Pith. sign in

REVIEW 2 major objections 4 minor 103 references

A Wigner-function framework yields a first-principles thermal nucleation rate in quantum field theory that generalizes Affleck and disagrees with Linde.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-13 04:30 UTC pith:US5KMK5G

load-bearing objection Clean first-principles derivation of a QFT nucleation rate that generalizes Affleck, keeps quantum prefactors, and asymptotes to Langer rather than Linde; immediately usable for analog experiments near T~m. the 2 major comments →

arxiv 2607.09233 v1 pith:US5KMK5G submitted 2026-07-10 hep-th cond-mat.quant-gashep-ph

Quantum field nucleating and Wigner functions

Joonas Hirvonen This is my paper
classification hep-th cond-mat.quant-gashep-ph
keywords false vacuum decayWigner functionsnucleation ratethermal phase transitionsAffleck formulaLanger theoryquantum field theoryanalog experiments
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a real-time description of metastable decay in quantum field theory from the flux of the Wigner quasiprobability distribution across a surface in field configuration space. The resulting nonperturbative rate formula covers both quantum tunneling and thermal over-the-barrier escape, and the author supplies an explicit four-step procedure that evaluates it at one-loop order. Applied to a simple real scalar theory at intermediate temperatures relevant to present analog experiments, the rate contains quantum trigonometric factors in the prefactor that are absent from Linde’s formula and asymptotes instead to Langer’s classical rate through high-temperature dimensional reduction. The same calculation shows that the high-temperature side of the so-called quantum-to-classical crossover remains quantum-mechanical even though the bounce background has classical symmetry. Readers care because the formula supplies the correct rate for table-top simulators and identifies which high-temperature expression should be used in early-universe calculations.

Core claim

The one-loop nucleation rate for over-the-barrier escape from a thermal metastable phase of a real scalar quantum field is Γ = (V/2π)(ΔE_cb/2πT)^{d/2} [det(-∂_μ^{2}+V''(ϕ_meta))/det_{+}(-∂_μ^{2}+V''(ϕ_cb))]^{1/2} exp(-βΔE_cb). This expression is the direct field-theoretic generalization of Affleck’s quantum-mechanical escape rate; at high temperature it recovers Langer’s rate via effective-field-theory dimensional reduction rather than Linde’s formula, and its prefactor retains genuine quantum effects even when the bounce possesses classical O(d) imes S^{1} symmetry.

What carries the argument

The nonperturbative nucleation rate written as the flux of the Wigner function through the critical surface ϕ_-=0, together with the four-step algorithm that constructs the equilibrium Wigner function on that surface and multiplies by the step-function off-equilibrium factor θ(π_--√|λ_-|ϕ_-).

Load-bearing premise

The metastable phase remains close enough to thermal equilibrium that the probability flow across the barrier can be read off from a simple step-function cut of the equilibrium Wigner distribution.

What would settle it

A classical or truncated-Wigner lattice simulation of the same scalar model that measures the intermediate-temperature nucleation rate and finds a prefactor matching Linde rather than the new formula, or an analog cold-atom experiment whose bubble-formation rate fails to display the predicted quantum trigonometric factors.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Analog experiments operating at T~m should employ the new one-loop formula rather than Linde’s rate.
  • Cosmological calculations previously based on Linde’s high-temperature expression need to be recomputed with the Affleck/Langer asymptotics.
  • The high-temperature side of the quantum-to-classical transition for thermal vacuum decay remains quantum-mechanical in the prefactor.
  • Extending the Wigner method with hard thermal loops can capture infrared quantum and fermionic corrections beyond existing Boltzmann treatments.
  • A two-loop analysis can decide whether the lower validity bound lies at √|λ_-|/π or √|λ_-|/(2π).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mismatch between Wigner-function singularities and the rate formula may signal that nonperturbative quantum effects set in earlier than one-loop suggests, altering the matching onto pure vacuum decay.
  • Existing cold-atom false-vacuum platforms can directly test the quantum prefactor by scanning temperature through the intermediate regime.
  • When equilibrium is not maintained, the step-function factor must be replaced by a dynamical solution of the Moyal equation, which may account for the order-of-magnitude deviations already seen in classical lattice studies.
  • The same flux construction applies immediately to charged or gauged systems once the corresponding Wigner function is known.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a real-time framework for metastable-state decay in QFTs based on the Wigner function. From the continuity equation for the Wigner function it obtains a nonperturbative flux formula for the escape rate (Eq. 2.7). Specializing to over-the-barrier nucleation from a thermal metastable phase, it expands the CTP path integral about the critical bubble, constructs the one-loop equilibrium Wigner function (Appendix B), multiplies by a step-function off-equilibrium factor that solves the leading Moyal equation, and evaluates the flux on the surface ϕ_-=0. The resulting one-loop rate (Eqs. 4.27–4.28) is the direct QFT generalization of Affleck’s formula; it contains quantum trigonometric factors in the prefactor and asymptotes, via high-T dimensional reduction, to Langer’s classical rate rather than to Linde’s formula. The intermediate-temperature window of validity is carefully delimited by the singularities of the Euclidean fluctuation spectrum (Secs. 5–6).

