THE SEMICLASSICAL RELATIVISTIC DARWIN POTENTIAL FOR SPINNING PARTICLES IN THE REST-FRAME INSTANT FORM: TWO-BODY BOUND STATES WITH SPIN 1/2 CONSTITUENTS

Abstract

We extend previous results on the extraction of the Darwin potential to all orders in c-2 from the radiation gauge Lienard–Wiechert solution for the system of N positive-energy scalar particles plus the electromagnetic field in the Wigner-covariant rest-frame instant form of dynamics to the case of N positive-energy spinning particles. This is done in the semiclassical approximation of using Grassmann-valued electric charges for regularizing the Coulomb self-energies and extracting the unique semiclassical action-at-a-distance interaction hidden in any Green function used for the Lienard–Wiechert solution. By describing semiclassically also the spin of the particles with Grassmann variables, by means of a semiclassical Foldy–Wouthuysen transformation applied the the Dirac-like constraints of the manifestly Lorentz covariant spinning particles, we determine the coupling of positive-energy spinning particles to the electric field in the semiclassical approximation. Then we follow the same procedure developed for scalar particles and, in the sector where there is no in-radiation, we determine the effective semiclassical interparticle potential. Besides the relativistic Darwin term there are spin-orbit and spin-spin terms in the potential. Quantization of the lowest order (in c-2) part of the closed form of the effective Hamiltonian in the case N = 2 reproduces exactly the standard result of the reduction of the Bethe–Salpeter equation for the bound states of two spin 1/2 constituents of arbitrary mass (hydrogen atom, positronium, muonium).