Anharmonic Waves in Field Theory

TL;DR

This paper introduces a framework for anharmonic waves in classical and quantum field theory, generalizing solutions beyond sinusoidal waves to include nonlinear interactions and harmonics, with implications for quantum fields and vacuum states.

Contribution

It develops a general class of anharmonic waves compatible with relativistic and quantum principles, challenging traditional concepts like orthogonality and superposition.

Findings

Allows zero frequency Fourier components, indicating non-zero vacuum expectation values.

Provides a method to define exact quantum fields and single-particle states.

Generalizes field expansions to include anharmonic effects.

Abstract

This work starts from the premise that sinusoidal plane waves cease to be solutions of field theories when turning on an interaction. A nonlinear interaction term generates harmonics analogous to those observed in nonlinear optical media. This calls for a generalization to anharmonic waves in both classical and quantum field theory. Three simple requirements make anharmonic waves compatible with relativistic field theory and quantum physics. Some non-essential concepts have to be abandoned, such as orthogonality, the superposition principle, and the existence of single-particle energy eigenstates. The most general class of anharmonic waves allows for a zero frequency term in the Fourier series, which corresponds to a quantum field with a non-zero vacuum expectation value. Anharmonic quantum fields are defined by generalizing the expansion of a field operator into creation and annihilation operators. This method provides a framework for handling exact quantum fields, which define exact single particle states.