  The equation for the quantum motion of a Brownian particle in a gaseous
environment is derived by means of S-matrix theory. This quantum version of the
linear Boltzmann equation accounts non-perturbatively for the quantum effects
of the scattering dynamics and describes decoherence and dissipation in a
unified framework. As a completely positive master equation it incorporates
both the known equation for an infinitely massive Brownian particle and the
classical linear Boltzmann equation as limiting cases.
