Two Hartree-Fock models for the vacuum polarization

Gravejat, Philippe; Hainzl, Christian; Lewin, Mathieu; Séré, Éric

doi:10.5802/jedp.87

Two Hartree-Fock models for the vacuum polarization
[Deux modèles de Hartree-Fock pour la polarisation du vide]

Gravejat, Philippe ¹ ; Hainzl, Christian ² ; Lewin, Mathieu ³ ; Séré, Éric ⁴

¹ Centre de Mathématiques Laurent Schwartz (UMR 7640) École Polytechnique F-91128 Palaiseau Cedex France
² Mathematisches Institut Auf der Morgenstelle 10 D-72076 Tübingen Germany
³ Centre National de la Recherche Scientifique and Laboratoire de Mathématiques (UMR 8088) Université de Cergy-Pontoise F-95000 Cergy-Pontoise France
⁴ Centre de Recherche en Mathématiques de la Décision (UMR 7534) Université Paris-Dauphine Place du Maréchal De Lattre de Tassigny F-75775 Paris Cedex 16 France

Journées équations aux dérivées partielles (2012), article no. 4, 31 p.

Résumé
Abstract

Nous présentons des résultats récents sur la dérivation et l’analyse de deux modèles de type Hartree-Fock pour la polarisation du vide. Nous portons une attention particulière à la construction variationnelle d’un vide polarisé auto-consistent, et à la pertinence physique de notre construction non perturbative vis-à-vis de la description perturbative donnée par l’électrodynamique quantique.

We review recent results about the derivation and the analysis of two Hartree-Fock-type models for the polarization of vacuum. We pay particular attention to the variational construction of a self-consistent polarized vacuum, and to the physical agreement between our non-perturbative construction and the perturbative description provided by Quantum Electrodynamics.

EuDML

DOI : 10.5802/jedp.87

Classification : 35Q41, 49S05, 81T16, 81V10
Keywords: Vacuum polarization, Dirac sea, Hartree-Fock approximation, Bogoliubov-Dirac-Fock model, Pauli-Villars regularization, charge renormalization, quantum electrodynamics
Mot clés : Polarisation du vide, mer de Dirac, approximation de type Hartree-Fock, modèle de Bogoliubov-Dirac-Fock, régularisation de Pauli-Villars, renormalisation de la charge, électrodynamique quantique

Affiliations des auteurs :

Gravejat, Philippe ¹ ; Hainzl, Christian ² ; Lewin, Mathieu ³ ; Séré, Éric ⁴

@article{JEDP_2012____A4_0,
     author = {Gravejat, Philippe and Hainzl, Christian and Lewin, Mathieu and S\'er\'e, \'Eric},
     title = {Two {Hartree-Fock} models for the vacuum polarization},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {4},
     pages = {1--31},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2012},
     doi = {10.5802/jedp.87},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jedp.87/}
}

TY  - JOUR
AU  - Gravejat, Philippe
AU  - Hainzl, Christian
AU  - Lewin, Mathieu
AU  - Séré, Éric
TI  - Two Hartree-Fock models for the vacuum polarization
JO  - Journées équations aux dérivées partielles
PY  - 2012
SP  - 1
EP  - 31
PB  - Groupement de recherche 2434 du CNRS
UR  - http://archive.numdam.org/articles/10.5802/jedp.87/
DO  - 10.5802/jedp.87
LA  - en
ID  - JEDP_2012____A4_0
ER  -

%0 Journal Article
%A Gravejat, Philippe
%A Hainzl, Christian
%A Lewin, Mathieu
%A Séré, Éric
%T Two Hartree-Fock models for the vacuum polarization
%J Journées équations aux dérivées partielles
%D 2012
%P 1-31
%I Groupement de recherche 2434 du CNRS
%U http://archive.numdam.org/articles/10.5802/jedp.87/
%R 10.5802/jedp.87
%G en
%F JEDP_2012____A4_0

Gravejat, Philippe; Hainzl, Christian; Lewin, Mathieu; Séré, Éric. Two Hartree-Fock models for the vacuum polarization. Journées équations aux dérivées partielles (2012), article  no. 4, 31 p. doi : 10.5802/jedp.87. http://archive.numdam.org/articles/10.5802/jedp.87/

Bibliographie
Cité par

[1] Bach, V.; Barbaroux, J.-M.; Helffer, B.; Siedentop, H. On the stability of the relativistic electron-positron field, Commun. Math. Phys., Volume 201 (1999) no. 2, pp. 445-460 | MR | Zbl

