Thelarge cosmological constant approximation to classical andquantum gravity: model examples

Abstract

We have recently introduced an approach for studying perturbativelyclassical and quantum canonical general relativity. The perturbativetechnique appears to preserve many of the attractive features of thenon-perturbative quantization approach based on Ashtekar's newvariables and spin networks. With this approach one can findperturbatively classical observables (quantities that have vanishingPoisson brackets with the constraints) and quantum states (states thatare annihilated by the quantum constraints). The relative ease withwhich the technique appears to deal with these traditionally hardproblems opensup several questions concerning how relevant the resultsproduced can possibly be. Among the questions is the issue of howuseful are results for large values of the cosmological constant andhow the approach can deal with several pathologies that are expectedto be present in the canonical approach to quantum gravity. With theaim of clarifying these points, and to make our construction asexplicit as possible, we study its application in several simplemodels. We consider Bianchi cosmologies, the asymmetric top,coupled harmonic oscillators with constant energy density and a simplequantum mechanical system with two Hamiltonian constraints. We findthat the technique satisfactorily deals with the pathologies of thesemodels and offers promise for finding (at least some) results even forsmall values of the cosmological constant. Finally, we briefly sketchhow the method would operate in the full four-dimensional quantumgeneral relativity case.