Annales Henri Poincaré

, Volume 14, Issue 5, pp 1305–1347 | Cite as

Semi- and Non-Relativistic Limit of the Dirac Dynamics with External Fields



We show how to approximate Dirac dynamics for electronic initial states by semi- and non-relativistic dynamics. To leading order, these are generated by the semi- and non-relativistic Pauli hamiltonian where the kinetic energy is related to \({\sqrt{m^2 + \xi^2}}\) and \({\frac{1}{2m}\xi^2}\) , respectively. Higher-order corrections can in principle be computed to any order in the small parameter \({{\upsilon /c}}\)which is the ratio of typical speeds to the speed of light. Our results imply the dynamics for electronic and positronic states decouple to any order in \({\upsilon /c \ll 1}\) . To decide whether to get semi- or non-relativistic effective dynamics, one needs to choose a scaling for the kinetic momentum operator. Then the effective dynamics are derived using space-adiabatic perturbation theory by Panati et al. (Adv Theor Math Phys 7:145–204, 2003) with the novel input of a magnetic pseudodifferential calculus adapted to either the semi- or non-relativistic scaling.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bjorken J.D., Drell S.D.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1998)Google Scholar
  2. 2.
    Bechouche P., Mauser N.J., Poupaud F.: (Semi)-Nonrelativistic limits of the Dirac equation with external time-dependent electromagnetic field. Commun. Math. Phys. 197, 405–425 (1998). doi: 10.1007/s002200050457 MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Bargmann V., Michel L., Telegdi V.L.: Precession of the polarisation of particles moving in a homogeneous electromagnetic field. Phys. Rev. Lett. 2, 435 (1959)ADSCrossRefGoogle Scholar
  4. 4.
    Brummelhuis R., Nourrigat J.: Scattering amplitude for Dirac operators. Commun. Partial Diff. Equ. 51(3), 231–261 (1999)MathSciNetGoogle Scholar
  5. 5.
    Cordes H.O.: A pseudodifferential Foldy–Wouthuysen transform. Commun. Partial Diff. Equ. 8(13), 1475–1485 (1983)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cordes H.O.: A precise pseudodifferential Foldy–Wouthuysen transform for the Dirac equation. J. Evol. Equ. 4(1), 128–138 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Douglas M., Kroll N.M.: Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. 82, 89–155 (1974)ADSCrossRefGoogle Scholar
  8. 8.
    De Nittis G., Lein M.: Applications of magnetic \({\Psi}\) DO techniques to SAPT—beyond a simple review. Rev. Math. Phys. 23, 233–260 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dimassi M., Sjöstrand J.: Spectral Asymtptotics in the Semi-Classical Limit, vol. 268. London Mathematical Society, London (1999)CrossRefGoogle Scholar
  10. 10.
    Folland G.B.: Harmonic Analysis on Phase Space. Princeton University Press, Princeton (1989)Google Scholar
  11. 11.
    Foldy L.L., Wouthuysen S.A.: On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Phys. Rev. 78(1), 29–36 (1950)ADSMATHCrossRefGoogle Scholar
  12. 12.
    Gesztesy F., Grosse H., Thaller B.: A rigorous approach to relativistic corrections of bound state energies for spin-1/2 particles. Annales de l’Institut Henri Poincaré (section A) 40(2), 159–174 (1984)MathSciNetMATHGoogle Scholar
  13. 13.
    Grigore D.R., Nenciu G., Purice R.: On the nonrelativistic limit of the Dirac Hamiltonian. Annales de l’Institut Henri Poincaré 51(3), 231–261 (1989)MathSciNetMATHGoogle Scholar
  14. 14.
    Heß B.A.: Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators. Phys. Rev. A 33(6), 3742–3748 (1986)ADSCrossRefGoogle Scholar
  15. 15.
    Hunziker W.: On the nonrelativistic limit of the Dirac theory. Commun. Math. Phys. 40(3), 215–222 (1975)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Hörmander L.: The Weyl Calculus of Pseudo-Differential Operators. Commun. Pure Appl. Math. XXXII, 359–443 (1979)CrossRefGoogle Scholar
  17. 17.
    Iftimie V., Măntoiu M., Purice R.: Magnetic pseudodifferential operators. Publ. Res. Inst. Math. Sci. 43(3), 585–623 (2007)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Iftimie V., Măntoiu M., Purice R.: Commutator criteria for magnetic pseudodifferential operators. Commun. Partial Diff. Equ. 35, 1058–1094 (2010)MATHCrossRefGoogle Scholar
  19. 19.
    Jansen G., Heß B.A.: Revision of the Douglas–Kroll transformation. Phys. Rev. A 39(11), 6016–6017 (1989)ADSCrossRefGoogle Scholar
  20. 20.
    Lein M.: Two-parameter asymptotics in magnetic Weyl calculus. J. Math. Phys. 51, 123519 (2010)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Lein, M.: Semiclassical Dynamics and Magnetic Weyl Calculus. Phd thesis, Technische Universität München, Munich, Germany (2011)Google Scholar
  22. 22.
    Mauser N.J.: Rigorous derivation of the Pauli equation with time-dependent electromagnetic field. VLSI Design 9, 415–426 (1999)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Măntoiu M., Purice R.: The magnetic Weyl calculus. J. Math. Phys. 45(4), 1394–1417 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    Panati G., Spohn H., Teufel S.: Space adiabatic perturbation theory. Adv. Theor. Math. Phys. 7, 145–204 (2003)MathSciNetGoogle Scholar
  25. 25.
    Reiher M.: Relativistic Douglas–Kroll theory. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2, 139–149 (2012)CrossRefGoogle Scholar
  26. 26.
    Reiher M., Wolf A.: Exact decoupling of the Dirac Hamiltonian. I. General theory. J. Chem. Phys. 121(5), 2037–2047 (2004)ADSCrossRefGoogle Scholar
  27. 27.
    Spohn H.: Semiclassical limit of the Dirac equation and spin precession. Ann. Phys. 282, 420–431 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Siedentop H., Stockmeyer E.: The Douglas–Kroll–Heß method: convergence and block-diagonalization of Dirac operators. Ann. Henri Poincaré 7(1), 45–58 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Teufel S.: Adiabatic Perturbation Theory in Quantum Dynamics. Springer, Berlin (2003)MATHCrossRefGoogle Scholar
  30. 30.
    Thaller B.: The Dirac Equation. Springer, Berlin (1992)Google Scholar
  31. 31.
    Treves F.: Topological vector spaces, distributions and kernels. Academic Press, Dublin (1967)MATHGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Excellence Cluster UniverseTechnische Universität MünchenGarchingGermany
  2. 2.Technische Universität München, Department für MathematikGarchingGermany
  3. 3.Eberhard Karls Universität Tübingen, Mathematisches InstitutTübingenGermany

Personalised recommendations