  We investigate the possibility that the semiclassical limit of quantum
mechanics might be correctly described by a classical dynamical theory, other
than standard classical mechanics. Using a set of classicality criteria
proposed in a related paper, we show that the time evolution of a set of
quantum initial data satisfying these criteria is fully consistent with the
predictions of a new theory of classical dynamics. The dynamical structure of
the new theory is given by the Moyal bracket. This is a Lie bracket that was
first derived as the dynamical structure of the Moyal-Weyl-Wigner formulation
of quantum mechanics. We present a new derivation of the Moyal bracket, this
time in the context of the semiclassical limit of quantum mechanics and thus
prove that both classical and quantum dynamics can be formulated in terms of
the same canonical structure.
