Gauss' Law and Non-Linear Plane Waves for Yang-Mills Theory

TL;DR

This paper derives exact non-linear plane wave solutions for classical Yang-Mills equations using Gauss's law, revealing periodic anharmonic waves with arbitrary mass and identifying a unique harmonic solution with complex phase and color patterns.

Contribution

It introduces a method to find exact non-linear plane wave solutions in SU(3) and SU(4) Yang-Mills theories, highlighting solutions absent in SU(2).

Findings

Solutions are expressed with Jacobi elliptic functions.

Existence of a unique harmonic plane wave with non-trivial phase, spin, and color.

Solutions depend on elliptic modulus values and are absent in SU(2).

Abstract

We investigate Non-Linear Plane-Wave solutions of the classical Minkowskian Yang-Mills (YM) equations of motion. By imposing a suitable ansatz which solves Gauss' law for the $SU(3)$ theory, we derive solutions which consist of Jacobi elliptic functions depending on an enumerable set of elliptic modulus values. The solutions represent periodic anharmonic plane waves which possess arbitrary non-zero mass and are exact extrema of the non-linear YM action. Among them, a unique harmonic plane wave with a non-trivial pattern in phase, spin and color is identified. Similar solutions are present in the $SU(4)$ case while are absent from the $SU(2)$ theory.