  A quantum linear Boltzmann equation is proposed, constructed in terms of the
operator-valued dynamic structure factor of the macroscopic system the test
particle is interacting with. Due to this operator structure it is a
non-Abelian linear Boltzmann equation and when expressed through the Wigner
function it allows for a direct comparison with the classical one. Considering
a Brownian particle the corresponding Fokker-Planck equation is obtained in a
most direct way taking the limit of small energy and momentum transfer. A
typically quantum correction to the Kramers equation thus appears, describing
diffusion in position and further implying a correction to Einstein's diffusion
coefficient in the high temperature and friction limit in which Smoluchowski
equation emerges.
