On the Low-Energy Ramifications and a Mathematical Extension of Loop Quantum Gravity

In this thesis we address two remaining open questions in loop quantum gravity. The first deals with the low-energy limit of the theory. We illustrate some of the conceptual difficulties and their resolution through the study of a toy model: the quantum mechanics of a point particle. We then find that this model can also be applied to the quantum mechanics of spatially isotropic, homogeneous cosmology within the framework of loop quantum cosmology (LQC). This leads us to extend our results to investigate, for the quantum constraint in LQC, the effective classical dynamics of the quantum theory. We find that we can calculate an effective Hamiltonian constraint, and we employ this to calculate the modifications to Friedmann's equations for a dust filled, spatially flat, isotropic universe. We then turn to a mathematical question, investigating the extension of integration theory on spaces of connections to connections with non-compact structure group. For groups that are the direct product of a compact group with a non-compact Abelian group, we demonstrate a fully satisfactory theory based on the almost periodic compactification of the group. This approach fails for other non-compact groups, and for the case of SL(2, <math> <f> <blkbd>R</blkbd></f> </math>) and SL(2, <math> <f> <blkbd>C</blkbd></f> </math>) we present a partial ‘no-go’ theorem that demonstrates that any successful integration theory for such spaces of connections with these gauge groups will of necessity be different in essential structure from the theory for compact and non-compact, Abelian groups.