  The rigorous derivation of the Uehling-Uhlenbeck equation from more
fundamental quantum many-particle systems is a challenging open problem in
mathematics. In this paper, we exam the weak coupling limit of quantum
N-particle dynamics. We assume the integral of the microscopic interaction is
zero and we assume W^{4,1} per-particle regularity on the coressponding BBGKY
sequence so that we can rigorously commute limits and integrals. We prove that,
if the BBGKY sequence does converge in some weak sense, then this weak-coupling
limit must satisfy the infinite quantum Maxwell-Boltzmann hierarchy instead of
the expected infinite Uehling-Uhlenbeck hierarchy, regardless of the statistics
the particles obey. Our result indicates that, in order to derive the
Uehling-Uhlenbeck equation, one must work with per-particle regularity bound
below W^{4,1}.
