Theoretical methods for the relativistic atomic many-body problem

Because of the well-understood nature of the electromagnetic interaction, the presence of a well-defined center of force, and the relative unimportance of nonperturbative effects, the atomic many-body problem is argued to be an ideal laboratory for the study of high-accuracy theoretical many-body techniques. In particular, the convergence of many-body perturbation theory (MBPT) for highly charged ions is so rapid that the relativistic generalization of the Schrödinger equation can be accurately solved with MBPT through second order. Relatively large radiative corrections in these ions require the integration of QED and MBPT, which can be accomplished by using $S$-matrix theory. However, to reach high accuracy for neutral atoms, methods based on summation of infinite classes of MBPT diagrams are required. Both RPA and Brueckner-orbital-type summations are needed to reach the one-percent level for heavy atoms, and to proceed further the need for the evaluation of new classes of diagrams involving triple excitations will be described.