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

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 $\varphi=κ/{r}$, 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.

Figures (1)

• Figure 1: Illustration of Predicting Spin-Dependent Coulomb Interaction Based on the YM Equations. The standard Coulomb potential is a simple but fundamental solution (i.e., $\mathcal{S}_{\rm Maxwell}$) of Maxwell's equations, hence Maxwell's equations predict the existence of the standard Coulomb potential in Nature. The YM equations are the natural generalizations of Maxwell's equations from the Abelian potentials to the non-Abelian ones, and the non-Abelian operators can be realized by the spin operators, then one can expect that there will be a counterpart solution of potentials (i.e., $\mathcal{S}_{\rm YM}$) that depends on spin. Thus based on the YM equations, one can predict a reasonable form of spin-dependent Coulomb potential, which naturally reduces the standard Coulomb potential if without considering the spin.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof