Foldy-Wouthyusen Transformation and Semiclassical Limit for Relativistic Particles in Strong External Fields

TL;DR

This paper develops a general Foldy-Wouthyusen transformation method for relativistic particles in strong fields, enabling a clear transition from quantum to classical descriptions and deriving equations of motion with full quantum-classical agreement.

Contribution

A novel, general FW transformation approach applicable to particles of any spin in strong external fields, facilitating semiclassical analysis and quantum-classical correspondence.

Findings

Derived explicit low-order terms in the FW Hamiltonian.

Established full agreement between quantum and classical equations of motion.

Applied method to spin-1/2 and scalar particles in electromagnetic fields.

Abstract

A general method of the Foldy-Wouthyusen (FW) transformation for relativistic particles of arbitrary spin in strong external fields has been developed. The use of the found transformation operator is not restricted by any definite commutation relations between even and odd operators. The final FW Hamiltonian can be expanded into a power series in the Planck constant which characterizes the order of magnitude of quantum corrections. Exact expressions for low-order terms in the Planck constant can be derived. Finding these expressions allows to perform a simple transition to the semiclassical approximation which defines a classical limit of the relativistic quantum mechanics. As an example, interactions of spin-1/2 and scalar particles with a strong electromagnetic field have been considered. Quantum and semiclassical equations of motion of particles and their spins have been deduced. Full agreement between quantum and classical theories has been established.