  The white noise approach to the investigation of the dynamics of a quantum
particle interacting with a dilute and in general non-equilibrium gaseous
environment in the low density limit is outlined. The low density limit is the
kinetic Markovian regime when only pair collisions (i.e., collisions of the
test particle with one particle of the gas at one time moment) contribute to
the dynamics. In the white noise approach one first proves that the appropriate
operators describing the gas converge in the sense of appropriate matrix
elements to certain operators of quantum white noise. Then these white noise
operators are used to derive quantum white noise and quantum stochastic
equations describing the approximate dynamics of the total system consisting of
the particle and the gas. The derivation is given ab initio, starting from the
exact microscopic quantum dynamics. The limiting dynamics is described by a
quantum stochastic equation driven by a quantum Poisson process. This equation
then applied to the derivation of quantum Langevin equation and linear
Boltzmann equation for the reduced density matrix of the test particle. The
first part of the paper describes the approach which was developed by L.
Accardi, I.V. Volovich and the author and uses the Fock-antiFock (or GNS)
representation for the CCR algebra of the gas. The second part presents the
approach to the derivation of the limiting equations directly in terms of the
correlation functions, without use of the Fock-antiFock representation. This
approach simplifies the derivation and allows to express the strength of the
quantum number process directly in terms of the one-particle $S$-matrix.
