Dirac’s magnetic monopole and the Kontsevich star product
Dirac’s magnetic monopole and the Kontsevich star product
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狄拉克磁单极子和康采维奇明星产品
DOI：
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发表时间：
2017
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通讯作者：
M. Soloviev
中科院分区：
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作者：
M. Soloviev
我们研究了在磁场中带电粒子的各种量子化方案之间的关系。量化映射定义在不变的几何条件下，适当的情况下，非平凡的拓扑结构，并构造两个运营商表示。在第一种情况下，量子算符作用于与霍普夫丛相关的非平凡复线丛截面的希尔伯特空间，而第二种方法则使用平凡四元数线丛截面的四元数希尔伯特模。我们表明，这两个量子化是自然相关的束态射，因此，诱导相同的相空间星星产品。我们得到明确的表达式对应于各种运营商订购的星产品的积分核，并计算其渐近展开到第三阶普朗克常数预测。我们还证明了对应于对称序的磁Weyl积的微分形式与泊松结构形变量子化的Kontsevich公式完全一致，并且可以用Kontsevich图表示。
We examine relationships between various quantization schemes for an electrically charged particle in the field of a magnetic monopole. Quantization maps are defined in invariant geometrical terms, appropriate to the case of nontrivial topology, and are constructed for two operator representations. In the first setting, the quantum operators act on the Hilbert space of sections of a nontrivial complex line bundle associated with the Hopf bundle, whereas the second approach uses instead a quaternionic Hilbert module of sections of a trivial quaternionic line bundle. We show that these two quantizations are naturally related by a bundle morphism and, as a consequence, induce the same phase-space star product. We obtain explicit expressions for the integral kernels of star-products corresponding to various operator orderings and calculate their asymptotic expansions up to the third order in the Planck constant ħ. We also show that the differential form of the magnetic Weyl product corresponding to the symmetric ordering agrees completely with the Kontsevich formula for deformation quantization of Poisson structures and can be represented by Kontsevich’s graphs.