Boundary Completion of Vacuum Persistence Probability

Hai-Qing Zhang; Yu Zhou

REVIEW 3 major objections 4 minor 33 references

Boundary Completion of Vacuum Persistence Probability

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Once endpoint vacuum wavefunctionals are included, the Green's function prescription agrees with the Bogoliubov formula, so vacuum persistence is prescription independent.

desk verdict 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. read the letter →

arxiv 2607.20936 v1 pith:BWHPLKQM submitted 2026-07-23 hep-th gr-qc

Yu Zhou , Hai-Qing Zhang This is my paper · ORCID

classification hep-thgr-qc MSC 81T2083C47 PACS 04.62.+v

keywords vacuumpersistenceprobabilityin-outeffectiveactionBogoliubovcoefficientsGreen'sfunctionprescriptionendpointwavefunctionalsSchrödingerwavefunctionalWronskianboundarytermsparticleproductionincurvedspacetime

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 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.

What carries the argument

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 $\alpha$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $P_{\rm vac}=e^{-2\operatorname{Im} W_B}$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The 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.

What 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.

Where 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.

There'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.

Who'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.

My 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Boundary completion by vacuum wavefunctionals, Eq. (13)] 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.
[In-out amplitude with endpoint states, Eqs. (8), (14)-(16)] 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.
[Conclusion, Eq. (19)] 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.

minor comments (4)

[Eq. (2)] 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.
[Throughout] The text uses 'wavefunctional' and 'wave functional' interchangeably; please choose one spelling and use it consistently.
[Supplemental Material, Eq. (S56)] 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.
[Eq. (19)] 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'.

Circularity Check

1 steps flagged · score 4.0 of 10

The endpoint completion cancels the obstruction by reusing the same in-out Green's function as the boundary two-point function, making the resolution partially circular.

self definitional [Boundary completion by vacuum wavefunctionals; Eq. (13) and Eqs. (14)-(17)]
"Restricting the in-out Feynman Green’s function to the endpoint surface and using θ(x,x) = 1/2 gives, ⟨φσ(x)φσ(y)⟩= 1/2 Σ_{k,k'} (α−1)_{kk'} [uout_{k′}(x)uin∗_k(y)+uout_{k′}(y)uin∗_k(x)]. Thus, the endpoint expectation value is not a new correlator. It is the same as the in-out Green’s function, restricted to the temporal boundary."

Eq. (13) identifies the endpoint two-point function with the very in-out Feynman Green's function whose coincident bulk limit, via Eq. (6), produced the boundary obstruction B_in+B_out in Eqs. (8)-(9). Inserting (13) into the Gaussian-kernel variation yields Wronskian terms built from the same (α^{-1}) mode sums as B_in/B_out (Eqs. (14)-(16)), so the cancellation in (17) is an algebraic identity. The endpoint contribution is therefore not an independent evaluation of the vacuum wavefunctionals; it is the same Green's function re-inserted with a boundary theta prescription. Any additional boundary contact or normal-ordering term in ⟨φφ⟩ would leave a residual term and invalidate Eq. (19).

full rationale

Overall, the paper contains a genuine calculation: the bulk Green's-function integral is reduced by the Green-Lagrange identity to a Bogoliubov variation plus endpoint Wronskians (Eqs. (7)-(9)), and that part is self-contained and independent of the conclusion. The circularity is localized in the completion step: the boundary two-point function is asserted to be the restricted in-out Green's function (Eq. (13)), and the endpoint Wronskian terms generated from it are the same objects as the obstruction terms, so the cancellation is built in. The self-citations (Refs. [15,24]) are motivational rather than load-bearing, and no fitting or imported uniqueness theorem is used. The reviewer's concern that Eq. (13) needs a separate derivation (e.g., potential boundary contact terms or a different coincident prescription for distinct boundary points) is a genuine correctness risk, but it is also what makes the completion step circular: if the identification is assumed, the Bogoliubov result follows by construction. Score 4 reflects the balance: the bulk obstruction derivation has independent content, while the central resolution is partly a rearrangement of the input propagator.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The derivation is a formal field-theoretic argument. The main inputs are the path integral representation with Gaussian vacuum wavefunctionals, the existence and completeness of the in/out mode bases, vanishing lateral boundary terms, and the large-mass reference condition. The only hand-set quantities are the arbitrary endpoint phases θ_in/out and the integration constant W0, neither of which affects the imaginary part.

free parameters (2)

