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  • Letter
  • Open Access

Relativistic quantum kinetic theory: Higher order contributions in assisted Schwinger pair production

James P. Edwards1, Naser Ahmadiniaz2, Sebastian M. Schmidt2,3, and Christian Kohlfürst2,*

  • *Contact author: c.kohlfuerst@hzdr.de
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Phys. Rev. D 112, L031901 – Published 6 August, 2025

DOI: https://doi.org/10.1103/mgst-vn3c

Abstract

Quantum kinetic theory is an important tool for studying nonequilibrium, nonperturbative and nonlinear interactions within an open quantum system, and as such is able to provide an unprecedented view on particle production in the relativistic, ultrahigh intensity regime of quantum electrodynamics. By reorganizing the relativistic quantum transport equations for Abelian plasmas and integrating them with a perturbative expansion, we significantly expand the scope for kinetic theories to further elucidate the peculiarities of particle production at a spectral level.

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References (115)

  1. P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. A. 117, 610 (1928).
  2. S. Laporta, High-precision calculation of the 4-loop contribution to the electron g-2 in QED, Phys. Lett. B 772, 232 (2017).
  3. S. I. Tomonaga, On a relativistically invariant formulation of the quantum theory of wave fields, Prog. Theor. Exp. Phys. 1, 27 (1946).
  4. J. Schwinger, Quantum electrodynamics. I. A covariant formulation, Phys. Rev. 74, 1439 (1948).
  5. J. Schwinger, Quantum electrodynamics. II. Vacuum polarization and self-energy, Phys. Rev. 75, 651 (1949).
  6. J. Schwinger, Quantum electrodynamics. III. The electromagnetic properties of the electron—radiative corrections to scattering, Phys. Rev. 76, 790 (1949).
  7. F. J. Dyson, The S matrix in quantum electrodynamics, Phys. Rev. 75, 1736 (1949).
  8. F. J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev. 75, 486 (1949).
  9. R. P. Feynman, The theory of positrons, Phys. Rev. 76, 749 (1949).
  10. R. P. Feynman, Mathematical formulation of the quantum theory of electromagnetic interaction, Phys. Rev. 80, 440 (1950).
  11. J. M. Raimond, M. Brune, and S. Haroche, Manipulating quantum entanglement with atoms and photons in a cavity, Rev. Mod. Phys. 73, 565 (2001).
  12. A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  13. A. Di Piazza, C. Muller, K. Z. Hatsagortsyan, and C. H. Keitel, Extremely high-intensity laser interactions with fundamental quantum systems, Rev. Mod. Phys. 84, 1177 (2012).
  14. A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S. Bulanov, Charged particle motion and radiation in strong electromagnetic fields, Rev. Mod. Phys. 94, 045001 (2022).
  15. M. Marklund and P. K. Shukla, Nonlinear collective effects in photon-photon and photon-plasma interactions, Rev. Mod. Phys. 78, 591 (2006).
  16. I. Levine et al. (TOPAZ Collaboration), Measurement of the electromagnetic coupling at large momentum transfer, Phys. Rev. Lett. 78, 424 (1997).
  17. N. Ahmadiniaz, C. Bähtz, A. Benediktovitch, C. Bömer, L. Bocklage, T. E. Cowan, J. Edwards, S. Evans, S. F. Viñas, H. Gies et al., Letter of intent: Towards a vacuum birefringence experiment at the Helmholtz international beamline for extreme fields, High Power Laser Sci. Eng. 13, e7 (2025).
  18. I. C. E. Turcu, F. Negoita, D. A. Jaroszynski, P. Mckenna, S. Balascuta, D. Ursescu, I. Dancus, M. O. Cernaianu, M. V. Tataru, P. Ghenuche et al., High field physics and QED experiments at ELI-NP, Rom. Rep. Phys. 68, S145 (2016), https://inspirehep.net/files/eb64dc541d20ce529e2ed2d759a9b492.
