Studies of the hyperfine splitting in hydrogen are strongly motivated by the level of accuracy achieved in recent atomic physics experiments, which yield finally model-independent informations about nuclear structure parameters with utmost precision. Considering the current status of the determination of corrections to the hyperfine splitting of the ground state in hydrogen, this thesis provides further improved calculations by taking into account the most recent value for the proton charge radius. Comparing theoretical and experimental data of the hyperfine splitting in hydrogen the proton-size contribution is extracted and a relativistic formula for this contribution is derived in terms of moments of the nuclear charge and magnetization distributions. An iterative scheme for the determination of the Zemach and magnetic radii of the proton is proposed. As a result, the Zemach and magnetic radii are determined and the values are compared with the corresponding ones deduced from data obtained in electron-proton scattering experiments. The extraction of the Zemach radius from a rescaled difference between the hyperfine splitting in hydrogen and in muonium is considered as well. Investigations of forbidden radiative transitions in few-electron ions within ab initio QED provide a most sensitive tool for probing the influence of relativistic electron-correlation and QED corrections to the transition rates. Accordingly, a major part of this thesis is devoted to detailed studies of radiative and interelectronic-interaction effects to the transition probabilities. The renormalized expressions for the corresponding corrections in one- and two-electron ions as well as for ions with one electron over closed shells are derived employing the two-time Green's function method. Numerical results for the correlation corrections to magnetic transition rates in He-like ions are presented. For the first time also the frequency-dependent contribution is calculated, which has to be accounted for preserving gauge invariance. One-loop QED corrections to the magnetic-dipole transition amplitude between the fine-structure levels 2p_{3/2} and 2p_{1/2} are calculated to all orders in \alpha Z. Taking into account consistently relativistic, interelectronic-interaction, and QED corrections to the magnetic-dipole transition amplitude allows for predictions of the lifetimes of the states (1s^2 2s^2 2p)^2P_{3/2} in B-like ions and (1s^2 2s 2p)^3P_2 in Be-like ions with utmost precision. The results of corresponding calculations are compared with experimental data obtained in recent measurements at the Heidelberg EBIT. Finally, for He-like ions with nonzero-spin nuclei the effect of hyperfine quenching on the lifetimes of the 2^3P_{0,2} states is investigated and again compared available experimental data.

Die vorliegende Arbeit beschäftigt sich mit der Bures-Geometrie auf Zustandsräumen über von-Neumann-Algebren. Diese basiert auf jenem Abstandsbegriff für normale Zustände, der von Bures im Jahre 1969 eingeführt wurde. Eng damit verbunden ist der Begriff der algebraischen Übergangswahrscheinlichkeit, der von Uhlmann 1976 vorgeschlagen wurde. An einem Beispiel wird gezeigt, dass man den Bures-Abstand unter Umständen nicht implementieren kann, wenn man einen der implementierenden Vektoren vorgeben will. Im weiteren wird der vom Bures-Abstand induzierte Paralleltransport von Vektoren entlang Loops von normalen Zuständen untersucht. Um die Holonomiegruppe im unendlichdimensionalen Fall zu untersuchen, werden Sätze über Produkte positiver Operatoren hergeleitet. Diese Sätze, die durchaus auch von eigenständigem Interesse sein könnten, werden mit Ergebnissen aus der Literatur verglichen. Schließlich wird der Bures-Abstand unter infinitesimalem Blickwinkel betrachtet. Die so entstehenden Bures-geodätischen Bögen werden untersucht. Speziell wird gefragt, ob gewisse Strata stets geodätisch konvex sind, also als Beispiel für Umgebungen dienen können. Um diese Frage am Ende negativ zu beantworten, werden mehrere Sätze über Sakaische Radon-Nikodym-Operatoren hergeleitet, die auch ohne Bezug zur Bures-Geometrie interessant sein könnten. Das entscheidende Gegenbeispiel nutzt Gohbergs Ergebnis zum Spektrum bestimmter Toeplitzoperatoren aus. Ein Nebeneffekt des beschriebenen Verfahrens ist, dass es auch zur Konstruktion von Operatoren mit hinreichend nichttrivialem Spektrum benutzt werden kann. / The present paper deals with Bures' geometry in the state space over von-Neumann algebras. This geometry is based on the distance introduced by Bures in 1969. Closely related with it is the concept of algebraic transition probability as proposed by Uhlmann in 1976. It is shown by an example that there are cases where one can not implement Bures' distance if one of the implementing vectors is given. In the following, the parallel transport of vectors along loops of normal states, which is induced by Bures' distance, is examined. In order to investigate the holonomy group in the infinite-dimensional case, theorems on products of positive operators are derived. These theorems, which could be of interest on their own, are compared with the literature. Finally, Bures' distance is examined infinitesimally. The thus arising Bures-geodesic arcs are investigated. Especially, it is asked whether certain strata are geodesically convex and can therefore serve as examples of neighbourhoods. In order to finally give a negative answer, several theorems on Sakai's Radon-Nikodym operators, which could also be of interest without a connection to Bures' geometry, are derived. The critical counterexample exploits Gohberg's result on the spectrum of certain Toeplitz operators. A by-product of the described procedure is that it can be used to construct operators which have a sufficiently non-trivial spectrum.

Studies of the hyperfine splitting in hydrogen are strongly motivated by the level of accuracy achieved in recent atomic physics experiments, which yield finally model-independent informations about nuclear structure parameters with utmost precision. Considering the current status of the determination of corrections to the hyperfine splitting of the ground state in hydrogen, this thesis provides further improved calculations by taking into account the most recent value for the proton charge radius. Comparing theoretical and experimental data of the hyperfine splitting in hydrogen the proton-size contribution is extracted and a relativistic formula for this contribution is derived in terms of moments of the nuclear charge and magnetization distributions. An iterative scheme for the determination of the Zemach and magnetic radii of the proton is proposed. As a result, the Zemach and magnetic radii are determined and the values are compared with the corresponding ones deduced from data obtained in electron-proton scattering experiments. The extraction of the Zemach radius from a rescaled difference between the hyperfine splitting in hydrogen and in muonium is considered as well. Investigations of forbidden radiative transitions in few-electron ions within ab initio QED provide a most sensitive tool for probing the influence of relativistic electron-correlation and QED corrections to the transition rates. Accordingly, a major part of this thesis is devoted to detailed studies of radiative and interelectronic-interaction effects to the transition probabilities. The renormalized expressions for the corresponding corrections in one- and two-electron ions as well as for ions with one electron over closed shells are derived employing the two-time Green's function method. Numerical results for the correlation corrections to magnetic transition rates in He-like ions are presented. For the first time also the frequency-dependent contribution is calculated, which has to be accounted for preserving gauge invariance. One-loop QED corrections to the magnetic-dipole transition amplitude between the fine-structure levels 2p_{3/2} and 2p_{1/2} are calculated to all orders in \alpha Z. Taking into account consistently relativistic, interelectronic-interaction, and QED corrections to the magnetic-dipole transition amplitude allows for predictions of the lifetimes of the states (1s^2 2s^2 2p)^2P_{3/2} in B-like ions and (1s^2 2s 2p)^3P_2 in Be-like ions with utmost precision. The results of corresponding calculations are compared with experimental data obtained in recent measurements at the Heidelberg EBIT. Finally, for He-like ions with nonzero-spin nuclei the effect of hyperfine quenching on the lifetimes of the 2^3P_{0,2} states is investigated and again compared available experimental data.