自旋统计定理与几何量子化
Spin-Statistics Theorem and Geometric Quantisation
摘要：
我们研究了自旋统计定理如何与量子化过程中引入的相空间几何结构（即U（1）束和连接）相关。这种关系可以在相对论和非相对论域（实际上对于包括内部对称性的任何对称群）中证明，而不需要量子场论，只要交换可以通过投射在联合系统相空间中的一类对称变换顺利实现。我们讨论了这一要求的解释，强调的事实是，全同粒子的任何区别完全来自坐标的选择。然后，我们研究我们的建设中的几何和相干状态路径积分量化方案。在附录中，我们将我们的结果应用到表现出连续“自旋”和“分数自旋”的奇异系统。这给出了自旋统计关系的新颖和不寻常的形式。
英文摘要：
We study how the spin-statistics theorem relates to the geometric structures on phase space that are introduced in quantisation procedures (namely a U(1) bundle and connection). The relation can be proved in both the relativistic and the non-relativistic domain (in fact for any symmetry group including internal symmetries) without quantum field theory, by the requirement that the exchange can be implemented smoothly by a class of symmetry transformations that project in the phase space of the joint system. We discuss the interpretation of this requirement, stressing the fact that any distinction of identical particles comes solely from the choice of coordinates. We then examine our construction in the geometric and the coherent-state-path-integral quantisation schemes. In the appendix we apply our results to exotic systems exhibiting continuous ``spin'' and ``fractional statistics''. This gives novel and unusual forms of the spin-statistics relation.
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