Significance. If correct, the result supplies a first-principles real-time derivation that places Affleck’s quantum-mechanical rate on a firm QFT footing and simultaneously clarifies why the high-T limit must recover Langer (via EFT) rather than Linde. The explicit one-loop formula is immediately usable for analog cold-atom experiments operating near T∼m, and the Wigner-function construction opens a systematic route to real-time corrections (off-equilibrium particles, infrared quantum fluctuations) that are inaccessible to Euclidean methods. The derivation is parameter-free and self-contained once the equilibrium ansatz is accepted.

major comments (2)
  1. Sec. 5: the one-loop Wigner function already diverges at T=√|λ_-|/π (from the anti-periodic zero modes), while the rate formula itself only diverges at the lower temperature T=√|λ_-|/(2π). The uniqueness-of-asymptotics argument offered for trusting the analytic continuation is plausible but not demonstrated. A two-loop computation of the Wigner function (or an explicit statement that the rate remains finite through the intermediate window) is needed before the lower edge of the validity window can be regarded as settled.
  2. Sec. 4, Eqs. (4.1)–(4.3) and (4.18): the entire rate rests on the assumption that the metastable phase remains close enough to equilibrium for the off-equilibrium factor to be the simple step function θ(π_--√|λ_-|ϕ_-). The paper itself cites 1+1 classical simulations that show O(1) deviations when thermal noise is absent. While this is correctly flagged as a limitation of applicability rather than an internal inconsistency, the manuscript should quantify (or at least bound) the size of the expected correction in d≥2 or under quantum evolution before claiming quantitative relevance for the analog experiments of Refs. [9,40].
minor comments (4)
  1. Eq. (4.28) and the sentence that follows: the Euclidean operators are said to have “periodic boundary conditions o even n eigenvalues of Eq. (5.3)”. A one-line clarification that the odd-n (anti-periodic) modes have already been integrated out into the Wigner function would remove a possible source of confusion.
  2. Appendix C: the critique of the truncated-Wigner literature is useful, but the claim that those works “promote zero-point fluctuations into classical propagating modes” would be stronger if a single concrete numerical discrepancy (e.g., from Ref. [77] or [108]) were quoted.
  3. Throughout: a few typographical slips remain (“possible possible”, “wenotethat”, missing spaces after periods). A careful proof-reading pass is warranted.
  4. Sec. 6: the thin-wall remark that wall-deformation modes become classical already at T∼m for large R is interesting; a short estimate of the corresponding correction to the prefactor would make the statement more quantitative.

Circularity Check

0 steps flagged

No significant circularity: the one-loop rate follows by direct evaluation of the Wigner flux under a standard equilibrium ansatz and the Liouville-compatible step-function factor; self-citations supply independent high-T and lattice checks rather than definitions of the central formula.

