization.","marker":"[32, 33]"}],"fun_headline_variants":["Boundary terms resolve vacuum persistence","Vacuum persistence made unique","Boundary completion settles vacuum persistence","One vacuum persistence after boundary completion","Endpoint wavefunctionals fix vacuum persistence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The cancellation assumes the boundary two-point function is exactly the in-out Feynman Green's function restricted to the Cauchy surface, with no extra local boundary counterterms or normal-ordering corrections; any such extra term would spoil the matching between the endpoint contributions.","fun_headline_variants_meta":{"raw":{"variants":["Boundary terms resolve vacuum persistence","Vacuum persistence made unique","Boundary completion settles vacuum persistence","One vacuum persistence after boundary completion","Endpoint wavefunctionals fix vacuum persistence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001028,"raw_usage":{"total_tokens":4316,"prompt_tokens":916,"completion_tokens":3400,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":3344}},"tokens_in":532,"tokens_out":3400,"duration_ms":21453,"temperature":1.0,"reasoning_tokens":3344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:30:29.810883+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a free scalar on a compact (1+1)-dimensional globally hyperbolic spacetime with a time-dependent mass profile, choose explicit in- and out-mode bases, and compute both sides of equation (18) within one regulator, including the endpoint Gaussian prefactor and kernel variations; if the imaginary part of the left-hand sum differs from $\\tfrac{1}{4}\\operatorname{Tr}\\log(\\alpha\\alpha^\\dagger)$ for any allowed endpoint phase or boundary basis, the boundary completion fails.","supporting_citations":[{"cited_title":"Ambjorn, R","cited_arxiv_id":null,"evidence_quote":"States the previously assumed equivalence of the two prescriptions, which the paper shows requires the endpoint boundary completion."},{"cited_title":"Zhou, H.-Q","cited_arxiv_id":null,"evidence_quote":"Previous cutoff analysis by the same authors that traced the de Sitter mismatch to regularization dependence; the present paper reinterprets that sensitivity as a boundary effect."}],"review_version":2}