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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"): $$Z(j_1,\dots,j_n):=\frac{\int_{\mathbb{R}^n} \exp\{-B(x_1,\dots,x_n)+\sum_kP_k(x)+\sum_{i=1}^nj_ix_i\}dx}{ \int_{\mathbb{R}^n} \exp\{-B(x_1,\dots,x_n)+\sum_kP_k(x)\}dx},$$ where $B$ is a positive definite quadratic form on $\mathbb{R}^n$, $P_k$ are homogeneous polynomials. 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.

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    $\begingroup$ I believe there are several papers by Jackson, Kempf, and Morales, where the combinatorics are worked out in some detail (cf. arxiv.org/abs/1612.00462, arxiv.org/abs/1805.09812, ...). IIRC, K. Yeats has worked on similar problems from an "abstract algebra" point of view (she has a rather detailed book on the combinatorics of Feynman diagrams and many lecture notes that you may find useful). Etingof (MIT lecture notes @ ocw.mit.edu/courses/mathematics/…) also mentions your result, but without proof. $\endgroup$ – Delmastro Jan 19 at 2:11
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Section 5 of Borcherds, Barnard, Lectures on Quantum Field Theory is a discussion of the 0-dimensional spacetime case, which gives finite dimensional integrals.

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