Order 1/N corrections to the time-dependent Hartree approximation for a system of N+1 oscillators

We solve numerically to order 1/N the time evolution of a quantum dynamical system of N oscillators of mass m coupled quadratically to a massless dynamic variable. We use Schwinger's closed time path (CTP) formalism to derive the equations. We compare two methods which differ by terms of order 1/N^2. The first method is a direct perturbation theory in 1/N using the path integral. The second solves exactly the theory defined by the effective action to order 1/N. We compare the results of both methods as a function of N. At N=1, where we expect the expansion to be quite innacurate, we compare our results to an exact numerical solution of the Schroedinger equation. In this case we find that when the two methods disagree they also diverge from the exact answer. We also find at N=1 that the 1/N corrected evolutions track the exact answer for the expectation values much longer than the mean field (N= \infty) result.