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Annales Henri Poincaré

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

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

  • Martin L. R. Fürst
  • Max Lein
Article

Abstract

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.

Keywords

Pseudodifferential Operator Selfadjoint Operator Symbol Class Moyal Product Pseudodifferential Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

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