  Building upon the recent pioneering work by Mazenko and by Das and Mazenko,
we develop a microscopic, non-equilibrium, statistical field theory for
initially correlated canonical ensembles of classical microscopic particles
obeying Hamiltonian dynamics. Our primary target is cosmic structure formation,
where initial Gaussian correlations in phase space are believed to be set by
inflation. We give an exact expression for the generating functional of this
theory and work out suitable approximations. We specify the initial
correlations by a power spectrum and derive general expressions for the
correlators of the density and the response field. We derive simple closed
expressions for the lowest-order contributions to the non-linear cosmological
power spectrum, valid for arbitrary wave numbers. We further calculate the
bispectrum expected in this theory within these approximations and the power
spectrum of cosmic density fluctuations to first order in the gravitational
interaction, using a recent improvement of the Zel'dovich approximation. We
show that, with a modification motivated by the adhesion approximation, the
non-linear growth of the density power spectrum found in numerical simulations
of cosmic structure evolution is reproduced well to redshift zero and for
arbitrary wave numbers even within first-order perturbation theory. Our results
present the first fully analytic calculation of the non-linear power spectrum
of cosmic structures.
