# Proof of 1-particle irreducible (1PI) diagrams

If we split the effective action into

$$Γ[Φ] =\frac{1}2ΦiG_0^{-1}Φ + Γ^{int} [Φ]$$

we can show that the full propagator is given by

$$G= i[iG − Σ]^{-1}$$

With

$$Σ=-Γ_{ΦΦ}^{int} [Φ]$$

Here $$Γ_{ΦΦ}$$ means double functional derivatives in relation to the mean field $$Φ$$.

How can we show that $$Σ$$ is made of only 1-particle irreducible diagrams?

1. The full (connected) propagator $$G_c~=~-\Gamma_2^{-1}$$ is (minus) the inverse Hessian of the proper/effective action $$\Gamma$$, cf. e.g. eq. (6) in my Phys.SE answer here.
2. The self-energy $$\Sigma~=~G_0^{-1}-G_c^{-1}$$ is an (amputated) 2-pt 1PI vertex, cf. e.g. my Phys.SE answer here. $$\Box$$