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Quantum field theory on curved manifolds

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Pith's one-line read This paper argues that the entanglement between distant wedges in the Unruh effect disappears when quantum field theory on curved spacetime is analyzed locally with exact WKB methods.

desk verdict A locality-first reformulation of Unruh particle production makes a strong no-entanglement claim, but the decisive coefficient is outsourced to earlier papers and the paper itself concedes it does not prove the absence of entanglement in global calculations. read the letter →

arxiv 2501.09919 v6 pith:KDRWVQUU submitted 2025-01-17 hep-th gr-qchep-ph

Tomohiro Matsuda This is my paper · ORCID
classification hep-thgr-qchep-ph PACS 04.62.+v
keywords UnruheffectexactWKBStokesphenomenontangent-spacevacuumdifferentialgeometryentanglementHawkingradiationSchwinger
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 sets out to show that particle production in quantum field theory on manifolds can be computed locally, using only the tangent space at each point, and that when this is done faithfully the widely discussed entanglement of the Unruh effect is not present. The author argues that the conventional Rindler calculation extrapolates an accelerating observer's moving frame far beyond its range of validity and defines a vacuum in a coordinate chart, steps that are inconsistent with the local structure of manifolds. If the paper is right, the thermal bath seen by an accelerating observer is not a global thermal state entangled between opposite Rindler wedges but a local particle-creation effect with a single-particle Boltzmann factor, while Hawking radiation remains a local pair-production process at the horizon. The broader stake is methodological: a naive global solution of a field equation can be wrong even when it is exact, and exact WKB supplies the missing local nonperturbative analysis.

What carries the argument

The central object is the distinction between a chart and a frame: the Rindler coordinate system is a chart, while the accelerating observer's local inertial system is a moving frame on the frame bundle. In the tangent space at the contact point the connection vanishes and the vacuum is defined there; exact WKB then gives a mathematically controlled account of Stokes phenomena in a neighborhood. The local vacuum-to-observer map uses the vierbein relation $dt=\cosh(a\tau)\,d\tau$, and its Stokes lines produce the Bogoliubov mixing that the paper identifies as the true local content of the Unruh effect.

What would settle it

Measure the coincidence rate of two Unruh-DeWitt detectors placed in the left and right Rindler wedges, separated so that no retarded signal can pass between them, and subtract single-detector local rates; the conventional entangled-wedge calculation predicts a positive left-right coincidence rate from the partner state, while the paper's local tangent-space calculation predicts none. A clearly nonzero excess would falsify the claim, and a null result would support it.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Unruh effect, computed in the standard differential-geometric formulation of quantum field theory on manifolds, contains no entanglement between the left and right Rindler wedges. The vacuum must be defined in the local tangent space at each point, not in a global coordinate chart such as the Rindler chart; the accelerating observer's moving frame has a finite range of validity proportional to $1/a$, and the conventional calculation extrapolates it to infinity and then traces over a distant partner state that was never produced locally. Using exact WKB, the author computes the Stokes phenomenon of the vacuum mode seen through the vierbein and obtains a local Bogoliubov coefficient whose single-particle suppression is $e^{-\pi\omega/2a}$; the usual thermal rate would be recovered only if one artificially inserts a correlated partner in the opposite wedge, which is the computational glitch. Hawking radiation is different: pair creation is localized at the horizon, so the same local machinery reproduces the standard Hawking result without invoking distant entanglement.

Load-bearing premise

The conclusion rests on the assertion that a field theory's vacuum must be defined in the local tangent space at each point rather than in a global chart such as the Rindler chart; if a global Minkowski vacuum and its restriction to the Rindler wedge are legitimate, the conventional entanglement calculation is allowed and the claimed disappearance does not follow.

