# One particle irreducible Feynman diagrams

In quantum field theory Feynman has invented a diagrammatic method to encode various terms in the Taylor decomposition of integrals of the following form below which I will write in a baby version as finite dimensional integral rather than path integral (and using “imaginary time”): $$Z(j_1,\dots,j_n):=\int_{\mathbb{R}^n} \exp\{B(x_1,\dots,x_n)+\sum_{i=1}^nj_ix_i\}dx/ \int_{\mathbb{R}^n} \exp\{B(x_1,\dots,x_n)\}dx,$$ where $$B$$ is a positive definite quadratic form on $$\mathbb{R}^n$$. Furthermore one can write $$Z(j)=exp\{ W(j)\}$$ and it is shown that $$W(j)$$ is a sum of terms corresponding only to connected diagrams.

In the context of path integrals there is a notion of effective action which in this context is defined as follows. Let $$\phi_i:=\frac{\partial W(j)}{\partial j_i}$$. Define the Legendre transform $$\Gamma(\phi):=\sum_{i=1}^n \phi_ij_i-W(j).$$

In QFT it is claimed that the Taylor decomposition of $$\Gamma(\phi)$$ is the sum of terms corresponding to connected one particle irreducible diagrams. I am wondering if a finite dimensional (baby) version of this claim is true, and if this is the case whether there is a reference to a detailed discussion of the finite dimensional case.