full rationale

The nonperturbative rate (2.7) is obtained from the continuity equation of the Wigner function after integrating out conjugate momenta; no free parameters are fitted. The one-loop evaluation proceeds by (i) the Euclidean saddle for the equilibrium density matrix around the critical bubble (standard, App. B), (ii) its Wigner transform (B.17), (iii) the off-equilibrium factor heta( heta(π_- - √|λ_-| φ_-) that solves the leading Moyal/Liouville equation (A.8) and enforces the metastable/stable boundary conditions (4.1–4.3, 4.18), and (iv) insertion into the surface integral (3.13) yielding (4.27)/(4.28). The high-T expansion (6.1) then matches the author’s earlier EFT free-energy construction and Langer’s rate by ordinary asymptotic analysis of the same determinants, not by redefinition. Self-citations ([56], [62], [65]) appear only as consistency checks or known limitations of the equilibrium assumption; they do not enter the derivation of the prefactor or the claim that the result generalizes Affleck rather than Linde. No uniqueness theorem is imported to forbid alternatives, no ansatz is smuggled via citation, and no data-fitting occurs. The only residual self-reference is the author’s prior EFT work used for the high-T comparison, which is non-load-bearing for the intermediate-T formula itself. Hence the derivation chain is self-contained against external benchmarks (Affleck, Langer, Euclidean one-loop determinants).

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard QFT tools (Wigner transform, CTP path integral, Euclidean density matrix, one-loop fluctuation determinants) plus the modeling assumption that the metastable phase is close enough to equilibrium for the step-function off-equilibrium factor to be accurate. No free parameters are fitted; no new particles or forces are introduced.

axioms (4)
  • standard math The Wigner function of a density matrix satisfies the Moyal equation, which integrates to a continuity equation for the probability current in configuration space (Eq. 2.5).
    Standard property of the Wigner transform; used to convert the decay rate into a surface flux (Sec. 2).
  • domain assumption Around the critical bubble the advanced-field nonlinearities can be neglected, yielding a linear real-time equation of motion for the retarded field (Eq. 3.8).
    Valid only when ϕ_a fluctuations remain suppressed; breaks down below T∼√|λ_-|/π (Sec. 5).
  • domain assumption The metastable phase is sufficiently close to thermal equilibrium that the nucleating Wigner function is the equilibrium Wigner function multiplied by a step function that selects trajectories originating from the metastable side (Eqs. 4.1–4.3, 4.18).
    Stated explicitly in Sec. 4; known to fail by O(1) factors in some classical 1+1 simulations.
  • standard math One-loop Euclidean fluctuation determinants around the critical bubble and the metastable minimum correctly normalize the Wigner function (Appendix B).
    Standard saddle-point evaluation of the thermal density matrix.

pith-pipeline@v1.1.0-grok45 · 26421 in / 2813 out tokens · 36954 ms · 2026-07-13T04:30:31.782400+00:00 · methodology

0 comments
read the original abstract

We present a novel real-time framework for the decay of metastable states in quantum field theories using Wigner functions. The framework introduces a nonperturbative nucleation rate formula that captures both quantum tunneling and over-the-barrier nucleation, alongside steps to evaluate it perturbatively. We apply it to a simple thermal example with direct relevance to current analog experiments. Our derived one-loop nucleation rate fundamentally differs from the widely cited high-temperature result by Linde: The prefactor contains quantum effects and also asymptotes to a differing form at high temperatures, where the quantum effects become negligible. Rather, the result is a generalization of Affleck's rate formula to quantum field theories, asymptoting to the effective field theory approach, and Langer's rate, at high temperatures. The example also reveals that the high-temperature side of the ``quantum-to-classical'' transition of thermal vacuum decay is still inherently quantum mechanical, even though the bounce background possesses the classical, $\mathrm{O}(d)\times S^1$, symmetry.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

103 extracted references · 64 linked inside Pith

  1. [1]

    Fialko, B

    O. Fialko, B. Opanchuk, A.I. Sidorov, P.D. Drummond and J. Brand,Fate of the false vacuum: towards realization with ultra-cold atoms,EPL110(2015) 56001 [1408.1163]

    Pith/arXiv arXiv 2015

  2. [2]

    Fialko, B

    O. Fialko, B. Opanchuk, A.I. Sidorov, P.D. Drummond and J. Brand,The universe on a table top: engineering quantum decay of a relativistic scalar field from a metastable vacuum, J. Phys. B50(2017) 024003 [1607.01460]

    Pith/arXiv arXiv 2017

  3. [3]

    Billam, K

    T.P. Billam, K. Brown and I.G. Moss,False-vacuum decay in an ultracold spin-1 Bose gas, Phys. Rev. A105(2022) L041301 [2108.05740]

    Pith/arXiv arXiv 2022

  4. [4]