Editorial extensions

If this is right

  • The Unruh effect is reclassified as a local particle-creation effect: an accelerating observer sees a single-particle Boltzmann suppression rather than a thermal density matrix entangled between the two Rindler wedges.
  • The factor-of-two discrepancy between local and conventional Unruh rates is explained as a spurious trace over a partner state in a distant wedge that the local calculation never produces.
  • Hawking radiation remains unaffected: it is pair production localized at the horizon, so the local calculation reproduces the standard result without distant entanglement.
  • The Schwinger effect with a slowly time-dependent electric field becomes a local Stokes phenomenon computed on each open set, with the production rate changing gradually rather than following a global scattering amplitude.
  • Defining a vacuum in a coordinate chart such as the Rindler chart is identified as mathematically illegitimate for Bogoliubov transformations; the vacuum must live in the local tangent space.

Reading between the lines

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

  • An extension the author leaves implicit is that other globally defined vacua, such as de Sitter static versus global coordinates, may harbor similar entanglement illusions if the tangent-space vacuum rule is taken seriously.
  • The paper's logic suggests a sharper test: coincidence measurements of two Unruh-DeWitt detectors placed in opposite Rindler wedges should show no left-right particle correlations beyond local noise, which would distinguish the local picture from the conventional thermofield-double picture.
  • A consequence worth exploring is that cavity or boundary-truncated Rindler systems, where the partner wedge is not available, might resolve their apparent Unruh thermality differently under the local prescription.
  • The argument implies that a state coherent over an entire coordinate chart has no operational meaning unless it is prepared by a local process; entanglement attributed to the Unruh effect would then need to be derived from local physics rather than imported by a global mode expansion.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper claims that, when quantum field theory on curved spacetime is formulated strictly in terms of differential geometry and local analysis, particle production in the Schwinger, Unruh, and Hawking settings can be computed locally using exact WKB and Stokes phenomena. Its central claim is that the conventional Unruh effect, including the thermal state and the entanglement between the left and right Rindler wedges, is an artifact of extrapolating the Rindler chart beyond its range of validity. The paper computes a local Bogoliubov coefficient e^{-πω/(2a)} from Stokes mixing of e^{±iω∫cosh(aτ)dτ}, obtains single-particle production without tracing out a distant wedge, and argues that the usual factor-of-two discrepancy arises only if one imposes entanglement by hand. The paper concludes that entanglement between distant wedges does not appear in the standard differential-geometric formulation.

Significance. If the central claim were established, it would overturn the standard thermal interpretation of the Unruh effect and replace it with local particle creation without left-right wedge correlations. The paper does take seriously a genuine question: whether entropic and thermal features of the Unruh effect require global boundary conditions or follow from local dynamics. It also makes a useful methodological point that local trivialization and the moving frame deserve more attention in calculations of particle production. Credit is due for explicitly invoking the exact WKB literature, for attempting to respect the Markov property in local calculations, and for distinguishing chart-based from frame-based definitions of vacuum. However, the paper does not provide a derivation of its main claim; the key Bogoliubov coefficient is imported from the author's earlier papers, and the decisive step from a local calculation to the absence of entanglement in the global Minkowski vacuum is asserted rather than proved.