  19. H. Abramowicz, U. Acosta, M. Altarelli, R. Aßmann, Z. Bai, T. Behnke, Y. Benhammou, T. Blackburn, S. Boogert, and O. Borysov et al., Conceptual design report for the LUXE experiment, Eur. Phys. J. Special Topics 230, 2445 (2021).
  20. V. Yakimenko, L. Alsberg, E. Bong, G. Bouchard, C. Clarke, C. Emma, S. Green, C. Hast, M. J. Hogan, J. Seabury, N. Lipkowitz, B. O’Shea, D. Storey, G. White, G. Yocky et al., FACET-II facility for advanced accelerator experimental tests, Phys. Rev. Accel. Beams 22, 101301 (2019).
  21. F. Gelis and N. Tanji, Schwinger mechanism revisited, Prog. Part. Nucl. Phys. 87, 1 (2016).
  22. A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, Advances in QED with intense background fields, Phys. Rep. 1010, 1 (2023).
  23. R. Ruffini, G. Vereshchagin, and S.-S. Xue, Electron–positron pairs in physics and astrophysics: From heavy nuclei to black holes, Phys. Rep. 487, 1 (2010).
  24. F. Dreisow, S. Longhi, S. Nolte, A. Tünnermann, and A. Szameit, Vacuum instability and pair production in an optical setting, Phys. Rev. Lett. 109, 110401 (2012).
  25. A. I. Berdyugin, N. Xin, H. Gao, S. Slizovskiy, Z. Dong, S. Bhattacharjee, P. Kumaravadivel, S. Xu, L. A. Ponomarenko, M. Holwill et al., Out-of-equilibrium criticalities in graphene superlattices, Science 375, 430 (2022).
  26. A. Schmitt, P. Vallet, D. Mele, M. Rosticher, T. Taniguchi, K. Watanabe, E. Bocquillon, G. Fève, J. M. Berroir, C. Voisin et al., Mesoscopic Klein-Schwinger effect in graphene, Nat. Phys. 19, 830 (2023).
  27. J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  28. F. Sauter, Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen theorie Diracs, Z. Phys. 69, 742 (1931).
  29. P. A. M. Dirac, A theory of electrons and protons, Proc. R. Soc. A. 126, 360 (1928).
  30. W. Heisenberg and H. Euler, Consequences of Dirac’s theory of positrons, Z. Phys. 98, 714 (1936); translated by W. Korolevski and H. Kleinert arXiv:physics/0605038.
  31. V. Weisskopf, The electrodynamics of the vacuum based on the quantum theory of the electron, Kong. Dan. Vid. Sel. Mat. Fys. Med. 14, 1 (1936); Early Quantum Electrodynamics: A Source Book, edited by A. I. Miller (Cambridge University Press, University College London, 1994).
  32. O. Gould and A. Rajantie, Thermal Schwinger pair production at arbitrary coupling, Phys. Rev. D 96, 076002 (2017).
  33. G. V. Dunne and C. Schubert, Worldline instantons and pair production in inhomogenous fields, Phys. Rev. D 72, 105004 (2005).
  34. G. V. Dunne, Q. Wang, H. Gies, and C. Schubert, Worldline instantons and the fluctuation prefactor, Phys. Rev. D 73, 065028 (2006).
  35. H. Gies and K. Klingmüller, Pair production in inhomogeneous fields, Phys. Rev. D 72, 065001 (2005).
  36. C. Kohlfürst, N. Ahmadiniaz, J. Oertel, and R. Schützhold, Sauter-Schwinger effect for colliding laser pulses, Phys. Rev. Lett. 129, 241801 (2022).
  37. H. Taya, T. Fujimori, T. Misumi, M. Nitta, and N. Sakai, Exact WKB analysis of the vacuum pair production by time-dependent electric fields, J. High Energy Phys. 03 (2021) 082.
  38. C. K. Dumlu and G. V. Dunne, Interference effects in Schwinger vacuum pair production for time-dependent laser pulses, Phys. Rev. D 83, 065028 (2011).
  39. E. Strobel and S. S. Xue, Semiclassical pair production rate for time-dependent electrical fields with more than one component: WKB-approach and world-line instantons, Nucl. Phys. B886, 1153 (2014).
  40. C. K. Dumlu and G. V. Dunne, Interference effects in Schwinger vacuum pair production for time-dependent laser pulses, Phys. Rev. D 83, 065028 (2011).
  41. C. K. Dumlu, Quantum kinetic approach and the scattering approach to vacuum pair production, Phys. Rev. D 79, 065027 (2009).
  42. S. M. Schmidt, D. Blaschke, G. Ropke, S. A. Smolyansky, A. V. Prozorkevich, and V. D. Toneev, A quantum kinetic equation for particle production in the Schwinger mechanism, Int. J. Mod. Phys. E 07, 709 (1998).
  43. Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola, Fermion pair production in a strong electric field, Phys. Rev. D 45, 4659 (1992).
  44. F. Hebenstreit, J. Berges, and D. Gelfand, Simulating fermion production in 1+1 dimensional QED, Phys. Rev. D 87, 105006 (2013).
  45. F. Hebenstreit, J. Berges, and D. Gelfand, Real-time dynamics of string breaking, Phys. Rev. Lett. 111, 201601 (2013).
  46. E. Brezin and C. Itzykson, Pair production in vacuum by an alternating field, Phys. Rev. D 2, 1191 (1970).
  47. E. S. Fradkin, D. M. Gitman, and S. M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum (Springer-Verlag, Berlin, 1991).
  48. S. P. Gavrilov and D. M. Gitman, Vacuum instability in external fields, Phys. Rev. D 53, 7162 (1996).
  49. I. A. Aleksandrov and C. Kohlfürst, Pair production in temporally and spatially oscillating fields, Phys. Rev. D 101, 096009 (2020).
  50. T. Heinzl, A. Ilderton, and M. Marklund, Finite size effects in stimulated laser pair production, Phys. Lett. B 692, 250 (2010).
  51. T. G. Blackburn and B. King, Higher fidelity simulations of nonlinear Breit–Wheeler pair creation in intense laser pulses, Eur. Phys. J. C 82, 44 (2022).
  52. A. Eckey, A. B. Voitkiv, and C. Müller, Strong-field Breit-Wheeler pair production with bremsstrahlung γ rays in the perturbative-to-nonperturbative-transition regime, Phys. Rev. A 105, 013105 (2022).
  53. M. M. Majczak, K. Krajewska, J. Z. Kamiński, and A. Bechler, Scattering matrix approach to dynamical Sauter-Schwinger process: Spin- and helicity-resolved momentum distributions, Phys. Rev. D 110, 116025 (2024).
  54. R. Schützhold, H. Gies, and G. Dunne, Dynamically assisted Schwinger mechanism, Phys. Rev. Lett. 101, 130404 (2008).
  55. G. Breit and J. A. Wheeler, Collision of two light quanta, Phys. Rev. 46, 1087 (1934).
  56. H. R. Reiss, Absorption of light by light, J. Math. Phys. (N.Y.) 3, 59 (1962).
  57. A. Ringwald, Pair production from vacuum at the focus of an X-ray free electron laser, Phys. Lett. B 510, 107 (2001).
  58. A. R. Bell and J. G. Kirk, Possibility of prolific pair production with high-power lasers, Phys. Rev. Lett. 101, 200403 (2008).
  59. G. V. Dunne, H. Gies, and R. Schützhold, Catalysis of Schwinger vacuum pair production, Phys. Rev. D 80, 111301(R) (2009).
  60. S. Villalba-Chávez and C. Müller, Signatures of the Schwinger mechanism assisted by a fast-oscillating electric field, Phys. Rev. D 100, 116018 (2019).
  61. C. Schneider and R. Schützhold, Dynamically assisted Sauter-Schwinger effect in inhomogeneous electric fields, J. High Energy Phys. 02 (2016) 164.
  62. H. Taya, Franz-Keldysh effect in strong-field QED, Phys. Rev. D 99, 056006 (2019).
  63. M. F. Linder, C. Schneider, J. Sicking, N. Szpak, and R. Schützhold, Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect, Phys. Rev. D 92, 085009 (2015).
  64. C. Schneider and R. Schützhold, Prefactor in the dynamically assisted Sauter-Schwinger effect, Phys. Rev. D 94, 085015 (2016).
  65. G. Torgrimsson, J. Oertel, and R. Schützhold, Sauter-Schwinger pair creation dynamically assisted by a plane wave, Phys. Rev. D 97, 096004 (2018).
  66. W. Franz, Einfluß eines elektrischen Feldes auf eine optische Absorptionskante, Z. Naturforsch. 13A, 484 (1958).
