Quantum Interaction φ 44 : the Construction of Quantum Field defined as a Bilinear Form

We construct the solution φ(t,x) of the quantum wave equation 2φ+m2φ+λ : φ3:= 0 as a bilinear form which can be expanded over Wick polynomials of the free in-field, and where :φ3(t,x) : is defined as the normal ordered product with respect to the free in-field. The constructed solution is correctly defined as a bilinear form on Dθ×Dθ, whereDθ is a dense linear subspace in the Fock space of the free in-field. On Dθ ×Dθ the diagonal Wick symbol of this bilinear form satisfies the nonlinear classical wave equation.