Exact Results in Quantum Theory

An important aspect in understanding the dynamics in LQG in the context of deparametrized models is to obtain a sufficient control on the quantum evolution generated by a given Hamiltonian operator. More specifically, we need to be able to compute the evolution of relevant physical states and observables with a relatively good precision. In this talk, I present an approximation method to deal with the physical Hamiltonian operators in the context of (time) deparametrized LQG models, and discuss two known models as examples. This method is using standard time-independent perturbation theory to define a perturbative expansion of a Hamiltonian operator, the small perturbation parameter being determined by the Barbero-Immirzi parameter. This method allows us to approximate transition amplitudes and to evolve expectation values of geometrical operators over a certain time interval.