The large cosmological constant approximation to classical and quantum gravity: model examples

We have recently introduced an approach for studying perturbatively classic al and quantum canonical general relativity. The perturbative technique app ears to preserve many of the attractive features of the non-perturbative qu antization approach based on Ashtekar's new variables and spin networks. Wi th this approach one can find perturbatively classical observables (quantit ies that have vanishing Poisson brackets with the constraints) and quantum states (states that are annihilated by the quantum constraints). The relati ve ease with which the technique appears to deal with these traditionally h ard problems opens up several questions concerning how relevant the results produced can possibly be. Among the questions is the issue of how useful a re results for large values of the cosmological constant and how the approa ch can deal with several pathologies that are expected to be present in the canonical approach to quantum gravity. With the aim of clarifying these po ints, and to make our construction as explicit as possible, we study its ap plication in several simple models. We consider Bianchi cosmologies, the as ymmetric top, coupled harmonic oscillators with constant energy density and a simple quantum mechanical system with two Hamiltonian constraints. We fi nd that the technique satisfactorily deals with the pathologies of these mo dels and offers promise for finding (at least some) results even for small values of the cosmological constant. Finally, we briefly sketch how the met hod would operate in the full four-dimensional quantum general relativity c ase.