major comments (4)
  1. [Sec. 2, Sec. 4] The paper's central conclusion depends on the assertion that a vacuum in field theory on a manifold must be defined in the local tangent space, not in a global chart such as the Rindler chart. This assertion is justified by an appeal to Lorentz symmetry and local trivialization, but it is not derived. The conventional Unruh calculation is a different well-defined construction: one takes the global Minkowski vacuum and restricts it to the algebra of the Rindler wedge. The paper never shows that this construction is illegitimate within the standard formulation; it only asserts that extrapolation is not rigorous. A concrete test would be to derive, within the paper's framework, the restriction of the global Minkowski vacuum to the Rindler-wedge algebra and show that it factorizes; without such a derivation, the no-entanglement claim remains an assumption.
  2. [Sec. 4, Eqs. (4.10)-(4.13)] The local Stokes analysis computes mixing of the phase factors e^{±iω∫(e(τ'))^t_τ dτ'} and interprets the result as a single-particle Boltzmann factor. Even if the Stokes coefficient is exactly e^{-πω/(2a)}, that coefficient is an amplitude for a mode on one side of a turning point; it does not by itself determine whether the global state factorizes across the left and right Rindler wedges. The paper's own footnote at line 30 concedes that it does not prove the absence of entanglement in conventional global calculations. The step from 'the local calculation does not invoke the distant wedge' to 'the distant wedge is unentangled' is the load-bearing step of the paper, and it is not supplied.
  3. [Sec. 4, paragraph containing 'For a local pair creation'] The reconciliation of the factor-of-two discrepancy is asserted rather than derived. The paper states that with entanglement the probability P_1 is squared because two particles are produced but only one is observed, giving P_entangled = P_1^2. This assumes that the local single-particle amplitude describes one member of a pair whose partner lies in the opposite wedge; but that is precisely the entanglement structure the paper claims to disprove. The relation between the local Stokes coefficient and the thermal density matrix of the Rindler wedge is not obtained from any quantum-field-theoretic calculation in this manuscript, so the claimed consistency with the conventional result by 'adding entanglement by hand' is not demonstrated.
  4. [Sec. 4, after Eq. (4.12)] The central Bogoliubov coefficient e^{-πω/(2a)} is not derived in this paper. The text says 'We have already analyzed the Stokes phenomenon of the above function in detail in Refs. [15,17]' and presents only the result. Since this coefficient is the quantitative basis for the paper's physical conclusion and for the claimed factor-of-two reconciliation, the manuscript should either reproduce the exact-WKB derivation or state precisely which theorem from the literature is being used and why it applies to the phase integral ω∫cosh(aτ)dτ. Without this, the central numerical claim is unverifiable from the present text.
minor comments (4)
  1. [Abstract; Sec. 1] There are several typographical errors: 'happned' in Sec. 1, 'shoule' and 'purpuses' in Sec. 3.2, 'Boltzman factor' in Sec. 4, and inconsistent capitalization 'Unruh Ef fect' in Sec. 5.
  2. [Sec. 4, Eq. (4.12)] The notation (e(τ'))^t_τ is not defined before it is used; the vierbein component being exponentiated should be specified explicitly, and the convention for its index placement should be stated.
  3. [Sec. 5, bullet list] The bullet 'The moving frame was not moving' is telegraphic and unclear; it should be rephrased as a complete statement about what the conventional calculation does to the moving frame.
  4. [Sec. 3.2, Eq. (3.16)] The local potential V(t)|_{U_i} is presented as the correct local description, but the justification for neglecting the transverse momentum k in the local inertial frame is only a one-line statement ('k ≃ 0 is also expected'); this deserves a more careful derivation because it affects the claimed local particle-production rate.

Circularity Check

2 steps flagged · score 8.0 of 10

The no-entanglement verdict is forced by the local-vacuum premise and rests on self-cited Stokes coefficients.

  1. self definitional [Section 4, after Eq. (4.13); see also footnote 18]
    "If we assume the entanglement between the distant wedge, our result can reproduce the usual Unruh effect calculations, but since our local calculation is rigorous as mathematics, there is no need to assume such entanglement with a distant wedge. ... In short, if we “assume” entanglement, our calculation coincides with the conventional calculation, but obviously our local calculation does not require the entanglement."

    The conclusion that entanglement does not appear is built into the calculation rather than derived from it. Eqs. (4.12)-(4.13) evolve only local modes in the observer's tangent space; no operator in the opposite wedge is introduced, so the amplitude contains no left-right correlation by construction. The paper states that the conventional result is recovered only if entanglement is “assumed” or “included by hand,” which means the factor-of-two discrepancy is an artifact of the setup, not a measured prediction. Footnote 18 further concedes that the paper does not prove absence of entanglement in global calculations. Hence the “illusion” claim reduces to the prior decision to forbid global vacua and trace-outs.

