On a vacuum state redefinition in QED corrections to energy shifts of heavy highly charged ions

Every interesting quantity to be investigated in the realm of bound-state quantum electrodynamics (BSQED), such as, for example, the Lamb shift, the (hyper)fine-splitting or the g-factor, is closely or remotely connected to the energy levels of the considered system. Therefore, as a prerequisite, it is mandatory to have the ability to accurately assess energy levels of increasingly sophisticated electronic configurations of atoms or ions. BSQED predictive powers are nowadays limited to either simple light systems, where an αZ expansion is justified, or heavy few-electron highly-charged ions, where specialized all-order methods in αZ are required, to reliably capture interelectronic interactions. The redefined vacuum state approach, which is frequently employed in the many-body perturbation theory, proved to be a powerful tool allowing analytical insights. This thesis elaborates on this approach within BSQED perturbation theory, based on the two-time Green's function method. In addition to a rather formal formulation, the particular example of a single-particle (electron or hole) excitation with respect to the redefined vacuum state is considered. Starting with simple one-particle Feynman diagrams, characterized by radiative corrections to identical single incoming and single outgoing state, first- and second-order many-electron contributions are derived, namely screened self-energy, screened vacuum-polarization, one-photon-exchange, and two-photon-exchange. The redefined vacuum state approach provides a straightforward and streamlined derivation facilitating its application to any electronic configuration. Moreover, based on the gauge invariance of the one-particle diagrams, various gauge-invariant subsets within analyzed many-electron QED contributions are identified. Building on the gathered expertise, the framework is extended to account for valence-hole excitation from the redefined vacuum state. Full first-order corrections in α are inferred, as a validation of the established formalism, and gauge-invariant subsets are highlighted. To showcase on the strength offered by the use of a vacuum state redefinition, a partial third-order interelectronic calculation is worked out, relying on the extension of the direct two-photonexchange gauge invariant subset. The resulting formulas are cross-checked with an independent perturbative derivation, and the cancellation of infrared divergences is demonstrated explicitly. The identification of gauge-invariant subsets in the framework of the proposed approach opens a way to tackle more complex diagrams, where the decomposition into simpler subsets is crucial for a successful calculation.