  67. L. V. Keldysh, Behaviour of non-metallic crystals in strong electric fields, J. Exp. Theor. Phys. (USSR) 33, 994 (1957) [Sov. Phys. JETP 6, 763 (1958)], http://www.jetp.ras.ru/cgi-bin/dn/e_006_04_0763.pdf.
  68. G. Torgrimsson, J. Oertel, and R. Schützhold, Doubly assisted Sauter-Schwinger effect, Phys. Rev. D 94, 065035 (2016).
  69. G. Torgrimsson, C. Schneider, and R. Schützhold, Dynamically assisted Sauter-Schwinger effect non-perturbative versus perturbative aspects, J. High Energy Phys. 06 (2017) 043.
  70. I. A. Aleksandrov, G. Plunien, and V. M. Shabaev, Dynamically assisted Schwinger effect beyond the spatially-uniform-field approximation, Phys. Rev. D 97, 116001 (2018).
  71. M. Orthaber, F. Hebenstreit, and R. Alkofer, Momentum spectra for dynamically assisted Schwinger pair production, Phys. Lett. B 698, 80 (2011).
  72. C. Fey and R. Schützhold, Momentum dependence in the dynamically assisted Sauter-Schwinger effect, Phys. Rev. D 85, 025004 (2012).
  73. A. Otto, H. Oppitz, and B. Kämpfer, Assisted vacuum decay by time dependent electric fields, Eur. Phys. J. A 54, 23 (2018).
  74. I. A. Aleksandrov, D. G. Sevostyanov, and V. M. Shabaev, Schwinger particle production: Rapid switch off of the external field versus dynamical assistance, Phys. Rev. D 111, 016010 (2025).
  75. R. Cabrera, A. G. Campos, D. I. Bondar, and H. A. Rabitz, Dirac open-quantum-system dynamics: Formulations and simulations, Phys. Rev. A 94, 052111 (2016).
  76. D. Vasak, M. Gyulassy, and H. T. Elze, Quantum transport theory for Abelian plasmas, Ann. Phys. (N.Y.) 173, 462 (1987).
  77. S. Lin, Quantum kinetic theory for quantum electrodynamics, Phys. Rev. D 105, 076017 (2022).
  78. P. Zhuang and U. Heinz, Relativistic quantum transport theory for electrodynamics, Ann. Phys. (N.Y.) 245, 311 (1996).
  79. P. Carruthers and F. Zachariasen, Relativistic quantum transport theory approach to multiparticle production, Phys. Rev. D 13, 950 (1976).
  80. I. Bialynicki-Birula, P. Górnicki, and J. Rafelski, Phase-space structure of the Dirac vacuum, Phys. Rev. D 44, 1825 (1991).
  81. Y. Hidaka, S. Pu, Q. Wang, and D. L. Yang, Foundations and applications of quantum kinetic theory, Prog. Part. Nucl. Phys. 127, 103989 (2022).
  82. P. Zhuang and U. Heinz, Equal-time hierarchies for quantum transport theory, Phys. Rev. D 57, 6525 (1998).
  83. S. Ochs and U. Heinz, Wigner functions in covariant and single-time formulations, Ann. Phys. (N.Y.) 266, 351 (1998).
  84. A. J. Macleod, J. P. Edwards, T. Heinzl, B. King, and S. V. Bulanov, Strong-field vacuum polarisation with high energy lasers, New J. Phys. 25, 093002 (2023).
  85. W. H. Furry, On bound states and scattering in positron theory, Phys. Rev. 81, 115 (1951).
  86. See Supplemental Material at http://link.aps.org/supplemental/10.1103/mgst-vn3c for technical details and background information; Refs. [87–90].
  87. X. L. Sheng, R. H. Fang, Q. Wang, and D. H. Rischke, Wigner function and pair production in parallel electric and magnetic fields, Phys. Rev. D 99, 056004 (2019).
  88. E. Akkermans and G. V. Dunne, Ramsey Fringes and time-domain multiple-slit interference from vacuum, Phys. Rev. Lett. 108, 030401 (2012).
  89. C. Kohlfurst, M. Mitter, G. von Winckel, F. Hebenstreit, and R. Alkofer, Optimizing the pulse shape for Schwinger pair production, Phys. Rev. D 88, 045028 (2013).
  90. J. Unger, S. Dong, R. Flores, Q. Su, and R. Grobe, Infinite-dimensional optimization applied to pair creation from the vacuum, Phys. Rev. A 99, 022128 (2019).
  91. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932).
  92. E. Calzetta and B. L. Hu, Nonequilibrium quantum fields: Closed-time-path effective action, Wigner function, and Boltzmann equation, Phys. Rev. D 37, 2878 (1988).
