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.

在两个球面上量化粒子的问题已经通过多种方法处理，包括 Isham 的全局方法，该方法基于在相空间上传递作用的辛对称群的酉表示。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，通过球体的磁通量丰富了这个模型。为了保持完全的一般性，我们使用梯形算子直接从对称代数构造希尔伯特空间，该空间显然是规范不变的。通过这种方式，我们可以代数地恢复量子化的完整分类，以及粒子的相应能谱。单极电荷的著名狄拉克量子化条件遵循经典和量子卡西米尔不变量匹配的要求。