  We derive and study two different formalisms used for non-equilibrium
processes: The coherent-state path integral, and an effective, coarse-grained
stochastic equation of motion. We first study the coherent-state path integral
and the corresponding field theory, using the annihilation process $A+A\to A$
as an example. The field theory contains counter-intuitive quartic vertices. We
show how they can be interpreted in terms of a first-passage problem.
Reformulating the coherent-state path integral as a stochastic equation of
motion, the noise generically becomes imaginary. This renders it not only
difficult to interpret, but leads to convergence problems at finite times. We
then show how alternatively an effective coarse-grained stochastic equation of
motion with real noise can be constructed. The procedure is similar in spirit
to the derivation of the mean-field approximation for the Ising model, and the
ensuing construction of its effective field theory. We finally apply our
findings to stochastic Manna sandpiles. We show that the coherent-state path
integral is inappropriate, or at least inconvenient. As an alternative, we
derive and solve its mean-field approximation, which we then use to construct a
coarse-grained stochastic equation of motion with real noise.
