Calculus of Feynman diagrams Feynman diagrams are wonderful tools to visualize particle interactions. They make deep connections/points of view in quantum mechanics intuitive (e. g. antiparticles = arrows against the direction of time) and give hints on how to decompose the integrals occuring by superposition. Often, one generates all possible Feynman diagrams to calculate probability amplitudes.
What I am missing is the ability to calculate with Feynman diagrams directly. In other words:

*

*Take a complicated problem

*Translate it into Feynman diagrams

*Do some cool graphtheoretical stuff directly on the Feynman diagrams to obtain a more simple diagram

*Retranslate them back into more simple/fewer formulae which otherwise would have been hard to derive.

Note that Feynman's rules "only" give assertions about how a diagram must look, not procedures on how to simplify it.
Is there such a Feynman calculus? Or is it somehow proven that such a calculus cannot exist? Perhaps, one could see commutative diagrams in Homological Algebra as having such a more direct calculus. (For example, the Snake lemma or results relating to the logical completeness of diagram chasing)
Disclaimer: Coming from mathematics, I read the Wikipedia article about Feynman diagrams some days ago and was instantly fascinated. Therefore, my physics background may not be the strongest one...
 A: Yes, there is such a thing, and Feynman diagrams were not invented by Feynman, but by mathematicians in the 19-th century, starting with Arthur Cayley. See, for example, my MO answer:
https://mathoverflow.net/questions/168888/who-invented-diagrammatic-algebra/260016#260016
A Feynman diagram is just an extremely compact yet precise notation for a contraction of tensors, say something like
$$
G_{i,r}=\sum_{j,\ell, p,q=1}^N C_{i,j}V_{j,\ell, p, q}C_{\ell, p}C_{q ,r}\ .
$$
If $C$ is a lattice propagator in position space and $V_{j,\ell,p,q}=\delta_{j,\ell}\delta_{j,p}\delta_{j,q}$, the above is just the tadpole diagram in the two-point function of the $\phi^4$ QFT. In the physics context, one wants to take $N\rightarrow\infty$ so summation over indices become integrals over continuous variables of position or momentum.
One can certainly do computation entirely in terms of diagrams, especially in the context of representation theory and invariant theory. This is the spirit of the book by Cvitanović, for instance.
You can also see examples of diagrammatic computations related to Pascal's Theorem in my article with Chipalkatti
"On the Reconstruction Problem for Pascal Lines",
Discrete & Computational Geometry volume 60, pages 381–405 (2018). Preprint version is here.
