Abstract
Use of the Dirac equation for a single particle in an electromagnetic field has led to many successful predictions. For more than one particle, the approach is usually through quantum field theory. In this contribution, we follow the original Dirac approach, but for a two-body problem, starting with a discussion of positronium. The problem can be reduced to that of solving a set of radial equations, but unfortunately obtaining an exact solution of them by means of a recurrence procedure, as was done for the hydrogen atom by Dirac, does not seem feasible. However, an evaluation of the relativistic corrections to the order (1/c2) is made for parapositronium and orthopositronium. These results are discussed and their relevance to other problems of this type - such as that of the quark-antiquark system - are mentioned, as solving these will be the eventual objective of future work.