  We construct a mathematical model analogous to quantum field theory, but
without the notion of vacuum and without measurable physical quantities. This
model is a direct mathematical generalization of scattering theory in quantum
mechanics to path integrals with multidimensional trajectories (whose
mathematical interpretation has been given in a previous paper). In this model
the normal ordering of operators in the Fock space is replaced by the
Weyl-Moyal algebra. This model shows to be useful in proof of various results
in quantum field theory: one first proves these results in the mathematical
model and then "translates" them into the usual language of quantum field
theory by more or less "ugly" procedures.
