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Abstract

In this thesis, the theory of the g-factor of bound electrons and muons is presented. For light muonic ions, we include one-loop self-energy as well as one- and two-loop vacuum polarization corrections with the interaction with the strong nuclear potential taken into account to all orders. Furthermore, we include effects due to nuclear structure and mass. We show that our theory for the bound-muon g-factor, combined with possible future bound-muon experiments, can be used to improve the accuracy of the muon mass by one order of magnitude. Alternatively, our approach constitutes an independent access to the controversial anomalous magnetic moment of the free muon. Furthermore, two-loop self-energy corrections to the bound-electron g-factor are investigated theoretically to all orders in the nuclear coupling strength parameter Z α (alpha). Formulas are derived in the framework of the two-time Green's function method, and the separation of divergences is performed by dimensional regularization. Our numerical evaluation by treating the nuclear Coulomb interaction in the intermediate-state propagators to zero and first order show that such two-loop terms are mandatory to take into account in stringent tests of quantum electrodynamics with the bound-electron g-factor, and in projected near-future determinations of fundamental constants.