The Covariant Evolution Operator and the Green’s-Operator Method

The third method we shall consider for numerical QED calculations on bound states is the Covariant-Evolution operator (CEO), developed during the last decade by the Gothenburg group (Lindgren et al., Phys. Rep. 389, 161–261, 2004) [130], (Lindgren et al., Can. J. Phys. 83, 183–218, 2005) [131], (Lindgren et al., Phys. Rev. A 73, 062–502, 2006) [132]. This procedure is based upon the non-relativistic time-evolution operator, discussed in Chap. 3, but made relativistically covariant in order to be applicable in relativistic calculations. This method has the advantage over the two methods discussed previously, the S-matrix procedure and the Green’s-function procedure, that it can be used perturbatively, and the perturbations can be included in the wave function—not only added to the energy. It then forms a convenient basis for a covariant relativistic many-body perturbation procedure, where electron correlation and quantum electrodynamics are systematically combined. For two-electron systems this is fully compatible with the Bethe–Salpeter equation. This question will be the main topic for the rest of the book.