  We have recently introduced an approach for studying perturbatively classical
and quantum canonical general relativity. The perturbative technique appears to
preserve many of the attractive features of the non-perturbative quantization
approach based on Ashtekar's new variables and spin networks. With this
approach one can find perturbatively classical observables (quantities that
have vanishing Poisson brackets with the constraints) and quantum states
(states that are annihilated by the quantum constraints). The relative ease
with which the technique appears to deal with these traditionally hard problems
opens several questions about how relevant the results produced can possibly
be. Among the questions is the issue of how useful are results for large values
of the cosmological constant and how the approach can deal with several
pathologies that are expected to be present in the canonical approach to
quantum gravity. With the aim of clarifying these points, and to make our
construction as explicit as possible, we study its application in several
simple models. We consider Bianchi cosmologies, the asymmetric top, the coupled
harmonic oscillators with constant energy density and a simple quantum
mechanical system with two Hamiltonian constraints. We find that the technique
satisfactorily deals with the pathologies of these models and offers promise
for finding (at least some) results even for small values of the cosmological
constant. Finally, we briefly sketch how the method would operate in the full
four dimensional quantum general relativity case.
