多连通空间中的量子力学
Quantum Mechanics in Multiply-Connected Spaces
摘要：
我们解释了为什么，在一个配置空间是多连通的，即，其基本群是非平凡的，有几种量子理论，对应于拓扑因子的不同选择。我们在玻姆力学的背景下这样做，玻姆力学是一种没有观察者的量子理论，量子形式主义可以从它导出。我们所做的工作可以看作是将$\mathbb{R}^{3N}$上的玻姆动力学推广到任意黎曼流形上，并对可能出现的动力学进行分类。这种方法为量子理论的拓扑特征提供了新的理解，例如全同粒子的对称化假设。在我们的分析中，我们采用了位形空间的泛覆盖空间上的波函数。
英文摘要：
We explain why, in a configuration space that is multiply connected, i.e., whose fundamental group is nontrivial, there are several quantum theories, corresponding to different choices of topological factors. We do this in the context of Bohmian mechanics, a quantum theory without observers from which the quantum formalism can be derived. What we do can be regarded as generalizing the Bohmian dynamics on $\mathbb{R}^{3N}$ to arbitrary Riemannian manifolds, and classifying the possible dynamics that arise. This approach provides a new understanding of the topological features of quantum theory, such as the symmetrization postulate for identical particles. For our analysis we employ wave functions on the universal covering space of the configuration space.
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