  2. self citation load bearing [Section 4, paragraph after Eq. (4.10)]
    "We have already analyzed the Stokes phenomenon of the above function in detail in Ref. [15,17]. As the details of the Stokes phenomenon of the solution are not the issue of this paper, we will present only the results here."

    The quantitative content of the paper's claim—the Stokes coefficient e^{-πω/2a} later identified as the Bogoliubov/Boltzmann factor—is not derived here. It is imported from Refs [15,17], both of which are by the same author(s) (Enomoto & Matsuda, JHEP 12 (2022) 037, and Matsuda, arXiv:2404.19160). This self-citation is load-bearing because it supplies the only numerical coefficient that distinguishes the local result from the conventional one; without it the paper has no local prediction to compare. The cited result is neither machine-checked nor independently reproduced in the present work, so the argument's central numerical premise rests on an unverified self-citation rather than on a self-contained derivation.

full rationale

Score 8. The paper makes a genuine attempt to redo particle production with differential geometry and exact WKB, and the local Stokes calculation for the Schwinger effect is largely self-contained. But the headline claim about the Unruh effect is forced by two inputs: (i) the assertion that a vacuum may be defined only in the local tangent space and that Rindler-chart/global-vacuum calculations are “conceptually unacceptable,” which makes the absence of the opposite wedge a premise rather than a result; and (ii) the exact-WKB Stokes coefficient for ∫ω cosh(aτ)dτ, taken from the author's own Refs [15,17]. The paper itself concedes in footnote 18 that it does not prove the absence of entanglement in conventional global calculations and calls the appearance of entanglement there an “indisputable fact.” Under the rubric, this is partial-to-strong circularity: the central negative claim reduces by construction to the local-vacuum choice plus a self-cited coefficient, so score 8 rather than 6. The caveat prevents score 10 because there is an independent (if imported) WKB computation and a consistent local interpretation; nevertheless, the claimed falsification of Unruh entanglement is not self-contained.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on nonstandard physical premises about where the vacuum lives, a textbook limit on accelerating frames, and an exact-WKB result imported from the author's own previous papers. No new particles or forces are introduced, and no parameters are fitted to data.

assumptions (4)
  • ad hoc to paper The vacuum of a field theory on a manifold must be defined in the local tangent space, not in a chart.
    Sec.2 states 'the vacuum must be defined in the tangent space as far as the vacuum respects the Lorentz symmetry'; this is the core assumption that excludes the Rindler chart from defining a vacuum.
  • domain assumption The moving frame of an accelerating observer is valid only within a range proportional to 1/a.
    Sec.2 cites Ref. [12] for the range a^{-1} and uses it to declare the conventional Rindler calculation an extrapolation. The paper treats this textbook statement as a rigorous restriction on quantum states.
  • ad hoc to paper The Stokes phenomenon of e^{±iω∫cosh(aτ)dτ} produces a Bogoliubov coefficient with factor e^{-πω/(2a)}.
    Sec.4 says 'the details of the Stokes phenomenon of the solution are not the issue of this paper' and refers to Refs [15,17]; this is the quantitative content on which the whole factor-of-two argument rests.
  • ad hoc to paper Covariant derivatives use the Lorentz frame, so the field equations cannot reveal the local Stokes phenomenon of the Unruh effect.
    Sec.4 asserts this mismatch to justify bypassing field equations and using the vierbein directly. It is a physical claim about which frame is physical, not a standard mathematical theorem.

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Cite this review

Pith. "Pith review of Quantum field theory on curved manifolds." pith.science (2026). https://pith.science/paper/KDRWVQUU

@misc{pith2026250109919,
  author       = {Pith},
  title        = {Pith review of: Quantum field theory on curved manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDRWVQUU}},
  note         = {Machine review of arXiv:2501.09919}
}
read the original abstract

This paper discusses how particle production from the vacuum can be explained by local analysis when the field theory is defined by differential geometry on curved manifolds. We have performed the local analysis in a mathematically rigorous way, respecting the Markov property. The exact WKB is used as a tool for extracting non-perturbative effect from the local system. After a serious application of the differential geometry and the exact WKB to particle production, we show that entanglement does not appear in the Unruh effect as far as the standard formulation by the differential geometry is valid. This result should not be attributed to a consistency problem between the ``entanglement state'' and the ``standard field theory by differential geometry'', but to the fact that the conventional calculation of the Unruh effect is done by extrapolation which is not consistent with the differential geometry. The situation is similar to that of the Dirac monopole, but topology is not relevant and the basis for building field theories in differential geometry is strongly involved.

