The enumeration of graphs in the Feynman-Dyson technique

As an aid to practical calculations, and as a first step in determining whether field theories that can be renormalized lead to convergent perturbation expansions, the number of graphs that can be drawn for each stage of approximation is investigated. Using the idea of a normal product of operators, a simple method is given for finding the number of Feynman-Dyson graphs, with n vertices, that can be drawn when the interaction Hamiltonian density is of a general type. By solving a difference equation approximately, it is shown that the number of graphs remaining after the removal of those containing subgraphs of a specified type is asymptotically equal to the total number of unmodified graphs. Also it is shown that the number of graphs that can be drawn with n vertices increases very rapidly with n, no matter how many external lines are present, and this rapid increase still occurs even when the class of irreducible graphs alone is considered. Thus the perturbation expansions of field theory cannot converge unless the matrix elements decrease with correspondingly great rapidity as n increases.