The $:{\varphi }_4^4:$ quantum field theory, I. Wave operator, holomorphity and Wick kernel

With the help of the complex structure and the wave operator of the nonlinear classical Klein-Gordon equation with the interaction $u_4^4$ we define the Wick kernel of the interacting quantum field in four-dimensional space-time and consider its properties. In particular, the diagonal of this Wick kernel is (real) solutions of the classical nonlinear Klein-Gordon equation with the interaction $u_4^4\mathrm{.}$