We consider the quantum mechanics of a charged particle in the presence of Dirac’s magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the Feynman–Kac formula expressing solutions of the imaginary time Schrödinger equation as stochastic integrals.

中文翻译：

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磁单极子的 Feynman-Kac 公式

我们考虑在狄拉克磁单极子存在的情况下带电粒子的量子力学。波函数是复杂线束的部分，而磁势是线束上的连接。我们使用连续特征函数展开来找到哈密顿量的基本自伴随的不变域。这导致了 Feynman-Kac 公式的证明，该公式将虚时间薛定谔方程的解表示为随机积分。