Categorical Semantics for Feynman Diagrams

TL;DR

This paper develops a categorical framework for Feynman diagrams, generalizing their traditional use to a semantic, compositional setting suitable for infinite-dimensional quantum reasoning, enabling new algebraic and diagrammatic insights.

Contribution

It introduces a novel categorical semantics for Feynman diagrams, extending their applicability and formal structure beyond traditional graph-based representations.

Findings

Categorical semantics for Feynman diagrams as morphisms in a dagger-compact category

Generalization to infinite-dimensional diagrammatic reasoning

Automatic superposition of diagrammatic combinations in composition

Abstract

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.