Eugene Wigner realised a long time ago that an irrdeucible representation is equatable to a "single particle". From this has developed the "many-particle" theory of Quantum systems known as the Racah-Wigner calculus. Feynman diagrams is a technique to "book-keep" for perturbative expansions in Quantum Field Theories ---a "many-particle" theory. The two disciplines must fit together. It is the intention of this talk to justify (in part) how this comes about in the realm of monoidal category theory. This talk is aimed at a very varied audience - from category theorists all the way through combinatorics to mathematical physicists.