[2] Blackett, P.M.S.; Occhialini, G.P.S. Some photographs of the tracks of penetrating radiation, Proc. Roy. Soc. Lond. A, Volume 139 (1933), pp. 699-726

[3] Casimir, H.B.G. On the attraction between two perfectly conducting plates, Proc. Kon. Nederland. Akad. Wetensch., Volume 51 (1948) no. 7, pp. 793-795 | Zbl

[4] Casimir, H.B.G.; Polder, D. The influence of retardation on the London-van der Waals forces, Phys. Rev., Volume 73 (1948) no. 4, pp. 360-372 | Zbl

[5] Chaix, P.; Iracane, D. From quantum electrodynamics to mean field theory: I. The Bogoliubov-Dirac-Fock formalism, J. Phys. B, Volume 22 (1989), pp. 3791-3814

[6] 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, Volume 22 (1989), pp. 3815-3828

[7] Dirac, P.A.M. The quantum theory of the electron, Proc. Roy. Soc. Lond. A, Volume 117 (1928), pp. 610-624

[8] Dirac, P.A.M. The quantum theory of the electron. II, Proc. Roy. Soc. Lond. A, Volume 118 (1928), pp. 351-361

[9] Dirac, P.A.M. A theory of electrons and protons, Proc. Roy. Soc. Lond. A, Volume 126 (1930), pp. 360-365

[10] Dyson, F.J. Advanced quantum mechanics, World Scientific, Hackensack, NJ, 2007 (Translated by D. Derbes) | MR | Zbl

[11] Engel, E.; Schwerdtfeger, P. Relativistic density functional theory: foundations and basic formalism, Relativistic electronic structure theory, part 1. Fundamentals (Theoretical and computational chemistry), Volume 11, Elsevier, Amsterdam, 2002, pp. 524-624

[12] Esteban, M.J.; Lewin, M.; Séré, É. Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc., Volume 45 (2008) no. 4, pp. 535-593 | MR

[13] Fock, V. Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems, Zts. f. Phys., Volume 61 (1930) no. 1-2, pp. 126-148

[14] Furry, W.H. A symmetry theorem in the positron theory, Phys. Rev., Volume 51 (1937) no. 2, pp. 125-129 | Zbl

[15] Gabrielse, G.; Hanneke, D.; Kinoshita, T.; Nio, M.; Odom, B. New determination of the fine structure constant from the electron g value and QED, Phys. Rev. Lett., Volume 97 (2006) no. 3, pp. 030802

[16] Gravejat, P.; Hainzl, C.; Lewin, M.; Séré, É. Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields, Preprint (2012) | MR

[17] Gravejat, P.; Lewin, M.; Séré, É. Ground state and charge renormalization in a nonlinear model of relativistic atoms, Commun. Math. Phys., Volume 286 (2009) no. 1, pp. 179-215 | MR | Zbl

[18] Gravejat, P.; Lewin, M.; Séré, É. Renormalization and asymptotic expansion of Dirac’s polarized vacuum, Commun. Math. Phys., Volume 306 (2011) no. 1, pp. 1-33 | MR | Zbl

[19] Greiner, W.; Reinhardt, J. Quantum electrodynamics, Springer-Verlag, Berlin, 2009 (Translated from the German) | MR | Zbl

[20] Hainzl, C.; Lewin, M.; Séré, É. Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation, Commun. Math. Phys., Volume 257 (2005) no. 3, pp. 515-562 | MR | Zbl

[21] Hainzl, C.; Lewin, M.; Séré, É. Self-consistent solution for the polarized vacuum in a no-photon QED model, J. Phys. A, Math. Gen., Volume 38 (2005) no. 20, pp. 4483-4499 | MR | Zbl

[22] Hainzl, C.; Lewin, M.; Séré, É. Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics, Arch. Rat. Mech. Anal., Volume 192 (2009) no. 3, pp. 453-499 | MR | Zbl

[23] 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, Volume 76 (2007) no. 5, pp. 052104

[24] Hainzl, C.; Lewin, M.; Solovej, J.-P. The mean-field approximation in quantum electrodynamics. The no-photon case, Commun. Pure Appl. Math., Volume 60 (2007) no. 4, pp. 546-596 | MR | Zbl

