Annales Henri Poincaré

, Volume 14, Issue 5, pp 1305-1347

First online:

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

  • Martin L. R. FürstAffiliated withExcellence Cluster Universe, Technische Universität MünchenTechnische Universität München, Department für Mathematik
  • , Max LeinAffiliated withEberhard Karls Universität Tübingen, Mathematisches Institut Email author 

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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.