  To avoid the computational burden of many-body quantum simulation, the
interaction of an electron with a photon (phonon) is typically accounted for by
disregarding the explicit simulation of the photon (phonon) degree of freedom
and just modelling its effect on the electron dynamics. For quantum models
developed from the (reduced) density matrix or its Wigner-Weyl transformation,
the modelling of collisions may violate complete positivity (precluding the
typical probabilistic interpretation). In this paper, we show that such quantum
transport models can also strongly violate the energy conservation in the
electron-photon (electron-phonon) interactions. After comparing collisions
models to exact results for an electron interacting with a photon, we conclude
that there is no fundamental restriction that prevents a collision model
developed within the (reduced) density matrix or Wigner formalisms to satisfy
simultaneously complete positivity and energy conservation. However, at the
practical level, the development of such satisfactory collision model seems
very complicated. Collision models with an explicit knowledge of the
microscopic state ascribed to each electron seems recommendable, since they
allow to model collisions of each electron individually in a controlled way
satisfying both conditions.
