Bosonic Loop Diagrams as Perturbative Solutions of the Classical Field Equations in $\phi^4$-Theory

TL;DR

This paper develops a perturbative approach to classical $^4$-theory solutions using Feynman diagrams, connecting classical solutions with quantum field theory diagrams, including loop diagrams via a stochastic background.

Contribution

It introduces a diagrammatic perturbation expansion for classical $^4$-theory solutions, incorporating loop diagrams through a classical stochastic background, bridging classical and quantum field theories.

Findings

Tree diagrams represent classical solutions with retarded Green's functions.

Loop diagrams are obtained by adding a stochastic background field.

Comparison with quantum Feynman diagrams shows structural similarities.

Abstract

Solutions of the classical $\phi^4$-theory in Minkowski space-time are analyzed in a perturbation expansion in the nonlinearity. Using the language of Feynman diagrams, the solution of the Cauchy problem is expressed in terms of tree diagrams which involve the retarded Green's function and have one outgoing leg. In order to obtain general tree diagrams, we set up a "classical measurement process" in which a virtual observer of a scattering experiment modifies the field and detects suitable energy differences. By adding a classical stochastic background field, we even obtain all loop diagrams. The expansions are compared with the standard Feynman diagrams of the corresponding quantum field theory.