  We treat quantum back-reaction in time dependent processes for quantum field
theory in various simplified models. The first example is a harmonic oscillator
whose frequency depends on a second quantum variable $x$. Beginning with a
classical analysis, we show how using a particular canonical transformation the
system can be described by an improved adiabatic expansion with a velocity
dependent force for $x$. We find an instability at a critical velocity that
prevents integrating out the oscillator degree of freedom in the new variables.
We extend this calculation to the quantum system and to field theory and
describe how to study fermions with similar techniques. Finally, we set up a
model with an abrupt change in the oscillator whose quantum mechanics can be
solved exactly so that one can study the effects of back-reaction of a fully
non-adiabatic change in a controlled setting. We comment on applications of
these general results to the physics of D-branes, inflation, and black holes in
AdS/CFT.
