Abstract
The stability of matter composed of electrons and static nuclei is investigated for a relativistic dynamics for the electrons given by a suitably projected Dirac operator and with Coulomb interactions. In addition there is an arbitrary classical magnetic field of finite energy. Despite the previously known facts that ordinary nonrelativistic matter with magnetic fields, or relativistic matter without magnetic fields, is already unstable when a, the fine structure constant, is too large, it is noteworthy that the combination of the two is still stableprovided the projection onto the positive energy states of the Dirac operator, whichdefines the electron, is chosen properly. A good choice is to include the magnetic field in the definition. A bad choice, which always leads to instability, is the usual one in which the positive energy states are defined by the free Dirac operator. Both assertions are proved here.
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H. A. Bethe and E. E. Salpeter. Quantum mechanics of one- and two-electron atoms. In S. Flügge, editor,Handbuch der Physik, XXXV, pages 88–436. Springer, Berlin, 1 edition, 1957.
M. S. Birman, L. S. Koplienko, and M. Z. Solomyak. Estimates for the spectrum of the difference between fractional powers of two self-adjoint operators.Soviet Mathematics 19(3): 1–6, 1975. Translation of Izvestija vyssich.
G. Brown and D. Ravenhall. On the interaction of two electrons.Proc. Roy. Soc. London A 208(A 1095):552–559, September 1951.
L. Bugliaro, J. Fröhlich, G. M. Graf, J. Stubbe, and C. Fefferman. A Lieb-Thirring bound for a magnetic Pauli Hamiltonian.Preprint, ETH-TH/96-31, 1996.
J. G. Conlon. The ground state energy of a classical gas.Commun. Math. Phys. 94(4): 439–458, August 1984.
I. Daubechies. An uncertainty principle for Fermions with generalized kinetic energy.Commun. Math. Phys. 90:511–520, September 1983.
F. J. Dyson and A. Lenard. Stability of matter I.J. Math. Phys. 8:423–434, 1967.
F. J. Dyson and A. Lenard. Stability of matter II.J. Math. Phys. 9:698–711, 1967.
W. D. Evans, P. Perry, and H. Siedentop. The spectrum of relativistic one-electron atoms according to Bethe and Salpeter.Commun. Math. Phys. 18:733–746, July 1996.
P. Federbush. A new approach to the stability of matter problem.I. J. Math. Phys. 16:347–351, 1975.
C. Fefferman. Stability of relativistic matter with magnetic fields.Proc. Nat. Acad. Sci. USA 92:5006–5007, 1995.
C. Fefferman, J. Fröhlich, and G. M. Graf. Stability of ultraviolet-cutoff quantum electrodynamics with non-relativistic matter.Texas Math. Phys. Preprint server 96–379, 1996.
J. Fröhlich, E. H. Lieb, and M. Loss. Stability of Coulomb systems with magnetic fields. I: The one-electron atom.Commun. Math. Phys. 104:251–270, 1986.
Y. Ishikawa and K. Koc. Relativistic many-body perturbation theory based on the no-pair Dirac-Coulomb-Breit Hamiltonian: Relativistic correlation energies for the noble-gas sequence through Rn (Z=86), the group-IIB atoms through Hg, and the ions of Ne isoelectronic sequence.Phys. Rev. A 50(6):4733–4742, December 1994.
H. J. A. Jensen, K. G. Dyall, T. Saue, and K. Faegri. Jr., Relativistic four-component multiconfigurational self-consistent-field theory for molecules: Formalism.J. Chem. Physics 104(11):4083–4097, March 1996.
E. H. Lieb. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities.Annals of Mathematics 118:349–374, 1983.
E. H. Lieb. On characteristic exponents in turbulence.Commun. Math. Phys. 92:473–480, 1984.
E. H. Lieb and M. Loss. Stability of Coulomb systems with magnetic fields. II: The manyelectron atom and the one-electron molecule.Commun. Math. Phys. 104:271–282, 1986.
E. H. Lieb, M. Loss, and H. Siedentop. Stability of relativistic matter via Thomas-Fermi theory.Helv. Phys. Acta 69:974–984, 1996.
E. H. Lieb, M. Loss, and J. P. Solovej. Stability of matter in magnetic fields.Phys. Rev. Lett. 75(6):985–989, August 1995.
E. H. Lieb and W. E. Thirring. Bound for the kinetic energy of Fermions which proves the stability of matter.Phys. Rev. Lett. 35(11):687–689, September 1975. Erratum:Phys. Rev. Lett. 36(16):11116, October 1975.
E. H. Lieb and W. E. Thirring. Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In E. H. Lieb, B. Simon, and A. S. Wightman, editors,Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. Princeton University Press, Princeton, 1976.
E. H. Lieb and H.-T. Yau. The stability and instability of relativistic matter.Common. Math Phys. 118:177–213, 1988.
J. Sucher. Foundations of the relativistic theory of many-electron atoms.Phys. Rev. A 22(2):348–362, August 1980.
J. Sucher. Foundations of the relativistic theory of many-electron bound states.International Journal of Quantum Chemistry 25:3–21, 1984.
J. Sucher. Relativistic many-electron Hamiltonians.Phys. Scripta 36:271–281, 1987.
W. Thirring, ed.The Stability of Matter. From Atoms to Stars, Selecta of Elliott H. Lieb, Springer-Verlag, Berlin, Heidelberg, New York, 1997.
E. H. Lieb, H. Siedentop, and J. P. Solovej. Stability of relativistic matter with magnetic fields.Phys. Rev. Lett. 79:1785, 1997.
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This paper is dedicated to Bernard Jancovici on the occasion of his 65th birthday.
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Lieb, E.H., Siedentop, H. & Solovej, J.P. Stability and instability of relativistic electrons in classical electromagnetic fields. J Stat Phys 89, 37–59 (1997). https://doi.org/10.1007/BF02770753
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DOI: https://doi.org/10.1007/BF02770753