An attempt to construct dynamical evolution in quantum field theory

Abstract

If we develop into perturbation series the evolution operator of the Heisenberg equation in the infinite dimensional Weyl algebra, say, for the $\phi^4$ model of field theory, then the arising integrals almost coincide with the usual Feynman diagram integrals. This fact leads to some mathematical definitions which, as it seemed to the author, defined dynamical evolution in quantum field theory in a mathematically rigorous way using the Weyl algebra. In fact the constructions of the paper are well defined in perturbation theory only in one-loop (quasiclassical) approximation. A variation of the construction is related with the Bogolyubov $S$-matrix $S(g)$.