S-matrix generating functional and effective action

It is shown how the effective action or proper-vertex generating functional, evaluated at its stationary configuration obeying an appropriate asymptotic boundary condition, gives the generating functional of S-matrix elements to all orders. Based on this connection, we discuss the structure of the loop expansion for S-matrix elements in some detail. The inverse problem of obtaining the classical equations from the S-matrix elements is also considered. Implications concerning string theories are emphasized.