  Including nucleon-nucleon correlations due to both Fermi statistics and
nuclear forces, we have developed a general formalism for calculating the
neutral-current neutrino-nucleon opacities in nuclear matter. We derive
corrections to the dynamic structure factors due to both density and spin
correlations and find that neutrino-nucleon cross sections are suppressed by
large factors around and above nuclear density. In addition, we find that the
spectrum of energy transfers in neutrino scattering is considerably broadened
by the interactions in the medium. An identifiable component of this broadening
comes from the absorption and emission of quanta of collective modes akin to
the Gamow-Teller and Giant Dipole resonances in nuclei (zero-sound; spin
waves), with \v{C}erenkov kinematics. Under the assumption that both the
charged-current and the neutral-current cross sections are decreased by
many-body effects, we calculate a set of ad hoc protoneutron star cooling
models to gauge the potential importance of the new opacities to the supernova
itself. We find that after many hundreds of milliseconds to seconds the driving
neutrino luminosities might be increased by from 10% to 100%. However, the
actual consequences, if any, of these new neutrino opacities remain to be
determined.
