Leading correction to the relativistic Foldy-Wouthuysen Hamiltonian

TL;DR

This work uses the exact Eriksen Foldy-Wouthuysen transformation to derive a leading correction to the relativistic FW Hamiltonian for a spin-1/2 particle in external fields within the weak-field limit. By performing an operator extraction of the square root in the Eriksen operator, the authors obtain a closed-form FW Hamiltonian H_FW = H1 + H2, where H1 = βε + E − (1/8){1/[ε(ε+m)], [O,[O,F]]} and H2 = (1/64){(2ε^2 − m^2)/(ε^4(ε+m)^2), [O^2,[O^2,F]]}, with F = E − iħ∂t and O the odd part of the Dirac Hamiltonian. The new term H2 is the leading relativistic correction in the weak-field regime, is proportional to ħ^2, and becomes relevant in relativistic scattering when the impact parameter is of order the Compton wavelength. The paper also builds a corresponding relativistic second-order wave equation, connecting first- and second-order descriptions, and analyzes concrete examples including a Dirac particle in electrostatic and nonuniform fields, with and without anomalous magnetic moments (Dirac-Pauli). The results reinforce the physical clarity of Eriksen-satisfying methods and illuminate their potential but typically small impact in laboratory settings, while highlighting their importance for scattering theory and NRQED connections.

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.