Combinatorial Species and Feynman Diagrams

A Feynman diagram is a graphical construction that describes certain interactions in physics. Most calculations with such diagrams reduce to consideration of connected Feynman diagrams. These in turn may be constructed from line-irreducible Feynman diagrams, those for which removal of a single line does not destroy connectivity. The purpose of this article is to exhibit the combinatorial nature of this construction in the framework of species of structures. The main result is a dissymmetry theorem for connected Feynman diagrams. This purely combinatorial theorem relates the species of connected diagrams to species with less symmetry, such as the species of connected diagrams with a designated line-irreducible subdiagram. There is also a discussion of the relation of this result to the Legendre transform.