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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