Unifying Quantum Electro-Dynamics and Many-Body Perturbation Theory

Abstract

Very accurate calculations have been performed on highly-charged helium like ions, using second-order QED (one- and two-photon exchange) or many-body perturbation technique (MBPT) with separately added first-order QED energy, and the results are in general in quite good agreement with the experimental results. We have recently developed a computational procedure, where first-order QED is combined with MBPT in a coherent fashion and included in the MBPT wave function, rather than just added to the energy, which leads to higher accuracy. The procedure we have developed is based upon the covariant-evolution-operator, introduced some time ago by us to perform QED calculations on quasi-degererate systems, where the standard S-matrix formalism was not applicable. The evolution operator is made covariant by allowing time to run over all (positive and negative) times at all vertices. Like the standard evolution operator, it becomes (quasi) singular, when an internal, bound state is (quasi)degenerate with a model state. Such degeneracies can be eliminated, leading to what is referred to as the Green’s operator, which is our main tool. With our new procedure we have for the first time evaluated the dominating QED effects beyond second order for a number of highly charged He-like ions, and we have found that these effects, which are small but in many cases quite significant, have been somewhat underestimated in previous works. In evaluating radiative effects (self-energy, vertex correction) we have apart from the Feynman gauge, which is the standard in such calculations, also for the first time used the Coulomb gauge, which led to surprising results. In the Feynman gauge there are huge cancellations between various contributions, which made the calculations numerically more unstable, a phenomenon not appearing in the Coulomb gauge. This has the consequence that meaningful results beyond second order could only be obtained with the Coulomb gauge. The Green’s-operator procedure was primarily intended for work on static systems, energy splittings etc. But we have lately shown that it can equally well be applied to dynamic problems, scattering cross sections, transition rates etc. Replacing the S-matrix—normally used in dynamic problems on free particles—by the Green’s operator, leads to an Optical theorem for bound states, making it possible to apply the same procedure on such systems.