Spin Vector Potential as an Exact Solution of the Yang-Mills Equations

TL;DR

This paper derives an exact solution to the Yang-Mills equations representing a spin-dependent gauge field, providing a theoretical foundation for the spin vector potential and linking spin physics with gauge theory.

Contribution

It introduces a new family of exact solutions to Yang-Mills equations that describe spin-dependent interactions, bridging spin physics and gauge theory.

Findings

Spin vector potential is an exact solution of Yang-Mills equations.

The solution describes a spin-dependent Coulomb interaction.

Schrödinger and Dirac equations with this interaction are exactly solvable.

Abstract

The spin vector potential, a gauge field generated by the intrinsic spin of a particle, has recently been proposed as a central element of spin Aharonov-Bohm effect. While its physical consequences have been explored, a fundamental and theoretical question remains: can it be systematically derived from a first-principle gauge theory? In this work, we prove that the spin vector potential $\vec{\mathcal{A}}= k (\vec{r} \times \vec{S})/{r^2}$, together with the Coulomb-type scalar potential $\varphi={\kappa}/{r}$, emerges as a new family of exact solutions to the non-Abelian Yang-Mills equations in vacuum. This solution, $\{\vec{\mathcal{A}}, \varphi\}$, describes a spin-dependent interaction that naturally reduces to the standard Coulomb interaction when spin effects are neglected. Moreover, we demonstrate that the Schr{\" o}dinger and Dirac equations incorporating this spin-dependent Coulomb interaction can be solved exactly. Our work not only provides a rigorous gauge-theoretical foundation for the previously proposed spin vector potential, but also establishes a direct link between spin physics and the Yang-Mills gauge theory. This opens new perspectives for understanding spin-dependent interactions, designing spin-dependent quantum phases, and exploring spin-mediated forces in quantum physics.