    B. Song, S. Dutta, S. Bhave, J.-C. Yu, E. Carter, N. Cooper et al.,Realizing discontinuous quantum phase transitions in a strongly correlated driven optical lattice,Nature Phys.18 (2022) 259 [2105.12146]

    Pith/arXiv arXiv 2022

  5. [5]

    Tian et al.,Supercooling of the A phase of3He,Nature Commun.14(2023) 148 [2208.14981]

    Y. Tian et al.,Supercooling of the A phase of3He,Nature Commun.14(2023) 148 [2208.14981]

    Pith/arXiv arXiv 2023

  6. [6]

    Viermann et al.,Quantum field simulator for dynamics in curved spacetime,Nature611 (2022) 260 [2202.10399]

    C. Viermann et al.,Quantum field simulator for dynamics in curved spacetime,Nature611 (2022) 260 [2202.10399]

    Pith/arXiv arXiv 2022

  7. [7]

    Zenesini, A

    A. Zenesini, A. Berti, R. Cominotti, C. Rogora, I.G. Moss, T.P. Billam et al.,Observation of false vacuum decay via bubble formation in ferromagnetic superfluids,Nature Physics (2023) 1 [2305.05225]

    Pith/arXiv arXiv 2023

  8. [8]

    Jenkins, J

    A.C. Jenkins, J. Braden, H.V. Peiris, A. Pontzen, M.C. Johnson and S. Weinfurtner,Analog vacuum decay from vacuum initial conditions,Phys. Rev. D109(2024) 023506 [2307.02549]

    Pith/arXiv arXiv 2024

  9. [9]

    Jenkins, I.G

    A.C. Jenkins, I.G. Moss, T.P. Billam, Z. Hadzibabic, H.V. Peiris and A. Pontzen, Generalized cold-atom simulators for vacuum decay,Phys. Rev. A110(2024) L031301 [2311.02156]. [12]QUEST-DMCcollaboration,A-B transition in superfluid 3He and cosmological phase transitions, in2023 International Symposium on Quantum Fluids and Solids, 1, 2024 [2401.07878]

    arXiv 2024

  10. [10]

    Darbha, M

    S. Darbha, M. Kornjača, F. Liu, J. Balewski, M.R. Hirsbrunner, P.L.S. Lopes et al.,False vacuum decay and nucleation dynamics in neutral atom systems,Phys. Rev. B110(2024) 155103 [2404.12360]

    Pith/arXiv arXiv 2024

  11. [11]

    Zhu et al.,Probing false vacuum decay on a cold-atom gauge-theory quantum simulator,2411.12565

    Z.-H. Zhu et al.,Probing false vacuum decay on a cold-atom gauge-theory quantum simulator,2411.12565

    Pith/arXiv arXiv

  12. [12]

    Cominotti, C

    R. Cominotti, C. Baroni, C. Rogora, D. Andreoni, G. Guarda, G. Lamporesi et al., Observation of Temperature Effects on False Vacuum Decay in Atomic Quantum Gases, Phys. Rev. Lett.135(2025) 183401 [2504.03528]

    arXiv 2025

  13. [13]

    Vodeb, J.-Y

    J. Vodeb, J.-Y. Desaules, A. Hallam, A. Rava, G. Humar, D. Willsch et al.,Stirring the false vacuum via interacting quantized bubbles on a 5,564-qubit quantum annealer,Nature Phys.21(2025) 386 [2406.14718]

    Pith/arXiv arXiv 2025

  14. [14]

    Osterholz, F

    P. Osterholz, F. Bensch, S. Tang, S.B. Sheela, B. Sbierski, I. Lesanovsky et al.,Collective cluster nucleation dynamics in quantum magnets,2512.04656

    arXiv

  15. [15]

    Luo et al.,Quantum simulation of bubble nucleation across a quantum phase transition, 2505.09607

    D. Luo et al.,Quantum simulation of bubble nucleation across a quantum phase transition, 2505.09607

    Pith/arXiv arXiv

  16. [16]

    Witten,Cosmic Separation of Phases,Phys

    E. Witten,Cosmic Separation of Phases,Phys. Rev. D30(1984) 272. – 22 –

    1984

  17. [17]

