  We continue the construction of the $:\phi^4_4:$ quantum field theory. In
this paper we consider the Wick kernel of the interacting quantum field. Using
the complex structure and the Fock-Bargmann-Berezin-Segal integral
representation we prove that this kernel defines a unique operator--valued
generalized function on the space $\Sc^\alpha(\R^4)$ for any $\alpha<6/5,$ i.e.
the constructed quantum field is the generalized operator-valued function of
localizable Jaffe class. The same assertion is valid for the outgoing quantum
field. These assertions about the quantum field allow to construct the Wightman
functions, the matrix elements of the quantum scattering operator and to
consider their properties (positivity, spectrality, Poincare invariance,
locality, asymptotic completeness, and unitarity of the quantum scattering).
