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On relativistic interaction of electric charges and external fields in quantum electrodynamics
On relativistic interaction of electric charges and external fields in quantum electrodynamics
The main subject of this thesis is the problem of introducing interactions into relativistic quantum mechanics. This problem has many facets, two of which will be discussed. The first one deals with a recent relativistically invariant integral equation for multi-time wave functions by Lienert. From a mathematical point of view this proposal is promising, since variants of it have been shown to be mathematically well-defined. In this thesis, firstly, previous results on existence and uniqueness of solutions of a variant of this equation for scalar particles are extended to include more realistic types of interaction. Secondly, a proof of existence and uniqueness of solutions of another variant that allows to treat spin 1/2 particles is provided. The second facet concerns interactions in the context of a variable number of particles. Following famous works of Dirac, Feynman and Schwinger, we treat external electrodynamic fields in an otherwise free Quantum Field Theory of electrons. In previous results, candidates for the time evolution operator have been constructed in this setting. This construction is unique up to a phase, which may depend on the external field. This phase affects the charge current density and should thus be identified. In this work, this problem is addressed by a geometric, which was inspired by and developed jointly with my supervisors, construction assuming a certain causality condition. Secondly, a compact formula for the scattering operator in terms of the corresponding one-particle scattering operator is provided and shown to be well-defined, assuming certain conditions on the external field. This formula is used to show that the second quantized scattering operator is an analytic function of the external field in a certain sense.
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Nöth, Markus Hartmut
2021
English
Universitätsbibliothek der Ludwig-Maximilians-Universität München
Nöth, Markus Hartmut (2021): On relativistic interaction of electric charges and external fields in quantum electrodynamics. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

The main subject of this thesis is the problem of introducing interactions into relativistic quantum mechanics. This problem has many facets, two of which will be discussed. The first one deals with a recent relativistically invariant integral equation for multi-time wave functions by Lienert. From a mathematical point of view this proposal is promising, since variants of it have been shown to be mathematically well-defined. In this thesis, firstly, previous results on existence and uniqueness of solutions of a variant of this equation for scalar particles are extended to include more realistic types of interaction. Secondly, a proof of existence and uniqueness of solutions of another variant that allows to treat spin 1/2 particles is provided. The second facet concerns interactions in the context of a variable number of particles. Following famous works of Dirac, Feynman and Schwinger, we treat external electrodynamic fields in an otherwise free Quantum Field Theory of electrons. In previous results, candidates for the time evolution operator have been constructed in this setting. This construction is unique up to a phase, which may depend on the external field. This phase affects the charge current density and should thus be identified. In this work, this problem is addressed by a geometric, which was inspired by and developed jointly with my supervisors, construction assuming a certain causality condition. Secondly, a compact formula for the scattering operator in terms of the corresponding one-particle scattering operator is provided and shown to be well-defined, assuming certain conditions on the external field. This formula is used to show that the second quantized scattering operator is an analytic function of the external field in a certain sense.