    Hogan,Gravitational radiation from cosmological phase transitions,Mon

    C.J. Hogan,Gravitational radiation from cosmological phase transitions,Mon. Not. Roy. Astron. Soc.218(1986) 629

    1986

  18. [18]

    Caprini and D.G

    C. Caprini and D.G. Figueroa,Cosmological Backgrounds of Gravitational Waves,Class. Quant. Grav.35(2018) 163001 [1801.04268]

    Pith/arXiv arXiv 2018

  19. [19]

    Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update,JCAP03(2020) 024 [1910.13125]

    C. Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update,JCAP03(2020) 024 [1910.13125]

    Pith/arXiv arXiv 2020

  20. [20]

    Hindmarsh, M

    M.B. Hindmarsh, M. Lüben, J. Lumma and M. Pauly,Phase transitions in the early universe,SciPost Phys. Lect. Notes(2021) 24 [2008.09136]

    Pith/arXiv arXiv 2021

  21. [21]

    Athron, C

    P. Athron, C. Balázs, A. Fowlie, L. Morris and L. Wu,Cosmological phase transitions: From perturbative particle physics to gravitational waves,Prog. Part. Nucl. Phys.135 (2024) 104094 [2305.02357]

    Pith/arXiv arXiv 2024

  22. [22]

    Y. Aoki, G. Endrodi, Z. Fodor, S.D. Katz and K.K. Szabo,The Order of the quantum chromodynamics transition predicted by the standard model of particle physics,Nature443 (2006) 675 [hep-lat/0611014]

    Pith/arXiv arXiv 2006

  23. [23]

    Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabo,The QCD transition temperature: Results with physical masses in the continuum limit,Phys. Lett. B643(2006) 46 [hep-lat/0609068]

    Pith/arXiv arXiv 2006

  24. [24]

    Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S.D. Katz, S. Krieg et al.,The QCD transition temperature: results with physical masses in the continuum limit II.,JHEP06(2009) 088 [0903.4155]. [28]Wuppertal-Budapestcollaboration,Is there still anyT c mystery in lattice QCD? Results with physical masses in the continuum limit III,JHEP09(2010) 073 [1005.3508]

    Pith/arXiv arXiv 2009

  25. [25]

    Bhattacharya et al.,QCD Phase Transition with Chiral Quarks and Physical Quark Masses,Phys

    T. Bhattacharya et al.,QCD Phase Transition with Chiral Quarks and Physical Quark Masses,Phys. Rev. Lett.113(2014) 082001 [1402.5175]

    Pith/arXiv arXiv 2014

  26. [26]

    Bazavov et al.,The chiral and deconfinement aspects of the QCD transition,Phys

    A. Bazavov et al.,The chiral and deconfinement aspects of the QCD transition,Phys. Rev. D85(2012) 054503 [1111.1710]

    Pith/arXiv arXiv 2012

  27. [27]

    Kajantie, M

    K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov,Is there a hot electroweak phase transition atmH ≳m W?,Phys. Rev. Lett.77(1996) 2887 [hep-ph/9605288]

    Pith/arXiv arXiv 1996

  28. [28]

    Kajantie, M

    K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov,The Electroweak phase transition: A Nonperturbative analysis,Nucl. Phys. B466(1996) 189 [hep-lat/9510020]

    Pith/arXiv arXiv 1996

  29. [29]

    Laine and K

    M. Laine and K. Rummukainen,A Strong electroweak phase transition up to m(H) is about 105-GeV,Phys. Rev. Lett.80(1998) 5259 [hep-ph/9804255]

    Pith/arXiv arXiv 1998

  30. [30]

    Nakai, M

    Y. Nakai, M. Suzuki, F. Takahashi and M. Yamada,Gravitational Waves and Dark Radiation from Dark Phase Transition: Connecting NANOGrav Pulsar Timing Data and Hubble Tension,Phys. Lett. B816(2021) 136238 [2009.09754]

    Pith/arXiv arXiv 2021

  31. [31]

    Ratzinger and P

    W. Ratzinger and P. Schwaller,Whispers from the dark side: Confronting light new physics with NANOGrav data,SciPost Phys.10(2021) 047 [2009.11875]. [36]NANOGra vcollaboration,Searching for Gravitational Waves from Cosmological Phase Transitions with the NANOGrav 12.5-Year Dataset,Phys. Rev. Lett.127(2021) 251302 [2104.13930]. – 23 –

