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

Abstract

Despite many attempts, gravity has vigorously resisted a unification with the laws of quantum mechanics. Besides a plethora of technical issues, one is also faced with many interesting conceptual problems. The study of quantum gravity in lower dimensional models ameliorates the technical difficulties while still preserving some of the conceptually fascinating characteristics of quantum gravity. In this thesis we analyze a very simple model of two dimensional quantum gravity. Although a rather extreme simplification of four dimensional quantum gravity, many of the fundamental issues are still relevant. The first fundamental aspect where we make a contribution is the problem of topology change of space. Particularly, we present an exactly solvable model which shows that it is possible to incorporate spatial topology changes in the path integral rigorously. We show that if the change in 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 models. The second conceptual topic we cover is the emergence of geometry from a background independent formulation of quantum gravity. We show that a classical geometry with constant negative curvature emerges from a path integral over noncompact manifolds. No initial singularity is present, so the model naturally is naturally compatible with the Hartle Hawking boundary condition. Furthermore, we demonstrate that under certain conditions the superimposed quantum fluctuations are small! The model is an interesting example where a classical background emerges from background independent quantum gravity. To conclude, we tackle 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 some results that indicate that such a path integral might be consistent, provided suitable causality restrictions are imposed. As a first step we extend the existing formalism of causal dynamical triangulations by a perturbative computation of amplitudes that include manifolds up to genus two. Further a toy model is presented where we make the approximation that the holes in the manifold are infinitesimally small. This simplification allows us perform an explicit sum over all genera and analyze the continuum limit exactly. Remarkably, the presence of the infinitesimal wormholes leads to a decrease in the effective cosmological constant, reminiscent of the suppression mechanism considered by Coleman and others in the four-dimensional Euclidean path integral.