A necessary condition for existence of S-matrix outside perturbation theory

TL;DR

This paper establishes a necessary condition for the existence of the quantum field theory S-matrix outside perturbation theory, linking it to the unitarity of a symplectic transformation in the Fock space within a quasiclassical approximation.

Contribution

It introduces a condition based on the unitarity of a tangent symplectic transformation that must be met for the S-matrix to exist outside perturbation theory.

Findings

The S-matrix can exist outside perturbation theory only if a certain symplectic transformation is unitarily implementable.

The condition for the S-matrix's existence is generally satisfied according to Maslov--Shvedov's results.

The work connects classical symplectic geometry with quantum field theory S-matrix existence.

Abstract

Using the Maslov--Shvedov method of complex germ, we show that quantum field theory S-matrix can exist outside perturbation theory in the principal order of quasiclassical approximation only under the condition that the tangent symplectic transformation to the evolution operator of non-linear classical field equation is unitarily implementable in the Fock space. However, the results of Maslov--Shvedov's book imply that this condition is seemingly always satisfied.