Figures

Figures reproduced from arXiv: 2501.09919 by the authors.

Figure 1
Figure 1. A local coordinate function ϕi is shown as a homeomorphism from Ui ∈ M to Vi ∈ R n . A chart is a pair (Ui , ϕ1). The Rindler coordinate is a typical chart on flat spacetime. The “chart” must be discriminated from the “frame”. Then, to describe the tangent space of a manifold, we define a tangent vector and a tangent space as follows: • A tangent vector: We introduce a tangent vector at p as X = X µ X µ ∂ ∂xµ . (2.1… view at source ↗
Figure 2
Figure 2. The tangent space TpM is spanned by {eµ} = {∂/∂xµ}, where {x µ} is the coordinate function of Ui . As is shown in Eq.(2.5) and the above picture, the vierbeins are essential for defining the Lie algebra. This means that the vacuum must be defined in the tangent space as far as the vacuum respects the Lorentz symmetry, even if it is placed at far distance as to describe the asymptotic state. The frame of an accelerat… view at source ↗
Figure 3
Figure 3. The typical “potential” of the equation is shown for [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The typical Stokes lines of the scattering problem by a qua [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: The upper picture shows the Stokes lines of the solution wh [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: The local vacuum defined in the tangent space is seen by an [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]

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Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 1 Pith paper

  1. [1]

    Quantised singularities in the electromagnetic field ,,

    P. A. M. Dirac, “Quantised singularities in the electromagnetic field ,,” Proc. Roy. Soc. Lond. A 133 (1931) no.821, 60-72

    work page 1931

  2. [2]

    Concept of Nonintegrable Phase Facto rs and Global Formulation of Gauge Fields,

    T. T. Wu and C. N. Yang, “Concept of Nonintegrable Phase Facto rs and Global Formulation of Gauge Fields,” Phys. Rev. D 12 (1975), 3845-3857

    work page 1975

  3. [3]

    On gauge invariance and vacuum polarization,

    J. S. Schwinger, “On gauge invariance and vacuum polarization,” P hys. Rev. 82 (1951), 664-679. 35

    work page 1951

  4. [4]

    Particle Creation by Black Holes,

    S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. P hys. 43 (1975), 199-220 [erratum: Commun. Math. Phys. 46 (1976), 206]

    work page 1975

  5. [5]

    Notes on black hole evaporation,

    W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D 14 (1976), 870

    work page 1976

  6. [6]

    Quantum Fields in Curved Space,

    N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,” Cambridge University Press, 1984

    work page 1984

  7. [7]

    Foundations of Differential Geome try Volume I and II

    S. Kobayashi and K. Nomizu, “Foundations of Differential Geome try Volume I and II”, Interscience Publishers, New York, NY, 1963

    work page 1963

  8. [8]

    Geometry, Topology and Physics,

    M. Nakahara, “Geometry, Topology and Physics,” CRC Press, 20 03

    work page

Show all 42 references
  1. [9]

    Mathematical Gauge Theory: With Applicat ions to the Stan- dard Model of Particle Physics

    Mark. J. D. Hamilton, “Mathematical Gauge Theory: With Applicat ions to the Stan- dard Model of Particle Physics”, Universitext, 2018

    2018

  2. [10]

    Penrose, J

    R. Penrose, J. Math. Phys. 8, 345 (1967)

    1967

  3. [11]

