General Static Solutions of the SU(2) Yang-Mills Equations from a Spin Vector Potential

TL;DR

This paper systematically classifies static solutions to source-free SU(2) Yang-Mills equations using a novel vector potential approach that incorporates spin operators, revealing new configurations beyond known solutions.

Contribution

It introduces the vector potential extraction approach (VPEA) to derive the most general static solutions with explicit spin dependence, expanding the solution space.

Findings

Derived the general form of spin vector potentials for static solutions.

Solved the resulting equations to classify all static solutions, including complex families.

Recovered known solutions and identified new static configurations with potential applications.

Abstract

We present a systematic study of static solutions to the source-free SU(2) Yang-Mills equations, in which the gauge potential explicitly depends on spin operators. By employing the \emph{vector potential extraction approach} (VPEA) -- which requires the total angular momentum operator (orbital plus spin) to satisfy the standard angular momentum algebra -- we derive the most general form of the spin vector potential. This leads to the static ansatz $\{ \vec{A} = [k_1(\hat{r}\times\vec{\Gamma}) + k_2\vec{\Gamma} + k_3(\vec{\Gamma}\cdot\hat{r})\hat{r}]/r, \varphi = f_1(r)\,(\vec{\Gamma}\cdot\hat{r}) + f_2(r)\}$, parametrized by three constants $\{k_1, k_2, k_3\}$ and two radial functions $\{f_1(r), f_2(r)\}$. Substituting this ansatz into the Yang-Mills equations and imposing the angular momentum constraints from the VPEA yields a set of consistency equations. Solving these equations provides a complete classification of static solutions, including both real and complex families. Known simple SU(2) static solutions are recovered as special cases. Our classification reveals new static configurations that could be valuable for non-perturbative studies and for models where spin degrees of freedom couple to non-Abelian gauge fields.