How are bound states handled in QFT?

The conventional way to handle bound states in relativistic quantum field theory is the Bethe-Salpeter equation. An old but very informative survey paper on the Bethe-Salpeter equation is

• M.M. Broido, Green functions in particle physics, Reports on Progress in Physics 32 (1969), 493-545.

The hydrogen atom is in QFT usually treated in an approximation where the proton is treated as an external Coulomb field (and some recoil effects are handled perturbatively). The basics are given in Weinbergs QFT book Vol. 1 (p.560 for the Bethe-Salpeter equation and Chapter 14 for 1-electron atoms). Weinberg notes on p.560 that

the theory of relativistic effects and radiative corrections in bound states is not yet in entirely satisfactory shape.

This quote from 1995 is still valid today, 20 years later.

On the other hand, quantum chemists use routinely relativistic quantum mechanical calculations for the prediction of properties of heavy atoms. For example, the color of gold or the fluidity of mercury at room temperature can be explained only through relativistic effects. They use the Dirac-Fock approximation of QED.

A bound state such as a hydrogen atom is 'hidden' in the given interaction as a pole in the scattering matrix for the two fields.