Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

Archive for Rational Mechanics and Analysis

Volumn 192, Issue 3, 2009, Pages 453-499

  Hainzl, Christian ^a   Lewin, Mathieu ^b   Séré, Éric ^c  

^a UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

^b UNIVERSITÉ DE CERGY PONTOISE   (France)

^c UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67349143641     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-008-0144-2     Document Type: Article

References (63)

1. C.D. Anderson 1933 The positive electron Phys. Rev. 43 491 494
2. J. Avron R. Seiler B. Simon 1994 The index of a pair of projections J. Funct. Anal. 120 220 237
3. V. Bach 1992 Error bound for the Hartree-Fock energy of atoms and molecules Commun. Math. Phys. 147 527 548
4. V. Bach J.-M. Barbaroux B. Helffer H. Siedentop 1999 On the stability of the relativistic electron-positron field Commun. Math. Phys. 201 445 460
5. V. Bach E.H. Lieb M. Loss J.P. Solovej 1994 There are no unfilled shells in unrestricted Hartree-Fock theory Phys. Rev. Lett. 72 2981 2983 (Pubitemid 24975302)
6. V. Bach E.H. Lieb J.P. Solovej 1994 Generalized Hartree-Fock theory and the Hubbard model J. Stat. Phys. 76 1-2 3 89 (Pubitemid 24824971)
7. Bhatia R.: Matrix analysis. Graduate Texts in Mathematics, vol. 169. Springer, New York (1997)
8. J. Borwein D. Preiss 1987 A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions Trans. Am. Math. Soc. 303 2 517 527
9. Chaix, P.: Une Méthode de Champ Moyen Relativiste et Application à l'Etude du Vide de l'Electrodynamique Quantique. PhD Thesis, University Paris VI (1990)
10. P. Chaix D. Iracane 1989 From quantum electrodynamics to mean field theory: I The Bogoliubov-Dirac-Fock formalism. J. Phys. B. 22 3791 3814
11. P. Chaix D. Iracane P.L. Lions 1989 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
12. K. Dietz B.A. Hess 1991 Hartree-Fock-type equations in relativistic quantum electrodynamics with non-linear gauge fixing J. Phys. E At. Mol. Opt. Phys. 24 1129 1142
13. J. Dolbeault M.J. Esteban E. Séré 2000 On the eigenvalues of operators with gaps Application to Dirac operators. J. Funct. Anal. 174 1 208 226
14. P.A.M. Dirac 1928 The quantum theory of the electron Proc. R. Soc. A 117 610 624
15. P.A.M. Dirac 1930 A theory of electrons and protons Proc. R. Soc. A 126 360 365
16. Dirac, P.A.M.: Théorie du positron. Solvay report, pp.203-212 (1934). Gauthier-Villars, Paris. XXV, 353 S
17. P.A.M. Dirac 1934 Discussion of the infinite distribution of electrons in the theory of the positron Proc. Camb. Philos. Soc. 30 150 163
18. E. Engel R.M. Dreizler 1987 Field-theoretical approach to a relativistic Thomas-Fermi-Dirac-Weisäcker model Phys. Rev. A 35 9 3607 3618
19. M.J. Esteban E. Séré 1999 Solutions of the Dirac-Fock equations for atoms and molecules Commun. Math. Phys. 203 499 530
20. M.J. Esteban E. Séré 2001 Nonrelativistic limit of the Dirac-Fock equations Ann. Henri Poincaré 2 5 941 961 (Pubitemid 33405839)
21. M.J. Esteban E. Séré 2002 A max-min principle for the ground state of the Dirac-Fock functional Contemp. Math. 307 135 141
22. L.L. Foldy E. Eriksen 1954 Some physical consequences of vacuum polarization Phys. Rev. 95 4 1048 1051
23. Ghoussoub N.: Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press, London (1993)
24. R. Glauber W. Rarita P. Schwed 1960 Vacuum polarization effects on energy levels in μ-mesonic atoms Phys. Rev. 120 2 609 613
25. L. Gomberoff V. Tolmachev 1971 Hartree-Fock approximation in quantum electrodynamics Phys. Rev. D 3 8 1796 1804
26. M. Griesemer E.H. Lieb M. Loss 2001 Ground states in non-relativistic quantum electrodynamics Invent. Math. 145 3 557 595 (Pubitemid 33725757)
27. C. Hainzl M. Lewin E. Séré 2005 Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation Commun. Math. Phys. 257 515 562 (Pubitemid 40900452)
28. C. Hainzl M. Lewin E. Séré 2005 Self-consistent solution for the polarized vacuum in a no-photon QED model J. Phys. A Math. Gen. 38 4483 4499 (Pubitemid 40669586)
29. C. Hainzl M. Lewin J.P. Solovej 2007 The mean-field approximation in quantum electrodynamics The no-photon case. Commun. Pure Appl. Math. 60 4 546 596
