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

TL;DR

This paper develops a rigorous approximation framework for Dirac dynamics in external fields, deriving semi- and non-relativistic limits with higher-order corrections, and demonstrating decoupling of electronic and positronic states for small velocities.

Contribution

It introduces a systematic method to derive semi- and non-relativistic effective dynamics from Dirac equations using space-adiabatic perturbation theory and magnetic pseudodifferential calculus.

Findings

Effective dynamics are generated by semi- and non-relativistic Pauli Hamiltonians.

Higher-order corrections in v/c can be computed systematically.

Electronic and positronic states decouple to all orders in v/c << 1.

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 $\xi^2 / 2m$, respectively. Higher-order corrections can in principle be computed to any order in the small parameter v/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 v/c << 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 with the novel input of a magnetic pseudodifferential calculus adapted to either the semi- or non-relativistic scaling.