  Historically it has been believed that a time-independent classical source
has no effect on the scattering of relativistic uncharged field, in contrast
with single particle quantum mechanics. In this work we show that the dynamics
is not trivial. We solve exactly for the scattering amplitudes and find that a
key ingredient is the production of particles from the unstable vacuum,
conceptually similar to the Schwinger mechanism. We compute exactly the
probabilities for the vacuum to decay in $n$ particles. The time dependence of
such probabilities displays interesting properties such as the quantum Zeno
effect and in particular has no regime where the exponential decay law is a
good approximation. We show that the trivial scattering found in the past is
the byproduct of the adiabatic switching of the interaction. In fact, it is not
possible to switch off the interaction (adiabatically or otherwise) at distant
times and recover the exact results. Finally, this non trivial vacuum behavior
is a source of particle production. We argue that such non-perturbative
calculations can be phenomenologically relevant for the production processes
that are suppressed at the lower orders in perturbation theory, for instance
dilaton production in a medium.
