Exact Foldy-Wouthuysen transformation of the Dirac-Pauli Hamiltonian in the weak-field limit by the method of direct perturbation theory

TL;DR

This paper rigorously derives the exact Foldy-Wouthuysen transformation of the Dirac-Pauli Hamiltonian in weak electromagnetic fields, confirming its agreement with classical physics through a perturbation theory approach.

Contribution

It provides a rigorous proof of the exact FW transformation for the Dirac-Pauli Hamiltonian in the weak-field limit using direct perturbation theory, aligning quantum and classical descriptions.

Findings

Exact FW transformations match Eriksen's method in special cases.

In weak fields, quantum Hamiltonian agrees with classical orbital and spin Hamiltonians.

Mathematical induction confirms the classical limit of the quantum Hamiltonian.

Abstract

We apply the method of direct perturbation theory for the Foldy-Wouthuysen (FW) transformation upon the Dirac-Pauli Hamiltonian subject to external electromagnetic fields. The exact FW transformations exist and agree with those obtained by Eriksen's method for two special cases. In the weak-field limit of static and homogeneous electromagnetic fields, by mathematical induction on the orders of $1/c$ in the power series, we rigorously prove the long-held speculation: the FW transformed Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.