  We consider the quantum many-body dynamics at the weak-coupling scaling. We
derive rigorously the quantum Boltzmann equation, which contains the classical
hard sphere model and, effectively, the inverse power law model, from the
many-body dynamics assuming a physical and optimal regularity bound. The
regularity bound we find, on the one hand, is satisfied by quasi-free solutions
and comes from calculations regarding the local Maxwellian solution, in which
we also prove that 2-body molecular chaos never happens unless $N=+\infty$; on
the other hand, it arises from the well-posedness threshold of the limiting
Boltzmann equation below which we prove ill-posedness. That is, the regularity
cannot be higher at the $N$-body level, cannot be lower in the limit, and is
hence a double criticality. To work with this borderline case, we analyze all
four sides, with respect to the Fourier transform, of the BBGKY hierarchy
sequence with new tools and techniques. We prove well-definedness, compactness,
convergence, and uniqueness of hierarchies right at the criticality to complete
an optimal derivation. In particular, we have proved that, for physical
$N$-particle solutions, the Boltzmann equation emerges as the mean-field limit
and time is hence irreversible, from first principles of quantum mechanics.
