Aharonov-Bohm效应、Dirac单极子和丛理论
Aharonov-Bohm Effect, Dirac Monopole, and Bundle Theory
摘要：
在丛理论的框架下讨论了磁荷$g={{1}\over{2}}$的Aharonov-Bohm（$A-B$）效应和Dirac（$D$）效应，揭示了它们之间的纯粹几何关系.如果$ξ_{A-B}$和$ξ_D$是各自的$U(1)$ -丛，我们证明了$ξ_{A-B}$同构于由包含相应的基空间$ι:(D_0^2)^*\to S^2$ }引起的$ξ_D$的拉回.当螺线管中的磁通量等于与电荷$\vert e\vert$相关的磁通量$Φ_0={{2π}\over{\vert e\vert}}$的整数倍时，$A-B$效应消失的事实在此反映为$ξ_D$中的狄拉克连接被$ι$拉回到$ξ_{A-B}$以及狄拉克量子化条件的结果。我们还证明了$ξ_D$中的陈类$c_1$的拉回在$ξ_{A-B}$中的必要消失。
英文摘要：
We discuss the Aharonov-Bohm ($A-B$) effect and the Dirac ($D$) monopole of magnetic charge $g={{1}\over{2}}$ in the context of bundle theory, exhibiting a purely geometric relation between them. If $ξ_{A-B}$ and $ξ_D$ are the respective $U(1)$-bundles, we show that $ξ_{A-B}$ is isomorphic to the pull-back of $ξ_D$ induced by the inclusion of the corresponding base spaces $ι:(D_0^2)^*\to S^2$}. The fact that the $A-B$ effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux $Φ_0={{2π}\over{\vert e\vert}}$ associated with the electric charge $\vert e\vert$, reflects here as a consequence of the pull-back by $ι$ of the Dirac connection in $ξ_D$ to $ξ_{A-B}$, and the Dirac quantization condition. We also show the necessary vanishing in $ξ_{A-B}$ of the pull-back of the Chern class $c_1$ in $ξ_D$.
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