  Recent body of work points out that the mean-field approximation, widely
employed to mimic the neutrino field within a neutrino-dense source, might give
different results in terms of flavor evolution with respect to the
correspondent many-body treatment. In this paper, we investigate whether such
conclusions derived within a constrained framework should hold in an
astrophysical context. We show that the plane waves, commonly adopted in the
many-body literature to model the neutrino field, provide results that are
crucially different with respect to the ones obtained using wavepackets of
finite size streaming with a non-zero velocity. The many-body approach
intrinsically includes coherent and incoherent scatterings. The mean-field
approximation, on the other hand, only takes into account the coherent
scattering in the absence of the collision term. Even if incoherent scatterings
are included in the mean-field approach, the nature of the collision term is
different from that in the many-body approach. Because of this, if only a
finite number of neutrinos is considered, as often assumed, the two approaches
naturally lead to different flavor outcomes. These differences are further
exacerbated by vacuum mixing. We conclude that existing many-body literature,
based on closed neutrino systems with a finite number of particles, is neither
able to rule out nor assess the validity of the mean-field approach adopted to
simulate the evolution of the neutrino field in dense astrophysical sources,
which are open systems.
