  The momentum dependence of the nucleon mean-field potential in a wide
momentum range can be an important factor to determine the $\Delta$ resonance
and pion production in intermediate-energy heavy-ion collisions. In particular,
in neutron-rich systems such as ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$
collisions, we need to carefully treat the momentum dependence because the
neutron and proton potentials can have different momentum dependence, as
characterized at low momenta by effective masses. In the present work, we
rigorously calculate the collision terms of $NN \leftrightarrow N \Delta$ and
$\Delta \leftrightarrow N \pi$ processes with the precise conservation of
energy and momentum under the presence of momentum-dependent potentials for the
initial and final particles of the process. The potentials affect not only the
threshold condition for the process but also the cross section in general as a
function of the momenta of the initial particles, which is treated in a natural
way in the present work. Calculations are performed by combining the nucleon
dynamics obtained by the antisymmetrized molecular dynamics (AMD) model with a
newly developed transport code which we call sJAM. The calculated results for
central ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ collisions at 270 MeV/nucleon
clearly show that the momentum dependence of the neutron and proton potentials
has a significant impact on the $NN \to N \Delta$ process, and this information
is strongly reflected in the charged pion ratio ($\pi^-/\pi^+$). We also
investigate the effects of the high-density symmetry energy and the isovector
part of the potential of $\Delta$ resonances on pion production, which we find
are relatively small compared to the effect of the momentum dependence of the
neutron and proton potentials.
