Relativistic treatment of many-particle systems

We now turn to the many-electron generalization of the Dirac equation. It must be said at the outset that no fully satisfactory relativistic Hamiltonian has yet been derived, and that difficulties remain at a fundamental level however, the somewhat ad hoc approach generally employed (including the reduction to Pauli form) leads to a large number 

In fact, the relativistically correct potentials are the retarded or Li nard-Wiechert potentials (Darwin, 1920 Feynman et al., 1964) 

The retarded potentials are discussed in detail in the references dted. Here we need only note that the classical Lagrangian for a pair of particles interacting through these potentials is given by (Darwin, 1920) 

The interaction term I is an approximation, good up to terms of the order 

The resultant Hamiltonian is found by straightforward but rather lengthy manipulation to be 

---