  We address the dynamics of a bosonic system coupled to either a bosonic or a
magnetic environment, and derive a set of sufficient conditions that allow one
to describe the dynamics in terms of the effective interaction with a classical
fluctuating field. We find that for short interaction times the dynamics of the
open system is described by a Gaussian noise map for several different
interaction models and independently on the temperature of the environment. In
order to go beyond a qualitative understanding of the origin and physical
meaning of the above short-time constraint, we take a general viewpoint and,
based on an algebraic approach, suggest that any quantum environment can be
described by classical fields whenever global symmetries lead to the definition
of environmental operators that remain well defined when increasing the size,
i.e. the number of dynamical variables, of the environment. In the case of the
bosonic environment this statement is exactly demonstrated via a constructive
procedure that explicitly shows why a large number of environmental dynamical
variables and, necessarily, global symmetries, entail the set of conditions
derived in the first part of the work.
