The covariant-evolution-operator method in bound-state QED

The methods of quantum-electrodynamical (QED) calculations on bound atomic systems are reviewed with emphasis on the newly developed covariant-evolution-operator method. The aim is to compare that method with other available methods and also to point out possibilities to combine that with standard many-body perturbation theory (MBPT) in order to perform accurate numerical QED calculations, including quasi-degeneracy, also for light elements, where the electron correlation is relatively strong. ackground, the time-independent many-body perturbation theory (MBPT) is briefly reviewed, particularly the method with extended model space. Time-dependent perturbation theory is discussed in some detail, introducing the time-evolution operator and the Gell–Mann–Low relation, generalized to an arbitrary model space. Three methods of treating the bound-state QED problem are discussed. The standard S-matrix formulation, which is restricted to a degenerate model space, is discussed only briefly. Two methods applicable also to the quasi-degenerate problem are treated in more detail, the two-times Greenʹs-function and the covariant-evolution-operator techniques. The treatment is concentrated on the latter technique, which has been developed more recently and which has not been discussed in more detail before. A comparison of the two-times Greenʹs-function and the covariant-evolution-operator techniques, which have great similarities, is performed. In the appendix a simple procedure is derived for expressing the evolution-operator diagrams of arbitrary order. ssibilities of merging QED in the covariant evolution-operator formulation with MBPT in a systematic way is indicated. With such a technique it might be feasible to perform accurate QED calculations also on light elements, which is presently not possible with the techniques available.