I've heard about two ways mathematicians describe Feynman diagrams:


* They can be seen as "string diagrams" describing various type of arrows (and/or compositions operations on them) in a monoidal closed category.
* They are combinatorial tools that allow one to give formulas for the asymptotic expansion of integrals of the form:


$$ \int\_{\mathbb{R}^n} g(x) e^{-S(x)/\hbar} $$ 


when $\hbar \rightarrow 0$ in terms of asymptotic expansions for $g$ and $S$ around $0$ (with $S$ having a unique minimum at $0$ and increasing quickly enough at $\infty$ and often with a very simple $g$, like a product of linear forms), as well as some variation of this idea, or for the slightly more subtle ``oscilating integral'' version of it, with $e^{-i S(x)/\hbar}$.


My question is: *is there a relation between the two ?*


I guess what I would like to see is a "high level" proof of the kind of formula we get in the second point in terms of monoidal categories which explains the link between the terms appearing in the expansion and arrows in a monoidal category... But maybe there is another way to understand it...



The two ways Feynman diagrams appear in mathematics
If I understand the question correctly, the search is for a calculation of the asymptotic expansion of Gaussian integrals using concepts and techniques from category theory. Here is one such calculation:


[Feynman diagrams via graphical calculus](https://arxiv.org/abs/math/0106001) (2001)



> 
> There is a very close connection between the graphical formalism for
>  ribbon categories and Feynman diagrams. Although this correspondence
>  is frequently implied, we know of no systematic exposition
>  in existing literature; the aim of this paper is to provide such an
>  account. In particular, in deriving Feynman diagrams expansion of
>  Gaussian integrals as an application of the graphical formalism for
>  symmetric monoidal categories, we discuss in detail how different
>  kinds of interactions give rise to different families of graphs and
>  show how symmetric and cyclic interactions lead to “ordinary” and
>  “ribbon” graphs respectively.
> 
> 
> 


