Exact SU(2) Yang-Mills Waves from a Simple Ansatz

TL;DR

This paper introduces a simple ansatz that yields exact, closed-form wave solutions of sourceless SU(2) Yang-Mills equations, revealing linear, nonlinear, and gauge solutions with distinct physical properties.

Contribution

The authors present a novel ansatz that reduces the nonlinear Yang-Mills equations to algebraic constraints, leading to three families of exact wave solutions with new physical insights.

Findings

Family I describes linear electromagnetic waves within SU(2)

Family II exhibits nonlinear self-interacting waves with a constant gauge-invariant offset

Family III provides pure gauge solutions with no field strengths

Abstract

We propose a simple ansatz to construct exact wave solutions of the sourceless SU(2) Yang-Mills equations in (3+1) dimensions. The ansatz employs a $y$-dependent rotated Pauli basis and assumes a phase $\theta=kz-\omega t$ dependence for the gauge potentials. Owing to this ansatz, the nonlinear field equations reduce to nine algebraic constraints, whose complete solution yields three families of exact waves. Family I describes linear (Abelian) electromagnetic waves embedded in the non-Abelian theory; all commutator terms vanish and the dispersion relation is $\omega=kc$. Family II represents genuinely nonlinear self-interacting waves that also propagate at the speed of light but exhibit a constant field offset, nonvanishing commutators, and do not obey superposition. The constant offset is gauge-invariant and gives rise to a non-zero time-averaged color-electric field. The energy density has nodes whose position ($\theta=0$ or $\theta=\pi$) is controlled by a discrete topological parameter $\xi\eta=\pm1$, providing an observable signature. Family III is a pure gauge solution with vanishing field strengths, valid for arbitrary $k$ and $\omega$ without any dispersion relation. All solutions are closed-form and provide new insights into how non-Abelian self-interactions fundamentally alter wave propagation.