REDUCED HAMILTONIANS FOR GRAVITY

In the first paper of this two part series we described a technique fo r passing from any dynamical system whose first order evolution equati ons are known, and whose bracket algebra is not degenerate, to a syste m of canonical variables and a non-zero Hamiltonian that generates the ir evolution. In this paper we advocate using the method to infer a ca nonical formalism, as a prelude to the quantization of gravity. As an example, we construct a reduced gravitational Hamiltonian in perturbat ion theory around a flat background on the manifold T3 x R. The result ing Hamiltonian is positive semidefinite and agrees with the ADM energ y in the limit that deviations from flat space remain localized as the toroidal radii become infinite. We also obtain closed form expression s for the reduced Hamiltonians of two mini-superspace truncations. Alt hough our results are classical they can be formally quantized to give the naive functional formulation of the quantum theory which is non-p erturbative, at least in principle. The marriage we advocate between t he old technique and canonical quantization seems to have profound imp lications for quantum gravity, especially as regards the conservation of energy, statistical mechanics, and the problem of time.