Quantum interaction :φ44:, the construction of quantum field defined as a bilinear form

Abstract

We construct the solution φ(t,x) of the quantum wave equation □φ+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, where Dθ is a dense linear subspace in the Fock space of the free in-field. On Dθ×Dθ the diagonal of the Wick symbol of this bilinear form satisfies the nonlinear classical wave equation.