The Number of Feynman Diagrams

The Feynman-Dyson formulation of a perturbation expansion for quantum field theory allows one to give a general combinatorial treatment to the Feynman diagrams involved. A very simple analysis for the total number of such diagrams, $T(n, \epsilon , \rho )$, in quantum electrodynamics, leads to: $T(n, \epsilon , \rho )=\frac{{\mathrm{}(n!)\mathrm{}}^2}{{(\epsilon !)}^2(n-\epsilon )!}.\frac{n!}{\rho !\left[\frac{1}{2}\right(n-\rho \left)\right]!2^{\frac{1}{2}(n-\rho )}},$ in which $n$ is the order of the perturbation and $\epsilon$, $\rho$ are the number of external electron and photon lines, respectively. The first factor is the number of different diagrams using only the electron lines and the second is that for the photon lines. In this total set of diagrams are many undesired ones; these are removed by means of generating functions. Relations which these functions satisfy are obtained, and from them one may readily find the exact numbers of diagrams desired, for not too large $n$. The generating functions are also used to find the asymptotic dependence on $n$, and it is found that this dependence is essentially unaffected by removing any specific type of graph. The sign alternations of the matrix elements in quantum electrodynamics are also considered in terms of similar generating functions. The generalization of the analysis to other types of interactions is also discussed.