    Nonadiabatic crossing of energy levels,

    C. Zener, “Nonadiabatic crossing of energy levels,” Proc. Roy. Soc. Lond. A 137 (1932) 696

    1932

  4. [12]

    Gravitation,

    C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravitation,” W. H . Freeman, 1973,

    1973

  5. [13]

    Monopole Pair Production in a Mag netic Field,

    I. K. Affleck and N. S. Manton, “Monopole Pair Production in a Mag netic Field,” Nucl. Phys. B 194 (1982), 38-64

    1982

  6. [14]

    Schwinger mechanism revisited,

    F. Gelis, and N. Tanji, “Schwinger mechanism revisited,” Prog. Pa rt. Nucl. Phys. 87, 1 (2015)

    2015

  7. [15]

    The Exact WKB analysis and the St okes phenomena of the Unruh effect and Hawking radiation,

    S. Enomoto and T. Matsuda, “The Exact WKB analysis and the St okes phenomena of the Unruh effect and Hawking radiation,” JHEP 12 (2022), 037

    2022

  8. [16]

    Nonperturbative particle production and differe ntial geometry,

    T. Matsuda, “Nonperturbative particle production and differe ntial geometry,” Int. J. Mod. Phys. A 38 (2023) no.28, 2350158

    2023

  9. [17]

    How to define the moving frame of the Unruh-DeW itt detector on manifolds

    T. Matsuda, “How to define the moving frame of the Unruh-DeW itt detector on manifolds” [arXiv:2404.19160 [hep-th]]. 36

    arXiv

  10. [18]

    Preheating of massive fermions after inflation: Analytical results,

    M. Peloso and L. Sorbo, “Preheating of massive fermions after inflation: Analytical results,” JHEP 0005 (2000) 016

    2000

  11. [19]

    The exact WKB and the Landau-Z ener transition for asymmetry in cosmological particle production,

    S. Enomoto and T. Matsuda, “The exact WKB and the Landau-Z ener transition for asymmetry in cosmological particle production,” JHEP 02 (2022), 131

    2022

  12. [20]

    The Exact WKB analysis for asymm etric scalar pre- heating,

    S. Enomoto and T. Matsuda, “The Exact WKB analysis for asymm etric scalar pre- heating,” JHEP 01 (2023), 088

    2023

  13. [21]

    Towards the th eory of reheating after inflation,

    L. Kofman, A. D. Linde and A. A. Starobinsky, “Towards the th eory of reheating after inflation,” Phys. Rev. D 56 (1997) 3258

    1997

  14. [22]

    On the theory of fermionic prehe ating,

    P. B. Greene and L. Kofman, “On the theory of fermionic prehe ating,” Phys. Rev. D 62 (2000), 123516

    2000

  15. [23]

    The exact WKB for cosmological p article production,

    S. Enomoto and T. Matsuda, “The exact WKB for cosmological p article production,” JHEP 03 (2021), 090

    2021

  16. [24]

    Resurgence, Physics and Numbers

    “Resurgence, Physics and Numbers” edited by F. Fauvet, D. M anchon, S. Marmi and D. Sauzin, Publications of the Scuola Normale Superiore

  17. [25]

    The return of the quartic oscillator – The complex WK B method

    A. Voros, “The return of the quartic oscillator – The complex WK B method”, Ann. Inst. Henri Poincare, 39 (1983), 211-338

    1983

  18. [26]

    Delabaere, H

    E. Delabaere, H. Dillinger and F. Pham: Resurgence de Voros et p eeriodes des courves hyperelliptique. Annales de l’Institut Fourier, 43 (1993), 163- 199

    1993

  19. [27]

    Observations on the JWKB trea tment of the quadratic barrier, Algebraic analysis of differential equations from microlocal anal- ysis to exponential asymptotics

    H. Shen and H. J. Silverstone, “Observations on the JWKB trea tment of the quadratic barrier, Algebraic analysis of differential equations from microlocal anal- ysis to exponential asymptotics”, Springer, 2008, pp. 237 - 250

    2008

  20. [28]

