  It is shown that the vacuum state of weakly interacting quantum field
theories can be described, in the Heisenberg picture, as a linear combination
of randomly distributed incoherent paths that obey classical equations of
motion with constrained initial conditions. We call such paths
"pseudoclassical" paths and use them to define the dynamics of quantum
fluctuations. Every physical observable is assigned a time-dependent value on
each path in a way that respects the uncertainty principle, but in consequence,
some of the standard algebraic relations between quantum observables are not
necessarily fulfilled by their time-dependent values on paths. We define
"collective observables" which depend on a large number of independent degrees
of freedom, and show that the dynamics of their quantum fluctuations can be
described in terms of unconstrained classical stochastic processes without
reference to any additional external system or to an environment. Our analysis
can be generalized to states other than the vacuum. Finally, we compare our
formalism to the formalism of coherent states, and highlight their differences.
