Gravity is the only one of the four fundamental interactions for which a quantum theory could not be constructed yet, and its quantisation remains one of the most challenging and difficult problems of modern mathematical physics. As it combines mathematical difficulties with deep conceptual questions about the nature of space and time, a full quantum theory of gravity in four dimensions currently does not seem within reach. In three dimensions, the theory simplifies dramatically while presenting the same conceptual challenges as in higher dimensions. 3d-gravity thus serves as a toy model which allows one to investigate these challenges in a mathematically simpler theory amenable to quantisation. However, despite considerable progress, many fundamental questions in 3d-quantum gravity remain open and need to be addressed to achieve a satisfactory quantisation of the theory. Among the most important ones are the relation between different quantisation approaches, the question how geometry emerges in the classical limit and the quantisation of 3d-gravity with Lorentzian signature. The research project aims to address these questions by investigating them in concrete and feasible examples. The first part of the project aims to clarify the relation between the spin foam model for Euclidean 3d-gravity with vanishing cosmological constant (Ponzano-Regge model) and the Hamiltonian quantisation formalism derived from the Chern-Simons formulation of the theory. Its second part is concerned with the classical limit of the theory in both quantisation formalisms and the question of how this limit gives rise to classical geometry. The third part of the project aims to construct a full quantum theory for Lorentzian 3d-gravity with vanishing cosmological constant and to use this theory to address concrete physics questions such as the role of time in the quantised theory and the quantisation of lengths and areas variables.
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