TY - JOUR

T1 - Semi- and Non-Relativistic Limit of the Dirac Dynamics with External Fields

AU - Fürst, Martin L.R.

AU - Lein, Max

N1 - Funding Information:
The authors would like to thank Harald Lesch, Radu Purice, Herbert Spohn and Stefan Teufel for useful discussions, comments, references and remarks as well as Friedrich Gesztesy for providing several useful references. M. L. appreciates financial support from the German-Israeli Foundation. M. F. acknowledges financial support from the DFG cluster of excellence “Origin and Structure of the Universe”.

PY - 2013/7

Y1 - 2013/7

N2 - 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 √ m2 ξ2 and 1/2m ξ2, respectively. Higher-order corrections can in principle be computed to any order in the small parameter υ/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 υ/c ≪. 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.

AB - 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 √ m2 ξ2 and 1/2m ξ2, respectively. Higher-order corrections can in principle be computed to any order in the small parameter υ/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 υ/c ≪. 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.

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U2 - 10.1007/s00023-012-0213-9

DO - 10.1007/s00023-012-0213-9

M3 - Article

AN - SCOPUS:84879982137

VL - 14

SP - 1305

EP - 1347

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 5

ER -