Quantum Interaction $ϕ^4_4$: the Construction of Quantum Field defined as a Bilinear Form

Abstract

We construct the solution $ϕ(t,{\bf x})$ of the quantum wave equation $\Boxϕ+ m^2ϕ+ λ:\!\!ϕ^3\!\!: = 0$ as a bilinear form which can be expanded over Wick polynomials of the free $in$-field, and where $:\!ϕ^3(t,{\bf 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 $D_θ\times D_θ$, where $D_θ$ is a dense linear subspace in the Fock space of the free $in$-field. On $D_θ\times D_θ$ the diagonal Wick symbol of this bilinear form satisfies the nonlinear classical wave equation.