I have a very basic question on quantum effective action. Consider the free scalar field with the action $$I[\phi]=\int d^4x \frac{1}{2}((\partial\phi)^2-m^2\phi^2).$$ How to compute explicitly the quantum effective action $\Gamma[\phi]$?
Especially I am interested in its interpretation as a sum of connected one-particle-irreducible graphs (which should be rather degenerate in the free case) as discussed in Section 16.1 of the Weinberg's book. Thus formula (16.1.17) says $$\Gamma[\phi_0]=\int_{1PI, connected}\left[\prod_x d\phi(x)\right]\exp\{iI[\phi+\phi_0]\}.\tag{16.1.17}$$ What does this formula mean for the free theory?
