  We introduce a novel compositional description of Feynman diagrams, with
well-defined categorical semantics as morphisms in a dagger-compact category.
Our chosen setting is suitable for infinite-dimensional diagrammatic reasoning,
generalising the ZX calculus and other algebraic gadgets familiar to the
categorical quantum theory community.
  The Feynman diagrams we define look very similar to their traditional
counterparts, but are more general: instead of depicting scattering amplitude,
they embody the linear maps from which the amplitudes themselves are computed,
for any given initial and final particle states. This shift in perspective
reflects into a formal transition from the syntactic, graph-theoretic
compositionality of traditional Feynman diagrams to a semantic,
categorical-diagrammatic compositionality.
  Because we work in a concrete categorical setting -- powered by non-standard
analysis -- we are able to take direct advantage of complex additive structure
in our description. This makes it possible to derive a particularly compelling
characterisation for the sequential composition of categorical Feynman
diagrams, which automatically results in the superposition of all possible
graph-theoretic combinations of the individual diagrams themselves.