Endpoint vacuum phases θ_in(m^2), θ_out(m^2) = arbitrary
Arbitrary real phases of the Gaussian vacuum wavefunctionals (Eq. 11, S64). They enter only the real combination Θ, shifting Re W but not Im W.
Integration constant W0 = Im W0 = 0 by large-mass limit
Mass-independent constant from integrating ∂_{m^2} W; the reference condition at large mass sets Im W0 = 0.

assumptions (5)

domain assumption The in-out amplitude admits a path-integral representation with endpoint vacuum wavefunctionals (Eq. 3).
Standard in-out formalism for free fields; assumed without derivation.
domain assumption The vacuum wavefunctionals are Gaussian with kernel Kσ built from the positive-frequency boundary data, with a nondegenerate boundary basis qσ (Eq. 10, S52).
Follows from the annihilation conditions for free fields; assumes the q-basis is complete and invertible.
domain assumption Lateral/spatial boundary contributions to the Gauss-law reduction vanish.
Assumed in Eq. (7) and S34 to keep only the temporal Cauchy surfaces.
domain assumption A common regulator can be introduced and removed after combining bulk and endpoint terms, with cancellations independent of the regulator.
All sums and determinants are treated in the regulated theory; the regulator is removed at the end.
domain assumption In the large-mass limit αα† → I, so Im W0 = 0.
Reference condition used to fix the mass-independent constant in the integrated effective action.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Boundary Completion of Vacuum Persistence Probability." pith.science (2026). https://pith.science/paper/BWHPLKQM

@misc{pith2026260720936,
  author       = {Pith},
  title        = {Pith review of: Boundary Completion of Vacuum Persistence Probability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWHPLKQM}},
  note         = {Machine review of arXiv:2607.20936}
}

read the original abstract

The vacuum persistence probability, encoded in the imaginary part of the in-out effective action, is a basic measure of vacuum instability and particle production. However, its evaluation may appear prescription-dependent: the Bogoliubov prescription and the Green's function prescription can yield different expressions. We show that this apparent ambiguity arises because the Green's function prescription omits the nontrivial contribution from the endpoint vacuum wavefunctionals, which should be present in the complete in-out amplitude. Including this contribution accomplishes the boundary completion of the Green's function prescription. The ambiguity is thereby resolved, and the complete result agrees with the Bogoliubov expression.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 21 canonical work pages

[1]

J. S. Schwinger, On gauge invariance and vacuum po- larization, Phys. Rev. 82 (1951) 664–679.doi:10.1103/ PhysRev.82.664

work page 1951
[2]

S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199–220, [Erratum: Commun.Math.Phys. 46, 206 (1976)].doi:10.1007/ BF02345020

work page 1975
[3]

Parker, Particle creation in expanding universes, Phys

L. Parker, Particle creation in expanding universes, Phys. Rev. Lett. 21 (1968) 562–564.doi:10.1103/ PhysRevLett.21.562

work page 1968
[4]

Parker, Quantized fields and particle creation in ex- panding universes

L. Parker, Quantized fields and particle creation in ex- panding universes. i, Phys. Rev. 183 (1969) 1057–1068. doi:10.1103/PhysRev.183.1057. URLhttps://link.aps.org/doi/10.1103/PhysRev. 183.1057

doi:10.1103/physrev.183.1057 1969
[5]

S. A. Fulling, Nonuniqueness of canonical field quanti- zation in riemannian space-time, Phys. Rev. D 7 (1973) 2850–2862.doi:10.1103/PhysRevD.7.2850. URLhttps://link.aps.org/doi/10.1103/PhysRevD. 7.2850

doi:10.1103/physrevd.7.2850 1973
[6]

B. S. DeWitt, Quantum field theory in curved spacetime, Physics Reports 19 (6) (1975) 295–357.doi:https: //doi.org/10.1016/0370-1573(75)90051-4. URLhttps://www.sciencedirect.com/science/ article/pii/0370157375900514

arXiv 1975
[7]

A. I. Nikishov, The vacuum-vacuum amplitude and bo- goliubov coefficients, Journal of Experimental and The- oretical Physics 96 (2) (2003) 180–192.doi:10.1134/1. 1560392. URLhttp://dx.doi.org/10.1134/1.1560392

doi:10.1134/1 2003
[8]

Parker, D

L. Parker, D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity, Cambridge Monographs on Mathematical Physics, Cambridge Uni- versity Press, 2009

work page 2009

Show all 33 references

[9]

