Foldy-Wouthuysen transformation for a Dirac-Pauli dyon and the Thomas-Bargmann-Michel-Telegdi equation

TL;DR

This paper derives the classical Hamiltonian for a spinning charged particle from the Dirac equation using the Foldy-Wouthuysen transformation, confirming consistency up to seventh order and extending results to dyons with electric and magnetic charges.

Contribution

It demonstrates the agreement between the relativistic quantum Dirac (and Dirac-Pauli) Hamiltonians and the classical Hamiltonian for particles with spin and electromagnetic interactions, including dyons.

Findings

Foldy-Wouthuysen transformation matches classical Hamiltonian up to 7th order.

Results extend to spin-1/2 dyons with electric and magnetic charges.

Quantum and classical descriptions are consistent for gyromagnetic ratio 2.

Abstract

The classical dynamics for a charged point particle with intrinsic spin is governed by a relativistic Hamiltonian for the orbital motion and by the Thomas-Bargmann-Michel-Telegdi equation for the precession of the spin. It is natural to ask whether the classical Hamiltonian (with both the orbital and spin parts) is consistent with that in the relativistic quantum theory for a spin-1/2 charged particle, which is described by the Dirac equation. In the low-energy limit, up to terms of the 7th order in $1/E_g$ ($E_g=2mc^2$ and $m$ is the particle mass), we investigate the Foldy-Wouthuysen (FW) transformation of the Dirac Hamiltonian in the presence of homogeneous and static electromagnetic fields and show that it is indeed in agreement with the classical Hamiltonian with the gyromagnetic ratio being equal to 2. Through electromagnetic duality, this result can be generalized for a spin-1/2 dyon, which has both electric and magnetic charges and thus possesses both intrinsic electric and magnetic dipole moments. Furthermore, the relativistic quantum theory for a spin-1/2 dyon with arbitrary values of the gyromagnetic and gyroelectric ratios can be described by the Dirac-Pauli equation, which is the Dirac equation with augmentation for the anomalous electric and anomalous magnetic dipole moments. The FW transformation of the Dirac-Pauli Hamiltonian is shown, up to the 7th order again, to be also in accord with the classical Hamiltonian.