The Semi-Classical Relativistic Darwin Potential for Spinning Particles in the Rest-Frame Instant Form: 2-Body Bound States with Spin 1/2 Constituents

TL;DR

This paper derives a relativistic semiclassical Darwin potential for spinning particles using Grassmann charges and Foldy-Wouthuysen transformation, matching Bethe-Salpeter results in the two-body case.

Contribution

It introduces a novel semiclassical approach to derive the relativistic Darwin potential for spinning particles in the rest-frame instant form.

Findings

Derived a unique acceleration-independent interaction in the semiclassical approximation.

Quantization reproduces Bethe-Salpeter equation results for two-body systems.

Established a connection between semiclassical potentials and quantum field theory results.

Abstract

In the semiclassical approximation of Grassmann-valued electric charges for regularizing Coulomb self-energies, we extract the unique acceleration-independent interaction hidden in any Lienard-Wiechert solution for the system of N positive-energy spinning particles plus the electromagnetic field in the radiation gauge of the rest-frame instant form. With the help of a semiclassical Foldy-Wouthuysen transformation, this allows us to find the relativistic semiclassical Darwin potential. In the 2-body case the quantization of the lowest order reproduces exactly the results from the reduction of the Bethe-Salpeter equation.