N. D. Birrell, P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, Cambridge, UK, 1984. doi:10.1017/CBO9780511622632

1984 doi
[10]

D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys.Rept.388(2003)279–360.arXiv:hep-th/0306138, doi:10.1016/j.physrep.2003.09.002

2003 arXiv
[11]

Ambjorn, R

J. Ambjorn, R. J. Hughes, N. K. Nielsen, Action Princi- ple of Bogolyubov Coefficients, Annals Phys. 150 (1983) 92.doi:10.1016/0003-4916(83)90005-2

1983 doi
[12]

S. P. Kim, Effective Action and Schwinger Pair Produc- tion in Strong QED, AIP Conf. Proc. 1150 (1) (2009) 95–100.arXiv:0902.3486,doi:10.1063/1.3192303

2009 arXiv
[13]

C.-M. Chen, S. P. Kim, C. A. R. Medina, Exact one- loop QED actions in global(A)dS 2 (3 2026).arXiv: 2603.05826

2026
[14]

M. F. Wondrak, W. D. van Suijlekom, H. Fal- cke, Gravitational pair production and black hole evaporation, Phys. Rev. Lett. 130 (2023) 221502. doi:10.1103/PhysRevLett.130.221502. URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.130.221502

2023 doi
[15]

Zhou, H.-Q

Y. Zhou, H.-Q. Zhang, Effective action and gravitational pair production in (a)ds spacetime (2024).arXiv:2410. 20949. URLhttps://arxiv.org/abs/2410.20949

2024 arXiv
[16]

Fecit, S

F. Fecit, S. A. Franchino-Viñas, F. D. Mazzitelli, Resummed effective actions and heat kernels: the Worldline approach and Yukawa assisted pair creation, JHEP 07 (2025) 041.arXiv:2501.17094,doi:10.1007/ JHEP07(2025)041

2025 arXiv
[17]

G. V. Dunne, Heisenberg–euler effective lagrangians: ba- sics and extensions, in: From Fields to Strings: Cir- cumnavigating Theoretical Physics: Ian Kogan Memo- rial Collection (In 3 Volumes), World Scientific, 2005, pp. 445–522

2005
[18]

L. H. Ford, Cosmological particle production: a review, Reports on Progress in Physics 84 (11) (2021) 116901. doi:10.1088/1361-6633/ac1b23. URLhttp://dx.doi.org/10.1088/1361-6633/ac1b23

2021 doi
[19]

E. W. Kolb, A. J. Long, Cosmological gravitational particle production and its implications for cosmo- logical relics, Rev. Mod. Phys. 96 (2024) 045005. doi:10.1103/RevModPhys.96.045005. URLhttps://link.aps.org/doi/10.1103/ RevModPhys.96.045005

2024 doi
[20]

Mottola, Particle creation in de sitter space, Phys

E. Mottola, Particle creation in de sitter space, Phys. Rev. D 31 (1985) 754–766.doi:10.1103/PhysRevD.31. 754. URLhttps://link.aps.org/doi/10.1103/PhysRevD. 31.754

1985 doi
[21]

E. T. Akhmedov, Real or Imaginary? (On pair cre- ation in de Sitter space), Mod. Phys. Lett. A 25 (2010) 2815–2823.arXiv:0909.3722,doi:10.1142/ S0217732310034043

2010 arXiv
[22]

Akhmedov, K

E. Akhmedov, K. Bazarov, D. Diakonov, U. Moschella, F. Popov, C. Schubert, Propagators and gaussian 6 effective actions in various patches of de sitter space, Physical Review D 100 (10) (Nov. 2019). doi:10.1103/physrevd.100.105011. URLhttp://dx.doi.org/10.1103/PhysRevD.100. 105011

2019 doi
[23]

E. T. Akhmedov, I. A. Belkovich, D. V. Diakonov, K. A. Kazarnovskii, On(dis)agreementbetweendifferentmeth- ods of calculation of the imaginary part of the effective actioninexpandingspace-times, Phys.Lett.B861(2025) 139256.arXiv:2411.02170,doi:10.1016/j.physletb. 2025.139256

2025
[24]

Zhou, H.-Q

Y. Zhou, H.-Q. Zhang, Imaginary part of the effective action in de Sitter spacetime with different regularization schemes, Phys. Rev. D 113 (2) (2026) 026007.arXiv: 2510.13712,doi:10.1103/tpdv-2q6v

2026
[25]

