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

1 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 semi-classical approximation of using Grassmann-valued electric charges for regularizing the Coulomb self-energies and extracting the unique semi-classical action-at-a-distance interaction hidden in any Green function used for the Lienard-Wiechert solution. By describing semi-classically also the spin of the particles with Grassmann-variables, by means of a semi-classical 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 semi-classical 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 semi-classical 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