  We develop a formalism that allows the study of correlations in space and
time in both the superfluid and Mott insulating phases of the Bose-Hubbard
Model. Specifically, we obtain a two particle irreducible effective action
within the contour-time formalism that allows for both equilibrium and out of
equilibrium phenomena. We derive equations of motion for both the superfluid
order parameter and two-point correlation functions. To assess the accuracy of
this formalism, we study the equilibrium solution of the equations of motion
and compare our results to existing strong coupling methods as well as exact
methods where possible. We discuss applications of this formalism to out of
equilibrium situations.
