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

TL;DR

This thesis explores a solvable two-dimensional quantum gravity model using causal dynamical triangulations, demonstrating topology change, emergent classical geometry, and potential consistency of topology fluctuations under causality constraints.

Contribution

It introduces an exactly solvable model incorporating topology change in 2D quantum gravity and shows how classical geometry emerges from quantum fluctuations.

Findings

Topology change can be included in the nonperturbative path integral.

Classical negatively curved geometry emerges from quantum path integrals.

Quantum fluctuations are small under certain conditions.

Abstract

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.