  We derive the evolution equations for a system of neutrinos interacting among
themselves and with a matter background, based upon the
Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. This theoretical
framework gives an (unclosed) set of first-order coupled integro-differential
equations governing the evolution of the reduced density matrices. By employing
the hierarchy, we first rederive the mean-field evolution equations for the
neutrino one-body density matrix associated with a system of neutrinos and
antineutrinos interacting with matter and with an anisotropic neutrino
background. Then, we derive extended evolution equations to determine neutrino
flavor conversion beyond the commonly used mean-field approximation. To this
aim we include neutrino-antineutrino pairing correlations to the two-body
density matrix. The inclusion of these new contributions leads to an extended
evolution equation for the normal neutrino density and to an equation for the
abnormal one involving the pairing mean-field. We discuss the possible impact
of neutrino-antineutrino correlations on neutrino flavor conversion in the
astrophysical and cosmological environments, and possibly upon the supernova
dynamics. Our results can be easily generalized to an arbitrary number of
neutrino families.
