量子化和超选择扇区III：多连通空间和不可区分粒子
Quantization and superselection sectors III: Multiply connected spaces and indistinguishable particles
摘要：
我们重新考虑（非相对论性）量子理论的不可分辨粒子的基础上里费尔的概念的C*-代数（“严格”）变形量子化。利用这种形式主义，我们把弥赛亚和格林伯格（1964）的算子方法与Souriau（1967）、Laidlaw和DeWitt（1971）、Leinaas和Myrheim（1977）等的位形空间方法联系起来。前者允许准统计，而后者显然只给玻色子和费米子留下了空间。就准统计学的可接受性而言，这似乎与运营商的方法相矛盾。为了解决这个问题，我们首先证明了第二种方法的拓扑非平凡配置空间是由第一种方法的观测量代数量子化的。其次，我们表明，后者的不可约表示可以实现向量丛的建设，其中包括线丛的第二种方法：表示高维束（定义parastatistics）不能被排除在先验。然而，我们表明，相应的粒子状态可能总是实现玻色子和/或费米子与不可观测的内部自由度。虽然这个结论是基于非相对论性量子力学，但它类似于代数量子场论中多普利歇-哈格-罗伯茨分析的严格结果，也类似于引导盖尔曼等人得出QCD的启发式论证。
英文摘要：
We reconsider the (non-relativistic) quantum theory of indistinguishable particles on the basis of Rieffel's notion of C*-algebraic (`strict') deformation quantization. Using this formalism, we relate the operator approach of Messiah and Greenberg (1964) to the configuration space approach due to Souriau (1967), Laidlaw and DeWitt (1971), Leinaas and Myrheim (1977), and others. The former allows parastatistics, whereas the latter apparently leaves room for bosons and fermions only. This seems to contradict the operator approach as far as the admissibility of parastatistics is concerned. To resolve this, we first prove that the topologically nontrivial configuration spaces of the second approach are quantized by the algebras of observables of the first. Second, we show that the irreducible representations of the latter may be realized by vector bundle constructions, which include the line bundles of the second approach: representations on higher-dimensional bundles (which define parastatistics) cannot be excluded a priori. However, we show that the corresponding particle states may always be realized in terms of bosons and/or fermions with an unobserved internal degree of freedom. Although based on non-relativistic quantum mechanics, this conclusion is analogous to the rigorous results of the Doplicher-Haag-Roberts analysis in algebraic quantum field theory, as well as to the heuristic arguments which led Gell-Mann and others to QCD.
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