Asymptotic Analysis in Relativistic Quantum Physics

This project deals with the Pauli equation, an extension of the Schrödinger equation that takes magnetic elds and spin into account. We deal with nonlinear time-dependent equations and the question of how the self-interaction with the electromagnetic elds generated by the moving charge can be modeled in a meaningful way. Furthermore, we consider the analysis of these models, i.e. questions of the existence of unique solutions for short or long times and the semiclassical limit from quantum mechanics to classical mechanics. The two-year phase abroad is carried out with T. Hou at Caltech (USA) and F. Golse at the École Polytechnique (France), supplemented by the international partners P. Germain (Imperial College London), Z. Zhou (Peking University) and Norbert J Mauser (Wolfgang Pauli Institute Vienna), where the return phase is located. Quantum mechanics, discovered in the 1920s, can be combined comparatively easily with special relativity to form relativistic quantum mechanics. The central equations are, for example, the relativistic Dirac equation for a particle with spin 1/2 and its antiparticle, the non-relativistic Schrödinger equation and the semi-relativistic Pauli equation. The interaction with the electromagnetic eld (generated by the moving charge) is described by Maxwell`s equations, which were discovered in 1861. They are "Lorentz-covariant", and therefore relativistic. It is therefore not clear whether the interaction of a non-relativistic particle, which obeys the Schrödinger equation, with the eld generated by itself can simply take place through Maxwell`s equations. The natural coupling here is that of the Dirac equation with Maxwell`s equation. In the "non-relativistic limit", which examines the case when the typical speed of the system becomes smaller and smaller in relation to the speed of light, it can be shown that the Dirac- Maxwell equation becomes the Schrödinger-Poisson equation, whereby the Poisson equation only describes the electrostatic interaction (without magnetic eld and spin). The so-called "semiclassical limit", on the other hand, investigates what happens when the system becomes "classical", i.e. is considered in the context of ever larger systems and on everyday length scales (this corresponds to the vanishing Planck`s constant). Quantum e ects can then be neglected. This project aims not only to answer open questions (e.g. what the Dirac-Maxwell equation converges to in the semiclassical limit), but also to identify the physically meaningful models and equations. Therefore, the project is equally dedicated to mathematics and physics. Furthermore, new mathematical methods for these questions will be developed (e.g. the "averaging lemmas" in collaboration with F. Golse). Part of the project is dedicated to numerical simulation.