    Pith/arXiv arXiv 2021

  32. [32]

    Bringmann, P.F

    T. Bringmann, P.F. Depta, T. Konstandin, K. Schmidt-Hoberg and C. Tasillo,Does NANOGrav observe a dark sector phase transition?,JCAP11(2023) 053 [2306.09411]

    Pith/arXiv arXiv 2023

  33. [33]

    Madge, E

    E. Madge, E. Morgante, C. Puchades-Ibáñez, N. Ramberg, W. Ratzinger, S. Schenk et al., Primordial gravitational waves in the nano-Hertz regime and PTA data — towards solving the GW inverse problem,JHEP10(2023) 171 [2306.14856]

    Pith/arXiv arXiv 2023

  34. [34]

    Gonçalves, D

    J. Gonçalves, D. Marfatia, A.P. Morais and R. Pasechnik,Supercooled phase transitions in conformal dark sectors explain NANOGrav data,Phys. Lett. B869(2025) 139829 [2501.11619]

    Pith/arXiv arXiv 2025

  35. [35]

    Garcia and J

    E.R. Garcia and J. Hofmann,Instanton theory and fluctuation corrections to the thermal nucleation rate of a ferromagnetic superfluid,2512.20734

    arXiv

  36. [36]

    Gibbs,On the equilibrium of heterogeneous substances,Transactions of the Connecticut Academy of Arts and Sciences3(1874) 343

    J.W. Gibbs,On the equilibrium of heterogeneous substances,Transactions of the Connecticut Academy of Arts and Sciences3(1874) 343

  37. [37]

    Becker and W

    R. Becker and W. Döring,Kinetische behandlung der keimbildung in übersättigten dämpfen, Annalen der physik416(1935) 719

    1935

  38. [38]

    Wigner,The transition state method,Transactions of the Faraday Society34(1938) 29

    E. Wigner,The transition state method,Transactions of the Faraday Society34(1938) 29

    1938

  39. [39]

    Zeldovich,On the Theory of New Phase Formation

    Y.B. Zeldovich,On the Theory of New Phase Formation. Cavitation, inSelected Works of Yakov Borisovich Zeldovich, Volume I: Chemical Physics and Hydrodynamics, J.P. Ostriker, G.I. Barenblatt, R.A. Sunyaev, A. Granik and E. Jackson, eds., vol. 1, (Princeton), pp. 120–137, Princeton University Press (1992), DOI

    1992

  40. [40]

    Cahn and J.E

    J.W. Cahn and J.E. Hilliard,Free energy of a nonuniform system. iii. nucleation in a two-component incompressible fluid,The Journal of chemical physics31(1959) 688

    1959

  41. [41]

    Langer,Theory of the condensation point,Annals Phys.41(1967) 108

    J.S. Langer,Theory of the condensation point,Annals Phys.41(1967) 108

    1967

  42. [42]

    Langer,Statistical theory of the decay of metastable states,Annals Phys.54(1969) 258

    J.S. Langer,Statistical theory of the decay of metastable states,Annals Phys.54(1969) 258

    1969

  43. [43]

    Langer,Metastable states,Physica73(1974) 61

    J.S. Langer,Metastable states,Physica73(1974) 61

    1974

  44. [44]

    Coleman,The Fate of the False Vacuum

    S.R. Coleman,The Fate of the False Vacuum. 1. Semiclassical Theory,Phys. Rev.D15 (1977) 2929

    1977

  45. [45]

    Callan, Jr

    C.G. Callan, Jr. and S.R. Coleman,The Fate of the False Vacuum. 2. First Quantum Corrections,Phys. Rev. D16(1977) 1762

    1977

  46. [46]

    Affleck,Quantum Statistical Metastability,Phys

    I. Affleck,Quantum Statistical Metastability,Phys. Rev. Lett.46(1981) 388

    1981

  47. [47]

    Linde,Decay of the False Vacuum at Finite Temperature,Nucl

    A.D. Linde,Decay of the False Vacuum at Finite Temperature,Nucl. Phys. B216(1983) 421

    1983

  48. [48]

