Path-integral representation for theSmatrix

We present a formulation of quantum field theory as a path-integral representation for elements of the $U$ or $S$ matrix in the coherent-state basis. These matrix elements are shown to serve as generating functionals for all the usual $S$-matrix elements between states of definite particle number. The coherent-state formalism for general bilinear quantum field theories is described and then incorporated into the construction of a path integral such that the formulation is independent of any canonical formalism. We discuss the relationship of this formulation to the usual path-integral formulation and show in what sense they are equivalent for canonical theories. We also discuss how this formulation is more general than the usual one in that it is well defined even for theories having no canonical form and for which a Lagrangian action and the usual path-integral formulation may not apply. Applications of this formulation to specific calculations are exhibited for the cases of a quantized field interacting with a given external source and for the renormalization of the simple quantum field theory model of a scalar meson field interacting with a nonrelativistic nucleon field.