Abstract
We consider the Hartree–Fock approximation of Quantum Electrodynamics, with the exchange term neglected. We prove that the probability of static electron–positron pair creation for the Dirac vacuum polarized by an external field of strength Z behaves as \({1-\exp(-\kappa Z^{2/3})}\) for Z large enough. Our method involves two steps. First, we estimate the vacuum expectation of general quasi-free states in terms of their total number of particles, which can be of general interest. Then, we study the asymptotics of the Hartree–Fock energy when \({Z \to+\infty}\) which gives the expected bounds.
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Anderson C.: The positive electron. Phys. Rev. 43, 491 (1933)
Araki H., Wyss W.: Representations of canonical anticommutation relations. Helv. Phys. Acta 37, 136 (1964)
Bach V., Barbaroux J.M., Helffer B., Siedentop H.: On the stability of the relativistic electron–positron field. Commun. Math. Phys. 201, 445–460 (1999)
Bach V., Lieb E.H., Solovej J.P.: Generalized Hartree–Fock theory and the Hubbard model. J. Statist. Phys. 76, 3–89 (1994)
Bulanov S.S., Mur V.D., Narozhny N.B., Nees J., Popov V.S.: Multiple colliding electromagnetic pulses: a way to lower the threshold of e + e − pair production from vacuum. Phys. Rev. Lett. 104, 220404 (2010)
Chaix P., Iracane D.: From quantum electrodynamics to mean field theory: I. The Bogoliubov–Dirac–Fock formalism. J. Phys. B 22, 3791–3814 (1989)
Chaix P., Iracane D., Lions P.-L.: From quantum electrodynamics to mean field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation. J. Phys. B 22, 3815–3828 (1989)
Dirac P.: A theory of electrons and protons. Proc. R. Soc. A 126, 360–365 (1930)
Dunne G.V.: New strong-field QED effects at extreme light infrastructure. EPJ D 55, 327–340 (2009). doi:10.1140/epjd/e2009-00022-0
Gravejat P., Lewin M., Séré É.: Ground state and charge renormalization in a nonlinear model of relativistic atoms. Commun. Math. Phys. 286, 179–215 (2009)
Guichardet A.: Produits tensoriels infinis et representations des relations d’anticommutation. Ann. Sci. Ecole Norm. Sup.(3) 83, 1–52 (1966)
Hainzl C.: On the vacuum polarization density caused by an external field. Ann. Henri Poincaré 5, 1137–1157 (2004)
Hainzl C., Lewin M., Séré É.: Existence of a stable polarized vacuum in the Bogoliubov–Dirac–Fock approximation. Commun. Math. Phys. 257, 515–562 (2005)
Hainzl C., Lewin M., Séré É.: Self-consistent solution for the polarized vacuum in a no-photon QED model. J. Phys. A 38, 4483–4499 (2005)
Hainzl C., Lewin M., Séré É.: Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics. Arch. Rational Mech. Anal. 192, 453–499 (2009)
Hainzl C., Lewin M., Séré É., Solovej J.P.: A minimization method for relativistic electrons in a mean-field approximation of quantum electrodynamics. Phys. Rev. A 76, 052104 (2007)
Hainzl C., Lewin M., Solovej J.P.: The mean-field approximation in quantum electrodynamics: the no-photon case. Comm. Pure Appl. Math. 60, 546–596 (2007)
Heisenberg W., Euler H.: Folgerungen aus der Diracschen Theorie des Positrons. Zeitschrift fur Physik 98, 714–732 (1936)
Jaksic V., Ogata Y., Pautrat Y., Pillet C.A.: Entropic fluctuations in quantum statistical mechanics: an introduction. Lecture Notes at Université de Cergy- Pontoise, Cergy-Pontoise (2011)
Klaus M., Scharf G.: Vacuum polarization in Fock space. Helv. Phys. Acta 50, 803–814 (1977)
Lieb E.H., Simon B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977)
Nenciu G.: On the adiabatic limit for dirac particles in external fields. Comm. Math. Phys. 76, 117–128 (1980)
Nenciu G.: Existence of the spontaneous pair creation in the external field approximation of QED Commun. Math. Phys. 109, 303–312 (1987)
Pickl, P.: Existence of spontaneous pair creation. PhD thesis, LMU München (2005)
Pickl P., Dürr D.: On adiabatic pair creation. Commun. Math. Phys. 282, 161–198 (2008)
Sauter F.: Über das verhalten eines elektrons im homogenen elektrischen feld nach der relativistischen theorie diracs. Zeitschrift für Physik A Hadrons and Nuclei 69, 742–764 (1931)
Scharf G., Seipp H.P.: Charged vacuum, spontaneous positron production and all that. Phys. Lett. B 108, 196–198 (1982)
Schwinger J.: On gauge invariance and vacuum polarization. Phys. Rev.(2) 82, 664–679 (1951)
Solovej, J.P.: Many body quantum mechanics. Lectures notes at LMU München (2007). http://www.mathematik.uni-muenchen.de/~lerdos/WS08/QM/solovejnotes.pdf
Tajima T.: Prospect for extreme field science. EPJ D 55, 519–529 (2009). doi:10.1140/epjd/e2009-00107-8
Thaller B.: The Dirac equation Texts and Monographs in Physics. Springer, Berlin (1992)
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Communicated by Jan Derezinski.
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Sabin, J. Static Electron–Positron Pair Creation in Strong Fields for a Non-linear Dirac Model. Ann. Henri Poincaré 14, 1349–1382 (2013). https://doi.org/10.1007/s00023-012-0221-9
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DOI: https://doi.org/10.1007/s00023-012-0221-9