  We propose a mathematically rigorous construction of the scattering matrix
and the interacting fields in models of relativistic perturbative quantum field
theory with massless fields and long-range interactions. We consider quantum
electrodynamics and a certain model of interacting scalar fields in which the
standard definition of the scattering matrix is not applicable because of the
infrared problem. We modify the Bogoliubov construction using the ideas of
Dollard, Kulish and Faddeev. Our modified scattering matrix and modified
interacting fields are constructed with the use of the adiabatic limit which is
expected to exist in arbitrary order of perturbation theory. In the paper we
prove this assertion in the case of the first- and the second-order corrections
to the modified scattering matrix and the first-order corrections to the
modified interacting fields. Our modified scattering matrix and modified
interacting fields are defined in the standard Fock space. However, the
particle interpretation of states in this space is non-standard. In particular,
the electrons and positrons are always surrounded by irremovable clouds of
photons. Moreover, the physical energy-momentum operators do not coincide with
the standard ones and their joint spectrum does not contain the mass
hyperboloid.
