| The issue of dynamics in a canonical approach to quantum gravity is still an open problem. In this thesis, I explore two different aspects of this dynamics.;In the first part, I compute the causal propagator in the context of canonical loop quantum gravity. For the Lorentzian signature, I find that the resultant power series can be expressed as a sum over branched, colored two-surfaces with an intrinsic causal structure. This leads to the definition of a general structure called "causal spin foam". I then describe some aspects of causal spin foams.;In the second part of the thesis, I present a general procedure for constructing new Hilbert spaces for loop quantum gravity on non-compact spatial manifolds. Given any fixed background state representing a non-compact spatial geometry, I use the Gel'fand-Naimark-Segal construction to obtain a representation of the algebra of observables. The resulting Hilbert space can be interpreted as describing fluctuation of compact support around this background state. I also give an example of a state which approximates classical flat space and can be used as a background state for our construction. |
