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 present Dijkgraaf-Witten theory. We start by considering a two-dimensional topological quantum field theory that can be used to prove Mednykh’s formula along the way. Subsequently...Show moreIn this thesis we present Dijkgraaf-Witten theory. We start by considering a two-dimensional topological quantum field theory that can be used to prove Mednykh’s formula along the way. Subsequently, we define the Dijkgraaf-Witten invariant as a partition function where we assign a specified weight to the 3-simplices of a compact, oriented and triangulated 3-manifold with boundary. The partition function depends on how we assign elements of a finite, discrete group G to all the oriented edges of the manifold. We prove that, whenever the triangulation of the boundary is fixed, the invariant does not depend on the triangulation of the manifold. Finally, we define a similar invariant where we model the weight of the 3-simplices to mimic the action of Chern-Simons theory. We demonstrate that by demanding invariance, we obtain the Dijkgraaf-Witten invariant.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