Quantum Interaction ${\varphi }_4^4$: the Construction of Quantum Field defined as a Bilinear Form

Abstract

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

32 pages, LaTeX