Topology Change and the Emergence of Geometry in Two Dimensional Causal Quantum Gravity

In this thesis we analyze a very simple model of two dimensional quantum gravity based on causal dynamical triangulations (CDT). We present an exactly solvable model which indicates that it is possible to incorporate spatial topology changes in the nonperturbative path integral. It is shown that if the change in spatial topology is accompanied by a coupling constant it is possible to evaluate the path integral to all orders in the coupling and that the result can be viewed as a hybrid between causal and Euclidian dynamical triangulation. The second model we describe shows how a classical geometry with constant negative curvature emerges naturally from a path integral over noncompact manifolds. No initial singularity is present, hence the quantum geometry is naturally compatible with the Hartle Hawking boundary condition. Furthermore, we demonstrate that under certain conditions the quantum fluctuations are small! To conclude, we treat the problem of spacetime topology change. Although we are not able to completely solve the path integral over all manifolds with arbitrary topology, we do obtain results that indicate that such a path integral might be consistent, provided suitable causality restrictions are imposed. How to read this thesis? After completion of this thesis considerable progress has been made and published in [1] and [2]. People interested in [1] and [2] can use this thesis as an introduction to the subject. The first chapter is written for a very general audience and can be skipped by most readers. In reference [1] spatial and spacetime topology change in two dimensional causal quantum gravity are formalized in the form of a string field theory. Subsequently, in [2] we uncovered a matrix model underlying the string field theory. In a forthcoming publication we show why the continuum amplitudes can be obtained from a matrix model. One will see that it emerges from a new continuum limit of the one matrix model.