Abstract
A new practical expression for the first-order electron self-energy correction is presented in the coordinate space. It is derived within the adiabatic formalism and decomposed into several parts. For each part a direct renormalization procedure is introduced which avoids the commonly used potential expansion and the Feynman regulated photon propagator. As a simple illustration, the direct renormalization method is applied to terms of the sum over intermediate states in the expansion for the bound electron self-energy of the hydrogen-like ions. It is shown that these renormalized terms give reasonable estimates for the self-energy shift in ground and excited states for different values of the nuclear charge Z including the 1S 1 2 state of the hydrogen-like uranium.
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