Fine Structure and Perihelion Advance

The evaluation of field-theoretical corrections to the properties of bound states can be greatly simplified by basing it on our relativistic Lippmann-Schwinger equation rather than the Bethe-Salpeter or Schrödinger equations. We have evaluated the fine structure to order ${\alpha }^4$ for a system of two scalar particles bound by the potential that arises from exchanging massless quanta of spin 0, 1, or 2. In each case fine structure is clearly related to seagull and/or graviton self-interaction diagrams, while neither the simple exchange diagram nor the box diagrams make any contribution. The simplification with respect to earlier calculations is particularly striking in the case of gravity. Also presented here is a calculation of the advance of the perihelion of Mercury within the framework of a recently developed classical-relativistic mechanics of two interacting point particles. The calculation is explicitly covariant.