Foldy-Wouthuysen Transformation in Strong Magnetic Fields and Relativistic Corrections for Quantum Cyclotron Energy Levels

TL;DR

This paper develops an iterative Foldy-Wouthuysen transformation for Dirac Hamiltonians in electromagnetic fields, including strong magnetic fields, to derive relativistic corrections for quantum cyclotron energy levels, emphasizing applications in Penning traps.

Contribution

It provides a detailed iterative method for the Foldy-Wouthuysen transformation in strong fields, including the anomalous magnetic moment, with new insights into electric field derivatives at higher orders.

Findings

Final expressions for Dirac Hamiltonian in combined electric and magnetic fields.

Time-derivative of electric field enters at seventh order in fine-structure constant.

Special focus on strong magnetic fields relevant for Penning trap experiments.

Abstract

We carry out a direct, iterative Foldy--Wouthuysen transformation of a general Dirac Hamiltonian coupled to an electromagnetic field, including the anomalous magnetic moment. The transformation is carried out through an iterative disentangling of the particle and antiparticle Hamiltonians, in the expansion for higher orders of the momenta. The time-derivative term from the unitary transformation is found to be crucial in supplementing the transverse component of the electric field in higher orders. Final expressions are obtained for general combined electric and magnetic fields, including strong magnetic fields. The time-derivative of the electric field is shown to enter only in the seventh order of the fine-structure constant if the transformation is carried out in the standard fashion. We put special emphasis on the case of strong fields, which are important for a number of applications, such as electrons bound in Penning traps.