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

TL;DR

This paper constructs a solution to the quantum wave equation with a cubic interaction as a bilinear form expanded over Wick polynomials, ensuring proper mathematical definition and connection to classical wave equations.

Contribution

It introduces a novel bilinear form approach to define solutions of the quantum $oxed{ ext{interacting}}$ wave equation with cubic nonlinearity, grounded in Wick polynomial expansion.

Findings

Solution defined as a bilinear form on a dense subspace of Fock space

Diagonal Wick symbol satisfies the classical nonlinear wave equation

Ensures rigorous mathematical foundation for quantum field with cubic interaction

Abstract

We construct the solution $\phi(t,{\bf x})$ of the quantum wave equation $\Box\phi + m^2\phi + \lambda:\!\!\phi^3\!\!: = 0$ as a bilinear form which can be expanded over Wick polynomials of the free $in$-field, and where $:\!\phi^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 $D_{\theta}\times D_{\theta}$, 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.