In this thesis, we will investigate the transformation of electromagnetic fields under conformal maps. When a conformal map is applied to such a field, the resulting field is again a valid...Show moreIn this thesis, we will investigate the transformation of electromagnetic fields under conformal maps. When a conformal map is applied to such a field, the resulting field is again a valid electromagnetic field. Even when the conformal map is complex, i.e. it mixes real and complex points of space, the resulting field is valid. To better understand complex conformal maps, we introduce Dirac spinors and Twistor space. Using these concepts, we find a nicer expression for a — possibly complex — conformal transformation. This could ease the calculation of the transformed electromagnetic field.Show less
In this thesis we will try to find explicit examples and characterisations of the quantum moduli algebras on ribbon graphs with one vertex. First, we will study the classical case of group gauge...Show moreIn this thesis we will try to find explicit examples and characterisations of the quantum moduli algebras on ribbon graphs with one vertex. First, we will study the classical case of group gauge theory, in which we identify the moduli algebra with the function algebra on the moduli space of flat connections. Secondly, the group gauge theory case will be extended to the group algebra case, for which we show that the quantum moduli algebra is isomorphic to the moduli algebra in the group gauge theory case. Thirdly, we will give a general construction of how to obtain quantum moduli algebras of semisimple finite-dimensional Hopf algebras, and we will identify this construction with the construction of the quantum moduli algebra in the group algebra case. Fourthly, we will be examining the situation in which our Hopf algebra is the Drinfel’d double of a group algebra. After giving some examples, we will show that the quantum moduli algebra in the case of the Drinfel’d double is isomorphic to the quantum moduli algebra in the group algebra case.Show less
The main subject of this thesis is a reformulation of Einstein’s equation. In this reformulation, the variable is not a metric, but a connection on a vector bundle. Nevertheless, we can associate a...Show moreThe main subject of this thesis is a reformulation of Einstein’s equation. In this reformulation, the variable is not a metric, but a connection on a vector bundle. Nevertheless, we can associate a Riemannian metric to a connection. This allows us to relate the new formulation to the usual formulation, i.e. this allows us to argue that the new formulation is in fact a reformulation of Einstein’s equation. Since the physically significant metrics are of Lorentzian signature, we also consider modifying the new formulation in an attempt to make it suitable for Lorentzian metrics.Show less
In this thesis we will investigate the Hopf map, a differentiable map from the three-sphere to the two-sphere. Its fibres, the inverse images of points on the sphere, are circles that are all...Show moreIn this thesis we will investigate the Hopf map, a differentiable map from the three-sphere to the two-sphere. Its fibres, the inverse images of points on the sphere, are circles that are all linked with every other fibre. Based on the Hopf map we will construct divergenceless vector fields that have a physical interpretation as the magnetic field in the theory of magnetohydrodynamics. The concept of linking relates to helicity in this theory, a quantity that will be used to exhibit self-stable configurations of plasma.Show less