[25] Hartree, D. The wave-mechanics of an atom with a non-coulomb central field. Part I, Proc. Camb. Philos. Soc., Volume 24 (1928) no. 1, pp. 89-312

[26] Heisenberg, W. Bemerkungen zur diracschen theorie des positrons, Zts. f. Phys., Volume 90 (1934) no. 3, pp. 209-231 | Zbl

[27] Heisenberg, W.; Euler, H. Folgerungen aus der diracschen theorie des positrons, Zts. f. Phys., Volume 98 (1936) no. 11-12, pp. 714-732 | Zbl

[28] Hunziker, W.; Sigal, I.M. The quantum N-body problem, J. Math. Phys., Volume 41 (2000) no. 6, pp. 3348-3509 | MR | Zbl

[29] Lamb, W.E.; Retherford, R.C. Fine structure of the hydrogen atom by a microwave method, Phys. Rev., Volume 72 (1947) no. 3, pp. 241-243

[30] Landau, L.D. On the quantum theory of fields. Bohr Volume, Pergamon Press, Oxford, 1955 (Reprinted in Collected papers of L.D. Landau. Pergamon Press, Oxford, 1965) | MR

[31] Landau, L.D.; Pomeranchuk, I.Y. On point interaction in quantum electrodynamics, Dokl. Akad. Nauk SSSR (N.S.), Volume 102 (1955), pp. 489-492 | MR | Zbl

[32] Lewin, M.; Exner, P. Renormalization of Dirac’s polarized vacuum, Mathematical results in quantum physics (Proceedings of the QMath11 Conference), World Scientific (2011), pp. 45-59 | MR | Zbl

[33] Lewin, M. A nonlinear variational problem in relativistic quantum mechanics, Proceeding of the sixth European Congress of Mathematics, European Mathematical Society (2013)

[34] Lieb, E.H. Variational principles for many-fermion systems, Phys. Rev. Lett., Volume 46 (1981) no. 7, pp. 457-459 | MR

[35] Lieb, E.H. Bound on the maximum negative ionization of atoms and molecules, Phys. Rev. A, Volume 29 (1984) no. 6, pp. 3018-3028

[36] Mandelshtam, L.I.; Tamm, I.E. The uncertainty relation between energy and time in nonrelativistic quantum mechanics, J. of Phys. (USSR), Volume 9 (1945), pp. 249-254 | MR | Zbl

[37] Nenciu, G.; Scharf, G. On regular external fields in quantum electrodynamics, Helv. Phys. Acta, Volume 51 (1978) no. 3, pp. 412-424 | MR

[38] Pauli, W.; Rose, M.E. Remarks on the polarization effects in the positron theory, Phys. Rev., Volume 49 (1936) no. 6, pp. 462-465

[39] Pauli, W.; Villars, F. On the invariant regularization in relativistic quantum theory, Rev. Modern Phys., Volume 21 (1949), pp. 434-444 | MR | Zbl

[40] Peskin, M.E.; Schroeder, D.V. An introduction to quantum field theory, Frontiers in Physics, 94, Westview Press, New-York, 1995 | MR

[41] Reed, M.; Simon, B. Methods of modern mathematical physics IV. Analysis of operators, Texts and Monographs in Physics, Academic Press, New-York, 1980 | MR | Zbl

[42] Sabin, J. Static electron-positron pair creation in strong fields for a nonlinear Dirac model, Preprint (2001)

[43] Schwinger, J. Quantum electrodynamics. I. A covariant formulation, Phys. Rev., Volume 74 (1948) no. 10, pp. 1439-1461 | MR | Zbl

[44] Serber, R. Linear modifications in the Maxwell field equations, Phys. Rev., Volume 48 (1935) no. 1, pp. 49-54

[45] Serber, R. A note on positron theory and proper energies, Phys. Rev., Volume 49 (1936) no. 7, pp. 545-550

[46] Solovej, J.-P. Proof of the ionization conjecture in a reduced Hartree-Fock model, Invent. Math., Volume 104 (1991) no. 2, pp. 291-311 | MR | Zbl

[47] Solovej, J.-P. The ionization conjecture in Hartree-Fock theory, Annals of Math., Volume 158 (2003) no. 2, pp. 509-576 | MR | Zbl

[48] Thaller, B. The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1992 | MR | Zbl

[49] Uehling, E.A. Polarization effects in the positron theory, Phys. Rev., Volume 48 (1935) no. 1, pp. 55-63 | Zbl

Cité par Sources :