论文标题
球体上的粒子：在存在磁性的情况下群体量化
Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole
论文作者
论文摘要
量化2个粒子上粒子的问题已通过多种方法处理，包括基于ISHAM的全局方法基于在相位空间上实现的符号对称群的单一表示。在这里，我们使用ISHAM的方案重新考虑了这个简单的模型，该模型通过磁通磁通量通过互合形形式的修改丰富了球体。为了维持完整的一般性，我们直接使用梯子运算符直接从对称代数构建希尔伯特空间，显然是规范的。通过这种方式，我们以代数的量化为代数，对粒子的相应能量光谱进行了完整的分类。单子电荷的著名狄拉克量化条件遵循了经典和量子Casimir不变性匹配的要求。在附录中，我们解释了这种方法与更常见的关系之间的关系，从一开始就假设了波浪函数的希尔伯特空间，该空间是球体上非平凡的线束的部分，并显示代数的Casimir不变性如何确定束拓扑。
The problem of quantizing a particle on a 2-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.