30. A. Hamm D. Schütte 1990 How to remove divergences from the QED-Hartree approximation J. Phys. A Math. Gen. 23 3969 3982
31. W. Heisenberg 1934 Bemerkungen zur Diracschen Theorie des Positrons Z. Phys. 90 209 223
32. W. Hunziker 1966 On the spectra of Schrödinger multiparticle Hamiltonians Helv. Phys. Acta 39 451 462
33. Kato T.: Perturbation Theory for Linear Operators. Springer, Heidelberg (1966)
34. Kato, T.: Notes on projections and perturbation theory. Technical report No 9, University of California (1955)
35. M. Klaus 1980 Non-regularity of the Coulomb potential in quantum electrodynamics Helv. Phys. Acta 53 36 39
36. M. Klaus G. Scharf 1977 The regular external field problem in quantum electrodynamics Helv. Phys. Acta 50 779 802
37. E.H. Lieb 1981 Variational principle for Many-Fermion systems Phys. Rev. Lett. 46 457 459
38. E.H. Lieb 1984 Bound on the maximum negative ionization of atoms and molecules Phys. Rev. A. 29 3018 3028
39. E.H. Lieb M. Loss 2003 Existence of atoms and molecules in non-relativistic quantum electrodynamics Adv. Theor. Math. Phys. 7 4 667 710
40. E.H. Lieb H. Siedentop 2000 Renormalization of the regularized relativistic electron-positron field Commun. Math. Phys. 213 3 673 683
41. E.H. Lieb B. Simon 1977 The Hartree-Fock theory for Coulomb systems Commun. Math. Phys. 53 185 194
42. Lions, P.-L.: The concentration-compactness method in the Calculus of Variations. The locally compact case. Part. I: Anal. non-linéaire, Ann. IHP 1, 109-145 (1984). Part. II: Anal. non-linéaire, Ann. IHP 1, 223-283 (1984)
43. P.-L. Lions 1987 Solutions of Hartree-Fock equations for Coulomb systems Commun. Math. Phys. 109 33 97
44. P.J. Mohr G. Plunien G. Soff 1998 QED corrections in heavy atoms Phys. Rep. 293 5&6 227 372 (Pubitemid 128186708)
45. G. Nenciu 1987 Existence of spontaneous pair creation in the external field approximation of Q.E.D. Commun Math. Phys. 109 303 312
46. E. Paturel 2000 Solutions of the Dirac equations without projector Ann. Henri Poincaré 1 1123 1157 (Pubitemid 33213511)
47. P.-G. Reinhard W. Greiner 1977 Quantum electrodynamics of strong fields Rep. Prog. Phys. 40 219 295
48. P.-G. Reinhard W. Greiner H. Arenhövel 1971 Electrons in strong external fields Nucl. Phys. A, 166 173 197
49. J. Reinhardt B. Müller W. Greiner 1981 Theory of positron production in heavy-ion collision Phys. Rev. A, 24 1 103 128
50. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. Academic Press, Inc., New York, 1980
51. S.N.M. Ruijsenaars 1977 On Bogoliubov transformations for systems of relativistic charged particles J. Math. Phys. 18 3 517 526
52. G. Scharf H.P. Seipp 1982 Charged vacuum, spontaneous positron production and all that Phys. Lett. 108B 3 196 198
53. J. Schwinger 1948 Quantum electrodynamics I. A covariant formulation Phys. Rev. 74 10 1439 1461
54. J. Schwinger 1949 Quantum electrodynamics II Vacuum polarization and self-energy. Phys. Rev. 75 4 651 679
55. J. Schwinger 1951 On Gauge invariance and vacuum polarization Phys. Rev. II. Ser. 82 5 664 679
56. 2 quantum field theory: upper bound on the pressure, Hamiltonian bound and linear lower bound Commun. Math. Phys. 45 99 114
57. Simon, B.: Trace ideals and their applications. London Mathematical Society Lecture Notes Series, vol. 35. Cambridge University Press, London, 1979
58. B. Swirles 1935 The relativistic self-consistent field Proc. R. Soc. A 152 625 649
59. Thaller B.: The Dirac Equation. Springer, Heidelberg (1992)
60. C. Tix 1997 Lower bound for the ground state energy of the no-pair Hamiltonian Phys. Lett. B 405 293 296 (Pubitemid 127376233)
61. C. Tix 1998 Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall. Bull Lond. Math. Soc. 30 3 283 290 (Pubitemid 128343972)
62. Van Winter, C.: Theory of finite systems of particles. I. The Green function. Mat.-Fys. Skr. Danske Vid. Selsk. 2(8), (1964)
63. G.M. Zhislin 1960 A study of the spectrum of the Schrödinger operator for a system of several particles (Russian) Trudy Moskov. Mat. Obšč. 9 81 120