  The paper is a survey of some author's results related with the
Maslov--Shvedov method of complex germ and with quantum field theory. The main
idea is that many results of the method of complex germ and of perturbative
quantum field theory can be made more simple and natural if instead of the
algebra of (pseudo)differential operators one uses the Weyl algebra (operators
with Weyl symbols) with the Moyal *-product. Section 1, devoted to quantum
mechanics, contains a closed mathematical description of the Maslov--Shvedov
method in the theory of Schrodinger equation, including the method of canonical
operator. In particular, it contains a new simple definition of the Maslov
index modulo 4. Section 2, devoted to quantum field theory, contains a
logically self-consistent exposition of the main results of perturbative
quantum field theory not using the subtraction of infinities from the quantum
Hamiltonian of free field and normal ordering of operators. It also contains a
result (dynamical evolution in quantum field theory in quasiclassical
approximation) close to the Maslov--Shvedov quantum field theory complex germ.