  93. F. Cooper, S. Habib, Y. Kluger, E. Mottola, J. P. Paz, and P. R. Anderson, Nonequilibrium quantum fields in the large-N expansion, Phys. Rev. D 50, 2848 (1994).
  94. I. A. Aleksandrov, A. Kudlis, and A. I. Klochai, Kinetic theory of vacuum pair production in uniform electric fields revisited, Phys. Rev. Res. 6, 043009 (2024).
  95. C. Kohlfürst, Pair production in circularly polarized waves, Phys. Rev. D 110, L111903 (2024).
  96. D. Seipt and A. G. R. Thomas, Kinetic theory for spin-polarized relativistic plasmas, Phys. Plasmas 30, 093102 (2023).
  97. G. Brodin and J. Zamanian, Quantum kinetic theory of plasmas, Rev. Mod. Plasma Phys. 6, 4 (2022).
  98. D. Kremp, Th. Bornath, M. Bonitz, and M. Schlanges, Quantum kinetic theory of plasmas in strong laser fields, Phys. Rev. E 60, 4725 (1999).
  99. G. Brodin, H. Al-Naseri, J. Zamanian, G. Torgrimsson, and B. Eliasson, Plasma dynamics at the Schwinger limit and beyond, Phys. Rev. E 107, 035204 (2023).
  100. F. Cooper and E. Mottola, Quantum back reaction in scalar QED as an initial-value problem, Phys. Rev. D 40, 456 (1989).
  101. J. Rau, Pair production in the quantum Boltzmann equation, Phys. Rev. D 50, 6911 (1994).
  102. J. Rau and B. Müller, From reversible quantum microdynamics to irreversible quantum transport, Phys. Rep. 272, 1 (1996).
  103. C. Best, P. Gornicki, and W. Greiner, The phase-space structure of the Klein-Gordon field, Ann. Phys. (N.Y.) 225, 169 (1993).
  104. R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, Pair creation and an X-ray free electron laser, Phys. Rev. Lett. 87, 193902 (2001).
  105. C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, Quantum effects with an X-ray free-electron laser, Phys. Rev. Lett. 89, 153901 (2002).
  106. F. Hebenstreit, R. Alkofer, G. V. Dunne, and H. Gies, Momentum signatures for Schwinger pair production in short laser pulses with a subcycle structure, Phys. Rev. Lett. 102, 150404 (2009).
  107. N. Ahmadiniaz, A. M. Fedotov, E. G. Gelfer, S. P. Kim, and C. Schubert, Generalized Gelfand-Dikii equation and solitonic electric fields for fermionic Schwinger pair production, Phys. Rev. D 108, 036019 (2023).
  108. Y. Kluger, E. Mottola, and J. M. Eisenberg, The quantum Vlasov equation and its Markov limit, Phys. Rev. D 58, 125015 (1998).
  109. S. Schmidt, D. Blaschke, G. Röpke, A. V. Prozorkevich, S. A. Smolyansky, and V. D. Toneev, Non-Markovian effects in strong-field pair creation, Phys. Rev. D 59, 094005 (1999).
  110. A. Ilderton, Physics of adiabatic particle number in the Schwinger effect, Phys. Rev. D 105, 016021 (2022).
  111. M. Diez, R. Alkofer, and C. Kohlfürst, Identifying time scales in particle production from fields, Phys. Lett. B 844, 138063 (2023).
  112. L. V. Keldysh, Ionization in the field of a strong electromagnetic wave, Sov. Phys. JETP 20, 1307 (1965) [J. Exp. Theor. Phys. (U.S.S.R.) 47, 1945 (1964)], https://inspirehep.net/files/6697e05d52e411291acc8238a780db45.
  113. G. Torgrimsson, Perturbative methods for assisted nonperturbative pair production, Phys. Rev. D 99, 096002 (2019).
  114. P. Copinger, J. P. Edwards, A. Ilderton, and K. Rajeev, Pair creation, backreaction, and resummation in strong fields, Phys. Rev. D 111, 036009 (2025).
  115. J. P. Edwards, N. Ahmadiniaz, S. M. Schmidt, and C. Kohlfürst, version 1, 10.14278/rodare.3881 (2025).

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Phys. Rev. D 112, L031901– Published 6 August, 2025
  • Received 4 April 2025
  • Accepted 16 July 2025
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