The scattering and production Green's functions for one nucleon and an arbitrary number of mesons are related by an infinite set of coupled linear integral equations. The first $N$ of these equations contain Green's functions involving $0, 1, 2, \cdots N$ external meson lines. The set of equations may be cut off at any point by making as assumption as to the structure of the Green's function with the highest number of external meson lines. In particular this function is approximated by decomposing it into products of lower-order Green's functions, the physical assumption being that one of the mesons interacts weakly with the remaining meson-nucleon system. This leads to a closed set of equations which are linear if vacuum polarization is neglected. Examples of successive approximations are derived. The formalism is also applied to the two-nucleon case and to the three-fields problem, the latter being treated in a manifestly guage-covariant manner.

• Received 15 April 1954

DOI:https://doi.org/10.1103/PhysRev.95.538