Bachelor thesis – Ludwig Peschik
Ludwig Peschik
Algebraic Properties of Quadrics over Finite Fields and their Symmetry Groups
[PDF]
finished 2019-02
supervised by Klaus Mecke
Abstract
Since its publication Einstein’s General Relativity has been one of the most influential classical works in the past century. It combines properties of the considered matter with the geometry of the background spacetime upon the matter exists. A few decades later the standard model of particle physics grew in acceptance due to its experimental verification which showed that a quantization of matter in the context of quantum field theory is necessary to describe all the fundamental interactions between particles on very tiny scales. Therefore, in the last decades scientist have been trying to combine the quantum nature of matter with the classical theory of general relativity. In a new approach by Klaus Mecke the field of real numbers is discarded as the under- lying number field of the theory in favour of a finite field and a projective space equipped with a biquadric is used instead of a Lorentzian spacetime manifold. Herein, the four-dimensionality of the directly observable universe and the signature of the Minkwoski metric come up quite naturally. In this thesis we want to study some algebraic properties of the standard quadric used in this new approach and, in particular, of the symmetry group of this quadric. This symmetry group is in close relation to the Lorentz group used in special relativity but we will see that it has some features which come up by using a finite field and a projective space instead of real numbers and a smooth manifold, respectively. In particular, we will discuss the application of the Cartan-Dieudonné theorem which describes a decomposition of the elements of such symmetry groups into reflections whose geometrical interpretation is a lot easier than the one of a general element of the symmetry group. At the end we also want to sketch an idea of finding a connection between the structure of the quadric and a field extension of the finite field.
Since its publication Einstein’s General Relativity has been one of the most influential classical works in the past century. It combines properties of the considered matter with the geometry of the background spacetime upon the matter exists. A few decades later the standard model of particle physics grew in acceptance due to its experimental verification which showed that a quantization of matter in the context of quantum field theory is necessary to describe all the fundamental interactions between particles on very tiny scales. Therefore, in the last decades scientist have been trying to combine the quantum nature of matter with the classical theory of general relativity. In a new approach by Klaus Mecke the field of real numbers is discarded as the under- lying number field of the theory in favour of a finite field and a projective space equipped with a biquadric is used instead of a Lorentzian spacetime manifold. Herein, the four-dimensionality of the directly observable universe and the signature of the Minkwoski metric come up quite naturally. In this thesis we want to study some algebraic properties of the standard quadric used in this new approach and, in particular, of the symmetry group of this quadric. This symmetry group is in close relation to the Lorentz group used in special relativity but we will see that it has some features which come up by using a finite field and a projective space instead of real numbers and a smooth manifold, respectively. In particular, we will discuss the application of the Cartan-Dieudonné theorem which describes a decomposition of the elements of such symmetry groups into reflections whose geometrical interpretation is a lot easier than the one of a general element of the symmetry group. At the end we also want to sketch an idea of finding a connection between the structure of the quadric and a field extension of the finite field.