  We use the in-in or Schwinger-Keldysh formalism to explore the construction
and interpretation of effective field theories for time-dependent systems
evolving out of equilibrium. Starting with a simple model consisting of a heavy
and a light scalar field taken to be in their free vacuum states at a finite
initial time, we study the effects from the heavy field on the dynamics of the
light field by analyzing the equation of motion for the expectation value of
the light background field. New terms appear which cannot arise from a local
action of an effective field theory in terms of the light field, though they
disappear in the adiabatic limit. We discuss the origins of these terms as well
as their possible implications for time dependent situations such as inflation.
