Feynman Diagrams: Modeling between Physics and Mathematics

The aim of the present paper is to analyze Feynman diagrams within the context of recent historical and philosophical debates about models in science and against the backdrop of other diagrammatic methods in mathematical physics when dealing with infinite or asymptotic series. Today's philosophical model debate largely defines itself by rejecting the traditional understanding of models as mathematical objects that fulfill the axioms of a theory and are isomorphic to an empirical phenomenon. Instead, it emphasizes the autonomy of models within, or even outside, an overarching theory. The example of Feynman diagrams shows that models thus conceived do not necessarily cease to be mathematical objects, if only in a heuristic or "theoretical" sense. Integrating Feynman diagrams into the mathematical tradition of infinite or asymptotic series allows one to avoid the dichotomy whether they represent mathematical or physical objects, or a mere tool mediating between them. Along those lines one does, however, not obtain a universal answer to the question as to what Feynman diagrams represent. A single Feynman diagram, in actual scientific practice, can stand for a single mathematical expression or for a physical phenomenon depending on whether the diagram stands for a single term in an infinite series or for a subseries that is given a physical interpretation. This reading of the representation problem also derives support from the historical fact that Feynman was initially motivated by the Breit-Schrödinger model of a quivering electron. Following Schrödinger, such fluctuations may be considered as a physical phenomenon in its own right that is mathematically construed from the macro-level without having a physically fully specified micro-theory.