Leading correction to the relativistic Foldy-Wouthuysen Hamiltonian

TL;DR

This paper derives a leading correction to the relativistic Foldy-Wouthuysen Hamiltonian for Dirac particles in weak fields, improving the description of relativistic scattering and connecting first and second order wave equations.

Contribution

It provides a rigorous derivation of a correction to the Foldy-Wouthuysen Hamiltonian using the Eriksen operator, enhancing the theoretical framework for relativistic particle scattering.

Findings

Derived a correction to the Foldy-Wouthuysen Hamiltonian in weak fields.

Established a connection between first and second order relativistic wave equations.

Improved the theoretical description of relativistic scattering processes.

Abstract

For Dirac particles interacting with external fields, we use the exact operator of the Foldy-Wouthuysen transformation obtained by Eriksen and rigorously derive a leading correction in the weak-field approximation to the known relativistic Foldy-Wouthuysen Hamiltonian. For this purpose, we carry out the operator extraction of a square root in the Eriksen operator. The derived correction is important for the scattering of relativistic particles. Since the description of this scattering by a relativistic wave equation of the second order is more convenient, we determine a general connection between relativistic wave equations of the first and second orders. For Dirac particles, the relativistic wave equation of the second order is obtained with a correction similar to that to the Foldy-Wouthuysen Hamiltonian.