Two-Body Problem in Quantum Field Theory

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Abstract
We study an approach to the relativistic two-body problem that represents a considerable improvement of the Bethe-Salpeter equation. The main advantages are the improved properties of the simple ladder approximation. (a) It has the correct static limit and is in this sense equivalent to the sum of all crossed ladders of the Bethe-Salpeter equation, (b) in the case of massless exchange it is possible to solve analytically and obtain the wave functions and the T matrix in closed form, (c) the system is closed, that is, the two-particle T matrix is unitary. (d) The wave functions admit a complete quantum-mechanical interpretation without compromise of relativistic covariance, the current is conserved, and gauge invariance is respected. (e) The evaluation of physical amplitudes is greatly simplified, as illustrated by positronium decay and the Lamb shift. Both are carried out with full relativistic covariance, and the Lamb-shift calculation in particular is greatly simplified and clarified in comparison with other methods. Most of the results are applicable only to the case of spin-0 or spin-½ particles interacting through the exchange of a single scalar or vector boson. In particular, the calculations of positronium decay and the Lamb shift are carried out with spinless particles. It is shown, however, that the introduction of photon spin presents no problem and that the Lamb shift can be calculated in a fully gauge-invariant manner.