    Resurgence, quantized canonical transformation s, and multiinstanton ex- pansions,

    F. Pham, “Resurgence, quantized canonical transformation s, and multiinstanton ex- pansions,” Algebraic Analysis, Vol. II, Academie Press, 1988, pp. 69 9-726

    1988

  21. [29]

    Approche de la R esurgence,

    B. Candelpergher, J. G. Nositicls and F. Pham, “Approche de la R esurgence,” Her- mann, 1993. 37

    1993

  22. [30]

    Resurgence de Voros et periodes des courbes hyper elliptiques,

    E. Delabaere, H. Dillinger and F. Pham, “Resurgence de Voros et periodes des courbes hyper elliptiques,” Ann. Inst. Fourier (Grenoble) 4 3 (1993), 163-1 99

    1993

  23. [31]

    Exact semi-classical exp ansions for one dimensional quantum oscillators,

    E. Delabaere, H. Dillinger and F. Pham, “Exact semi-classical exp ansions for one dimensional quantum oscillators,” J. Math. Phys. 38 (1997), 6126- 6184

    1997

  24. [32]

    Algebraic analysis of differential equations: From microlocal analysis to exponential asymptotics fe stschrift in honor of Takahiro Kawai

    T. Aoki, H. Majima, Y. Takei, N. Tose (Eds.) “Algebraic analysis of differential equations: From microlocal analysis to exponential asymptotics fe stschrift in honor of Takahiro Kawai”

  25. [33]

    Delabaere and F

    E. Delabaere and F. Pham, “Resurgent methods in semi-classica l asymptotics, Ann. Inst. H. Poincare, 71(1999),1-94

    1999

  26. [34]

    Algebraic Analysis of Singular Perturbat ion Theory,

    T. Kawai and Y. Takei, “Algebraic Analysis of Singular Perturbat ion Theory,” Iwanami Series in Modern Mathematics, 2005

    2005

  27. [35]

    Virtual Turning Points

    N. Honda, T. Kawai and Y. Takei, “Virtual Turning Points”, Spr inger (2015)

    2015

  28. [36]

    Exact WKB analysis of the vacuum pair production by time-dependent electric fields,

    H. Taya, T. Fujimori, T. Misumi, M. Nitta and N. Sakai, “Exact WKB analysis of the vacuum pair production by time-dependent electric fields,” JHEP 03 (2021), 082

    2021

  29. [37]

    Unstable vacuum and fermion total reflection by the Klein step,

    H. Nakazato and M. Ochiai, “Unstable vacuum and fermion total reflection by the Klein step,” PTEP 2022 (2022) no.7, 073B02

    2022

  30. [38]

    Schwinger Effect in Inflaton-Driven Electric Field,

    H. Kitamoto, “Schwinger Effect in Inflaton-Driven Electric Field,” Phys. Rev. D 98 (2018) no.10, 103512

    2018

  31. [39]

    Nonuniqueness of canonical field quantization in Rie mannian space- time,

    S. A. Fulling, “Nonuniqueness of canonical field quantization in Rie mannian space- time,” Phys. Rev. D 7 (1973), 2850-2862

    1973

  32. [40]

    Scalar particle production in Schwarzschild and Rindler metrics,

    P. C. W. Davies, “Scalar particle production in Schwarzschild and Rindler metrics,” J. Phys. A 8 (1975), 609-616

    1975

  33. [41]

    A WKB-Like App roach to Unruh Radiation,

    A. de Gill, D. Singleton, V. Akhmedova and T. Pilling, “A WKB-Like App roach to Unruh Radiation,” Am. J. Phys. 78 (2010), 685-691

    2010

  34. [42]

    Can Quantum-Mechanical D escription of Phys- ical Reality Be Considered Complete?

    A. Einstein, B. Podolsky,N. Rosen, “Can Quantum-Mechanical D escription of Phys- ical Reality Be Considered Complete?”, Phy. Rev. 47(1935), 10. 38

    1935

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