Path integral representation of S-matrix

By integration over lattice sequences of intermediate states which are coherent states of the field operators, a path integral representation for the elements of the U- and S-matrix is constructed. The U- and S-matrices are more directly related to the physical quantities than are Green's functions of the usual formulation, and they can be constructed even when a Lagrangian does not exist. The method is illustrated by integrating the scalar meson model with weakly recoiling nucleons--its renormalized S-matrix elements are obtained