Renormierung in der Schleifenquantengravitation

In this thesis, we will investigate the quantisation ambiguities that arise during the canonical quantisation of General Relativity (GR), and we will develop a method of Hamiltonian renormalisation to fix the mentioned ambiguities. ''Quantisation'' is an ansatz to obtain a fundamental (i.e. quantum) theory from a classical description. However, there is no way to ''derive'' this more complex framework if we only know a special case of it, i.e. its classical pendant. Hence, quantisation contains a lot of possible choices and must be supplemented by mathematical consistency and experimental evidence. In the case of gravity, whose effects are described by GR, no experiments are known which reveal properties of its quantum nature. Thus, we must rely purely on mathematical rigour to obtain a version of Quantum Gravity (QG). A promising candidate for this endeavour is ''Loop Quantum Gravity'' (LQG), a modern version of the canonical or Hamiltonian approach. During its development over the past 30 years, it achieved to describe a well-defined canonical quantum field theory. LQG presented a unique Hilbert space representation and the quantisation of constraints as operators acting thereon. During quantisation one must make certain choices by introducing ''regularisation parameters''. However, the details of the operators will be influenced by these choices even in the limit of vanishing regularisation parameters. Their varying physical predictions present an unsatisfactory situation as it is not clear which of those describe the real world. The present work adapts the techniques from the covariant ''renormalisation group'' from Quantum Field Theory (QFT) to LQG. The philosophy of this machinery is that continuum theories should provide a consistent description, no matter with what resolution one looks at the system under consideration. This tool has been used in the context of defining other quantum field theories via the path integral framework. It turned out to be very successful and is, as of today, one of the main tools for studying weak and strong interaction in the Standard Model of particle physics. But, since the mathematical language of canonical QG is vastly different, we will translate the renormalisation group from covariant QFT. Afterwards, we will test it on a simple model, i.e. the massive free scalar field in arbitrary dimensions, and we will present a detailed analysis of it. This includes robustness of the fixed point under different choices of the renormalisation map and the fixed point's range of attraction. Also, we study properties of the discrete projections such as the perfect lattice Laplacian and restoration of continuum symmetries. Finally, we will show the non-trivial impact of quantisation ambiguities in the context of LQG, which can already be seen in cosmological models describing our universe at large scales. Despite these drawbacks, the recent developments in the field allowed to pinpoint the caveats due to their mathematically precise formulation and suggest renormalisation techniques as a possible future improvement. It turns out that the new framework of direct Hamiltonian renormalisation serves as a good candidate to resolve the quantisation ambiguities plaguing QG.