    Andreassen, D

    A. Andreassen, D. Farhi, W. Frost and M.D. Schwartz,Direct Approach to Quantum Tunneling,Phys. Rev. Lett.117(2016) 231601 [1602.01102]

    Pith/arXiv arXiv 2016

  49. [49]

    W.-Y. Ai, B. Garbrecht and C. Tamarit,Functional methods for false vacuum decay in real time,JHEP12(2019) 095 [1905.04236]

    Pith/arXiv arXiv 2019

  50. [50]

    Garbrecht and N

    B. Garbrecht and N. Wagner,Instantons meet resonances: Unifying two seemingly distinct approaches to quantum tunneling,2512.04907

    arXiv

  51. [51]

    Gould and J

    O. Gould and J. Hirvonen,Effective field theory approach to thermal bubble nucleation, Phys. Rev. D104(2021) 096015 [2108.04377]. – 24 –

    Pith/arXiv arXiv 2021

  52. [52]

    Hirvonen, J

    J. Hirvonen, J. Löfgren, M.J. Ramsey-Musolf, P. Schicho and T.V.I. Tenkanen,Computing the gauge-invariant bubble nucleation rate in finite temperature effective field theory,JHEP 07(2022) 135 [2112.08912]

    Pith/arXiv arXiv 2022

  53. [53]

    Löfgren, M.J

    J. Löfgren, M.J. Ramsey-Musolf, P. Schicho and T.V.I. Tenkanen,Nucleation at Finite Temperature: A Gauge-Invariant Perturbative Framework,Phys. Rev. Lett.130(2023) 251801 [2112.05472]

    Pith/arXiv arXiv 2023

  54. [54]

    Ekstedt,Bubble nucleation to all orders,JHEP08(2022) 115 [2201.07331]

    A. Ekstedt,Bubble nucleation to all orders,JHEP08(2022) 115 [2201.07331]

    Pith/arXiv arXiv 2022

  55. [55]

    Steingasser, M

    T. Steingasser, M. König and D.I. Kaiser,Finite-temperature instantons from first principles,Phys. Rev. D110(2024) L111902 [2310.19865]

    Pith/arXiv arXiv 2024

  56. [56]

    Steingasser and D.I

    T. Steingasser and D.I. Kaiser,Toward quantum tunneling from excited states: Recovering imaginary-time instantons from a real-time analysis,Phys. Rev. D111(2025) 096009 [2402.00099]

    Pith/arXiv arXiv 2025

  57. [57]

    Hirvonen,Real-time nucleation and off-equilibrium effects in high-temperature quantum field theories,Phys

    J. Hirvonen,Real-time nucleation and off-equilibrium effects in high-temperature quantum field theories,Phys. Rev. D111(2025) 116020 [2403.07987]

    Pith/arXiv arXiv 2025

  58. [58]

    Pîrvu, A

    D. Pîrvu, A. Shkerin and S. Sibiryakov,Thermal false vacuum decay in (1+1)-dimensions: Evidence for non-equilibrium dynamics,""(2024) [2408.06411]

    Pith/arXiv arXiv 2024

  59. [59]

    Pîrvu, A

    D. Pîrvu, A. Shkerin and S. Sibiryakov,Thermal False Vacuum Decay Is Not What It Seems,""(2024) [2407.06263]

    arXiv 2024

  60. [60]

    Hirvonen and O

    J. Hirvonen and O. Gould,Langer’s Nucleation Rate Reproduced on the Lattice,Phys. Rev. Lett.136(2026) 081601 [2505.22732]

    Pith/arXiv arXiv 2026

  61. [61]

    Shoji and M

    Y. Shoji and M. Yamaguchi,Symmetry of Bounce Solutions at Finite Temperature, 2511.05950

    arXiv

  62. [62]

    Barni and T

    G. Barni and T. Steingasser,On quantum tunnelling in the presence of Noether charges, 2604.08660

    Pith/arXiv arXiv

  63. [63]

    Barni and J.R

    G. Barni and J.R. Espinosa,Globally Charged Vacuum Decay,2606.21653

    Pith/arXiv arXiv

  64. [64]

    Farakos, K

    K. Farakos, K. Kajantie, K. Rummukainen and M.E. Shaposhnikov,3-D physics and the electroweak phase transition: Perturbation theory,Nucl. Phys. B425(1994) 67 [hep-ph/9404201]

