  We use the stochastic limit method to study long time quantum dynamics of a
test particle interacting with a dilute Bose gas. The case of arbitrary
form-factors and an arbitrary, not necessarily equilibrium, quasifree low
density state of the Bose gas is considered. Starting from microscopic dynamics
we derive in the low density limit a quantum white noise equation for the
evolution operator. This equation is equivalent to a quantum stochastic
equation driven by a quantum Poisson process with intensity $S-1$, where $S$ is
the one-particle $S$ matrix. The novelty of our approach is that the equations
are derived directly in terms of correlators, without use of a Fock-antiFock
(or Gel'fand-Naimark-Segal) representation. Advantages of our approach are the
simplicity of derivation of the limiting equation and that the algebra of the
master fields and the Ito table do not depend on the initial state of the Bose
gas. The notion of a causal state is introduced. We construct master fields
(white noise and number operators) describing the dynamics in the low density
limit and prove the convergence of chronological (causal) correlators of the
field operators to correlators of the master fields in the causal state.
