  In this paper, we present a new method to calculate the n-Loop n-particle
irreducible effective action. The key is an organizational trick that involves
the introduction of a set of fictitious bare vertices that are set to zero at
the end of the calculation. Using these fictitious vertices, we prove that the
Schwinger-Dyson equations are the same as the equations of motion obtained from
the n-particle irreducible effective action, up to the level at which they
respect the symmetries of the original theory. This result allows us to obtain
the effective action directly from the Schwinger-Dyson equations, which are
comparatively easy to calculate. As a check of our method, we reproduce the
known results for the n-Loop n-particle irreducible effective action with n = 4
and n = 5. We also use the technique to calculate the 6-Loop 6-particle
irreducible effective action.
