Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole
Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole
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DOI：
10.1088/1751-8121/abf961
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发表时间：
2020-11
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影响因子：
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通讯作者：
Rodrigo Andrade e Silva;T. Jacobson
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文献类型：
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作者：
Rodrigo Andrade e Silva;T. Jacobson
在两个球上量子化粒子的问题已经被许多方法处理过，包括Isham的基于相空间上传递作用的辛对称群的么正表示的全局方法。在这里，我们使用Isham的方案重新考虑这个简单的模型，通过修改辛形式，通过球的磁通量来丰富这个简单的模型。为了保持完全的通用性，我们使用梯形算子，直接从明显规范不变的对称代数构造希尔伯特空间。通过这种方式，我们从代数上恢复了量子化的完全分类，以及粒子的相应能谱。著名的单极电荷狄拉克量子化条件源于经典Casimir不变量和量子Casimir不变量匹配的要求。在附录中，我们解释了这种方法与更常见的方法之间的关系，后者从一开始就假设波函数的希尔伯特空间是球面上非平凡线丛的部分，并展示了代数的Casimir不变量如何决定丛的拓扑。
The problem of quantizing a particle on a two-sphere has been treated by numerous approaches, including Isham’s global method based on unitary representations of a symplectic symmetry group that acts transitively on the phase space. Here we reconsider this simple model using Isham’s scheme, enriched by a magnetic flux through the sphere via a modification of the symplectic form. To maintain complete generality we construct the Hilbert space directly from the symmetry algebra, which is manifestly gauge-invariant, using ladder operators. In this way, we recover algebraically the complete classification of quantizations, and the corresponding energy spectra for the particle. The famous Dirac quantization condition for the monopole charge follows from the requirement that the classical and quantum Casimir invariants match. In an appendix we explain the relation between this approach and the more common one that assumes from the outset a Hilbert space of wave functions that are sections of a nontrivial line bundle over the sphere, and show how the Casimir invariants of the algebra determine the bundle topology.