Unendlicher Panpsychismus. Kraft und Substanz in der Philosophie des Individuums von Leibniz.

Mit dem Kraftbegriff behandelt die vorliegende Dissertation ein Schlüsselthema der Leibnizschen Philosophie des Individuums. Zwei zentrale und seit jeher weitgehend getrennt geführte Interpretationsdiskurse über Leibniz werden anhand der differenzierten Analyse dieses metaphysischen Kernbegriffs zusammengeführt. Dies ist zum einen der Diskurs über die Leibnizsche Metaphysik und Monadenlehre und zum anderen der über die physikalische Dynamik und die Prinzipien seiner naturwissenschaftlichen Auffassung. Der Begriff der Kraft wird von Leibniz als ein grundlegendes Konstitutivum verwendet, das in den verschiedenen Wirklichkeitsbereichen eine spezifische Auslegung in entsprechende ontologische Charakteristika erfährt. Aus diesem Grund ist dieser Begriff der methodische Ausgangspunkt zur Untersuchung des Leibnizschen Individuums oder der Monade, wo seelische Wirklichkeit, Bewusstsein und Materialität gleichermaßen metaphysisch und ontologisch begründet werden. Das Verhältnis von Körper und Seele bei Leibniz, sein naturwissenschaftlicher Denkansatz und ein spezifischer Begriff von Freiheit erhält von hier aus seinen Sinn. Denn Leibniz versteht die körperhafte Verfasstheit oder die Materialität des Individuums ebenso als dessen essentiellen Bestandteil wie das Seelische in Form von Perzeption und Apperzeption. Die Monade zeichnet sich also sowohl durch eine interne als auch externe Aktivität aus. Dies hat insbesondere die Konsequenz, dass für Leibniz der organische Körper und die mit diesem gegebene Lebendigkeit, einmal geschaffen, so unvergänglich ist wie die mit diesem geschaffene seelische Wirklichkeit. In der Konsequenz führt dies Leibniz darauf, das Universum als eine bis ins Unendliche gehende Verschachtelung von in einem Nexus stehenden organischen Individuen zu interpretieren. Das Universum wird begreifbar als eine unendlich panorganische und panpsychische Welt.

Use of computer algebra in the calculation of Feynman diagrams

The increasing precision of current and future experiments in high-energy physics requires a likewise increase in the accuracy of the calculation of theoretical predictions, in order to find evidence for possible deviations of the generally accepted Standard Model of elementary particles and interactions. Calculating the experimentally measurable cross sections of scattering and decay processes to a higher accuracy directly translates into including higher order radiative corrections in the calculation. The large number of particles and interactions in the full Standard Model results in an exponentially growing number of Feynman diagrams contributing to any given process in higher orders. Additionally, the appearance of multiple independent mass scales makes even the calculation of single diagrams non-trivial. For over two decades now, the only way to cope with these issues has been to rely on the assistance of computers. The aim of the xloops project is to provide the necessary tools to automate the calculation procedures as far as possible, including the generation of the contributing diagrams and the evaluation of the resulting Feynman integrals. The latter is based on the techniques developed in Mainz for solving one- and two-loop diagrams in a general and systematic way using parallel/orthogonal space methods. These techniques involve a considerable amount of symbolic computations. During the development of xloops it was found that conventional computer algebra systems were not a suitable implementation environment. For this reason, a new system called GiNaC has been created, which allows the development of large-scale symbolic applications in an object-oriented fashion within the C++ programming language. This system, which is now also in use for other projects besides xloops, is the main focus of this thesis. The implementation of GiNaC as a C++ library sets it apart from other algebraic systems. Our results prove that a highly efficient symbolic manipulator can be designed in an object-oriented way, and that having a very fine granularity of objects is also feasible. The xloops-related parts of this work consist of a new implementation, based on GiNaC, of functions for calculating one-loop Feynman integrals that already existed in the original xloops program, as well as the addition of supplementary modules belonging to the interface between the library of integral functions and the diagram generator.

Existence of solutions and asymtptotic limits of the Euler-Poisson and the quantum drift-diffusion systems

My work concerns two different systems of equations used in the mathematical modeling of semiconductors and plasmas: the Euler-Poisson system and the quantum drift-diffusion system. The first is given by the Euler equations for the conservation of mass and momentum, with a Poisson equation for the electrostatic potential. The second one takes into account the physical effects due to the smallness of the devices (quantum effects). It is a simple extension of the classical drift-diffusion model which consists of two continuity equations for the charge densities, with a Poisson equation for the electrostatic potential. Using an asymptotic expansion method, we study (in the steady-state case for a potential flow) the limit to zero of the three physical parameters which arise in the Euler-Poisson system: the electron mass, the relaxation time and the Debye length. For each limit, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimates. For a vanishing electron mass or a vanishing relaxation time, this method gives us a new approach in the convergence of the Euler-Poisson system to the incompressible Euler equations. For a vanishing Debye length (also called quasineutral limit), we obtain a new approach in the existence of solutions when boundary layers can appear (i.e. when no compatibility condition is assumed). Moreover, using an iterative method, and a finite volume scheme or a penalized mixed finite volume scheme, we numerically show the smallness condition on the electron mass needed in the existence of solutions to the system, condition which has already been shown in the literature. In the quantum drift-diffusion model for the transient bipolar case in one-space dimension, we show, by using a time discretization and energy estimates, the existence of solutions (for a general doping profile). We also prove rigorously the quasineutral limit (for a vanishing doping profile). Finally, using a new time discretization and an algorithmic construction of entropies, we prove some regularity properties for the solutions of the equation obtained in the quasineutral limit (for a vanishing pressure). This new regularity permits us to prove the positivity of solutions to this equation for at least times large enough.