We show that the Aharonov-Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labeling charged superselection sectors. In the present paper we show that this "topological" quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov-Bohm effect. To confirm these abstract results we quantize the Dirac field in presence of a background flat potential and show that the Aharonov-Bohm phase gives an irreducible representation of the fundamental group of the spacetime labeling the charged sectors of the Dirac field. We also show that non-Abelian generalizations of this effect are possible only on space-times with a non-Abelian fundamental group.

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Aharonov-Bohm 超选扇区

我们表明，Aharonov-Bohm 效应在局部可观测物的超选择扇区的弯曲时空上的 QFT 设置中找到了自然的描述。将超选择扇区的分析从闵可夫斯基时空扩展到任意全局双曲时空，揭示了标记带电超选择扇区的新量子数的存在。在本文中，我们展示了这个“拓扑”量子数相当于背景平坦电位的存在，当沿着路径传输时，如 Aharonov-Bohm 效应一样，它控制着电荷的行为。为了证实这些抽象结果，我们在背景平坦电位存在的情况下量化了狄拉克场，并表明阿哈罗诺夫-玻姆相给出了时空基本群的不可约表示，标记了狄拉克场的带电扇区。我们还表明，这种效应的非阿贝尔推广只有在具有非阿贝尔基本群的时空上才有可能。