Exact solutions of classical scalar field equations

TL;DR

This paper presents exact classical solutions for quartic scalar field theories that produce mass-like behavior from nonlinear terms, and explores their implications for quantum field theory, including the spectrum and triviality at large coupling.

Contribution

It introduces a class of exact solutions demonstrating how nonlinearities generate mass and influence the quantum theory, extending understanding of scalar field dynamics.

Findings

Mass contributions arise from nonlinear terms while maintaining wave-like behavior.

The quantum propagator is consistent at large coupling, indicating triviality in the limit.

An infinite spectrum of classical solutions leads to equivalent quantum theories at high coupling.

Abstract

We give a class of exact solutions of quartic scalar field theories. These solutions prove to be interesting as are characterized by the production of mass contributions arising from the nonlinear terms while maintaining a wave-like behavior. So, a quartic massless equation has a nonlinear wave solution with a dispersion relation of a massive wave and a quartic scalar theory gets its mass term renormalized in the dispersion relation through a term depending on the coupling and an integration constant. When spontaneous breaking of symmetry is considered, such wave-like solutions show how a mass term with the wrong sign and the nonlinearity give rise to a proper dispersion relation. These latter solutions do not change the sign maintaining the property of the selected value of the equilibrium state. Then, we use these solutions to obtain a quantum field theory for the case of a quartic massless field. We get the propagator from a first order correction showing that is consistent in the limit of a very large coupling. The spectrum of a massless quartic scalar field theory is then provided. From this we can conclude that, for an infinite countable number of exact classical solutions, there exist an infinite number of equivalent quantum field theories that are trivial in the limit of the coupling going to infinity.