I’ve heard about two ways mathematician describe Feynman diagrams:

• They can be seen as “string diagrams” describing various type of arrows (and/or compositions operations on them) in monoidal closed category.

• They are combinatorial tools that allows to give formulas the asymptotical expansion of integrals for the form:

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

when $ \hbar \rightarrow 0$ in terms of assymptotical expansion for $ g$ and $ S$ around $ 0$ (with $ S$ having a unique maximum at $ 0$ and decreasing quickly 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…

PS: I had no ideas what should be the tags of this question, so if anyone want to put tags that he thinks are appropriate he is more than welcome !