P. R. Anderson, E. Mottola, D. H. Sanders, Decay of the de sitter vacuum, Phys. Rev. D 97 (2018) 065016. doi:10.1103/PhysRevD.97.065016. URLhttps://link.aps.org/doi/10.1103/PhysRevD. 97.065016

2018 doi
[26]

A. I. Nikishov, Pair production by a constant external field, Zh. Eksp. Teor. Fiz. 57 (1969) 1210–1216

1969
[27]

Fukushima, S

K. Fukushima, S. Minato, In-in formalism with resum- mation in a constant electric field: propagators including nontrivial boundary wavefunctions, JHEP 05 (2026) 139. arXiv:2512.19337,doi:10.1007/JHEP05(2026)139

2026
[28]

Weinberg, The Quantum theory of fields

S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations, Cambridge University Press, 2005.doi: 10.1017/CBO9781139644167

2005 doi
[29]

J. S. Schwinger, On the Green’s functions of quantized fields. 1., Proc. Nat. Acad. Sci. 37 (1951) 452–455.doi: 10.1073/pnas.37.7.452

1951 doi
[30]

See Supplemental Material for conventions, the deriva- tion of the in-out Green’s function representation, the re- duction of the bulk Green’s function integral to endpoint terms, and the evaluation of the endpoint wavefunctional contribution
[31]

C. Baer, N. Ginoux, F. Pfaeffle, Wave equations on lorentzian manifolds and quantization (2008).arXiv: 0806.1036. URLhttps://arxiv.org/abs/0806.1036

2008 arXiv
[32]

D. Long, G. Shore, The schrödinger wave func- tional and vacuum states in curved spacetime, Nuclear Physics B 530 (1-2) (1998) 247–278. doi:10.1016/s0550-3213(98)00408-8. URLhttp://dx.doi.org/10.1016/S0550-3213(98) 00408-8

1998 doi
[33]

Boundary Completion of Vacuum Persistence Probability

D. Long, G. Shore, The schrödinger wave functional and vacuum states in curved spacetime ii: Boundaries and foliations, Nuclear Physics B 530 (1-2) (1998) 279–303. doi:10.1016/s0550-3213(98)00409-x. URLhttp://dx.doi.org/10.1016/S0550-3213(98) 00409-X 7 Supplemental Material fo...

1998 doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

J. S. Schwinger, On gauge invariance and vacuum po- larization, Phys. Rev. 82 (1951) 664–679.doi:10.1103/ PhysRev.82.664

work page 1951

[2] [2]

S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199–220, [Erratum: Commun.Math.Phys. 46, 206 (1976)].doi:10.1007/ BF02345020

work page 1975

[3] [3]

Parker, Particle creation in expanding universes, Phys

L. Parker, Particle creation in expanding universes, Phys. Rev. Lett. 21 (1968) 562–564.doi:10.1103/ PhysRevLett.21.562

work page 1968

[4] [4]

Parker, Quantized fields and particle creation in ex- panding universes

L. Parker, Quantized fields and particle creation in ex- panding universes. i, Phys. Rev. 183 (1969) 1057–1068. doi:10.1103/PhysRev.183.1057. URLhttps://link.aps.org/doi/10.1103/PhysRev. 183.1057

doi:10.1103/physrev.183.1057 1969

[5] [5]

S. A. Fulling, Nonuniqueness of canonical field quanti- zation in riemannian space-time, Phys. Rev. D 7 (1973) 2850–2862.doi:10.1103/PhysRevD.7.2850. URLhttps://link.aps.org/doi/10.1103/PhysRevD. 7.2850

doi:10.1103/physrevd.7.2850 1973

[6] [6]

B. S. DeWitt, Quantum field theory in curved spacetime, Physics Reports 19 (6) (1975) 295–357.doi:https: //doi.org/10.1016/0370-1573(75)90051-4. URLhttps://www.sciencedirect.com/science/ article/pii/0370157375900514

arXiv 1975

[7] [7]

A. I. Nikishov, The vacuum-vacuum amplitude and bo- goliubov coefficients, Journal of Experimental and The- oretical Physics 96 (2) (2003) 180–192.doi:10.1134/1. 1560392. URLhttp://dx.doi.org/10.1134/1.1560392

doi:10.1134/1 2003

[8] [8]

Parker, D

L. Parker, D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity, Cambridge Monographs on Mathematical Physics, Cambridge Uni- versity Press, 2009

work page 2009