  An important aspect in understanding the dynamics in the context of
deparametrized models of LQG 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 article, we introduce an
approximation method to deal with the physical Hamiltonian operators in
deparametrized LQG models, and apply it to models in which a free Klein-Gordon
scalar field or a non-rotational dust field is taken as the physical time
variable. This method is based on using standard time-independent perturbation
theory of quantum mechanics to define a perturbative expansion of the
Hamiltonian operator, the small perturbation parameter being determined by the
Barbero-Immirzi parameter $\beta$. This method allows us to define an
approximate spectral decomposition of the Hamiltonian operators and hence to
compute the evolution over a certain time interval. As a specific example, we
analyze the evolution of expectation values of the volume and curvature
operators starting with certain physical initial states, using both the
perturbative method and a straightforward expansion of the expectation value in
powers of the time variable. This work represents a first step towards
achieving the goal of understanding and controlling the new dynamics developed
in [25, 26].
