Treatment of atomic quasi-degeneracy in bound-state QED. Application to the 1s2p states of He-like ions

Abstract

In the case of quasi-degeneracy, i.e., when there are states close in energy to the state considered, standard perturbation theory may lead to convergence problem. A typical example is the relativistic treatment of the fine structure of light heliumlike ions. In Many-BodyPerturbation Theory (MBPT) this problem can be treated by including the quasi-degenerate states in an extended model space. The mixing is then treated to all orders already in the zeroth-order wave function by diagonalizing an effective hamiltonian in this subspace. Only non-quasi-degenerate states are treated by perturbation.Standard QED treatment of bound systems is based upon the S -matrix formalism. Here an extended model space cannot be used, the reason being that only matrix elements diagonal in energy can be determined, while the effective hamiltonian needed contains also non-diagonal elements between the states in the model space. In the present paper a modification of the bound-state QED procedure is described, based upon a covariant form of the time-evolution operator, rather than the S -matrix. In this way also elements non-diagonal in energy can be evaluated. The operator appearing in the final procedure is free from singularities, and no special procedure is needed for evaluating the so-called Model-Space Contribution, caused by model-space states appearing as the intermediate states.The procedure has been applied to the fine structure of helium-like ions, and numerical second-order QED results, including quasi-degenerate levels, are presented for the first time. The formalism is closely related to MBPT and may open the possibility to combine QED and MBPT in a more systematic fashion.