The topic of this thesis is quantum gravity in 1 + 1 dimensions. We will focus
on two formalisms, namely Causal Dynamical Triangulations (CDT) and Dy-
namical Triangulations (DT). Both theories regularize the gravity path integral
as a sum over triangulations. The difference lies in the class of triangulations
considered. While the CDT triangulations have a natural Lorentzian structure,
in DT the triangulations are Euclidean.
The thesis is built up around three papers, reproduced as Chapter 3, 4 and 6.
The outline is as follows: The rst two chapters provides background material
on path integral quantization and the CDT formalism. In Chapter 3 we consider
a generalization of CDT (introduced in Ref. [43]) and show that the continuum
limit is the same as for plain CDT. This provides evidence for the robustness of
the CDT universality class. Chapter 4 provides an analysis of CDT coupled to
Yang-Mills theory. In Chapter 5 we review the DT formalism and some basic
aspects of Liouville Theory. We put special emphasis on some subtleties of the
continuum limit. Finally, Chapter 6 contains a discussion on mixing between
geometrical and matter degrees of freedom, when DT is coupled to non-unitary
CFTs.
Most of the material in Chapter 1, 2 and 5 is not new, and we attempt to
provide relevant references. Chapter 3 and 4 is co-authored with J. Ambjrn,
while Chapter 6 is co-authored with J. Ambjrn, A. Gorlich and H.-G. Zhang.