    Pith/arXiv arXiv 1994

  65. [65]

    Kajantie, M

    K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov,Generic rules for high temperature dimensional reduction and their application to the standard model,Nucl. Phys. B458(1996) 90 [hep-ph/9508379]

    Pith/arXiv arXiv 1996

  66. [66]

    Braaten and A

    E. Braaten and A. Nieto,Effective field theory approach to high temperature thermodynamics,Phys. Rev.D51(1995) 6990 [hep-ph/9501375]

    Pith/arXiv arXiv 1995

  67. [67]

    Braaten and A

    E. Braaten and A. Nieto,Free energy of QCD at high temperature,Phys. Rev. D53(1996) 3421 [hep-ph/9510408]

    Pith/arXiv arXiv 1996

  68. [68]

    Hirvonen,Intuitive method for constructing effective field theories,2205.02687

    J. Hirvonen,Intuitive method for constructing effective field theories,2205.02687

    Pith/arXiv arXiv

  69. [69]

    Calzetta, A

    E. Calzetta, A. Roura and E. Verdaguer,Vacuum decay in quantum field theory,Phys. Rev. D64(2001) 105008 [hep-ph/0106091]

    Pith/arXiv arXiv 2001

  70. [70]

    Braden, M.C

    J. Braden, M.C. Johnson, H.V. Peiris, A. Pontzen and S. Weinfurtner,New Semiclassical Picture of Vacuum Decay,Phys. Rev. Lett.123(2019) 031601 [1806.06069]. – 25 –

    Pith/arXiv arXiv 2019

  71. [71]

    Hertzberg and M

    M.P. Hertzberg and M. Yamada,Vacuum Decay in Real Time and Imaginary Time Formalisms,Phys. Rev. D100(2019) 016011 [1904.08565]

    Pith/arXiv arXiv 2019

  72. [72]

    H. Wang, R. Qin and L. Bian,False Vacuum Decay across the Quantum-to-Thermal Crossover: A Comparison of Real-Time Observables,2506.18334

    Pith/arXiv arXiv

  73. [73]

    Tranberg and G

    A. Tranberg and G. Ungersbäck,Bubble nucleation and quantum initial conditions in classical statistical simulations,JHEP09(2022) 206 [2206.08691]

    Pith/arXiv arXiv 2022

  74. [74]

    Moore and K

    G.D. Moore and K. Rummukainen,Electroweak bubble nucleation, nonperturbatively,Phys. Rev. D63(2001) 045002 [hep-ph/0009132]

    Pith/arXiv arXiv 2001

  75. [75]

    Moore, K

    G.D. Moore, K. Rummukainen and A. Tranberg,Nonperturbative computation of the bubble nucleation rate in the cubic anisotropy model,JHEP04(2001) 017 [hep-lat/0103036]

    Pith/arXiv arXiv 2001

  76. [76]

    Gould, S

    O. Gould, S. Güyer and K. Rummukainen,First-order electroweak phase transitions: A nonperturbative update,Phys. Rev. D106(2022) 114507 [2205.07238]

    Pith/arXiv arXiv 2022

  77. [77]

    Batini, A

    L. Batini, A. Chatrchyan and J. Berges,Real-time dynamics of false vacuum decay,Phys. Rev. D109(2024) 023502 [2310.04206]

    Pith/arXiv arXiv 2024

  78. [78]

    Gould, A

    O. Gould, A. Kormu and D.J. Weir,Nonperturbative test of nucleation calculations for strong phase transitions,Phys. Rev. D111(2025) L051901 [2404.01876]

    arXiv 2025

  79. [79]

    Blaizot and E

    J.-P. Blaizot and E. Iancu,The Quark gluon plasma: Collective dynamics and hard thermal loops,Phys. Rept.359(2002) 355 [hep-ph/0101103]

    Pith/arXiv arXiv 2002

  80. [80]

    Nair,Hard thermal loops, gauged WZNW action and the energy of hot quark - gluon plasma,Phys

    V.P. Nair,Hard thermal loops, gauged WZNW action and the energy of hot quark - gluon plasma,Phys. Rev. D48(1993) R3432 [hep-ph/9307326]

    Pith/arXiv arXiv 1993

Showing first 80 references.