Necessary Condition for the Existence of the $S$ Matrix Outside the Scope of Perturbation Theory

Using the Maslov–Shvedov complex-germ method due to Maslov–Shvedov, we obtain a necessary condition for the existence of the quantum-field $S$ matrix outside the scope of perturbation theory in the leading order of semiclassical approximation. This condition consists in that the tangent symplectic transformation to the evolution operator of the nonlinear classical field equation is realized by a unitary transformation of Fock space. It follows from the results of the book of Maslov and Shvedov that this condition always holds.