This paper investigates the quantum states that emerge from the transformation design of conformal mapping on the two-dimensional sphere. Three results are reported. First, the construction of form-invariant spherical harmonics labelled by the fractional quantum number through a scalar potential interaction is given. Second, the form-invariant equation of the charge-monopole system is studied. Rather than the half-integer classification of the monopole harmonics, the quantization of the monopole harmonics here can be any fractional number specified by the conformal index. The gauge equivalent condition of the vector potentials which result in the invariant equation shows that the monopole field and the quantization condition of the pole strength due to Dirac can be generalized to more general vector fields and values in the conformal space. Finally, we explore the quadratic conformal image of the charged particle coupling to the constant monopole field on the sphere. It is shown...

中文翻译：

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球上量子态的形式不变解

本文研究了二维球形共形映射变换设计中出现的量子态。报告了三个结果。首先，给出了通过标量势相互作用用分数量子数标记的形式不变球谐函数的构造。其次，研究了电荷单极子系统的形式不变方程。除了单极子谐波的半整数分类，这里的单极子谐波的量化可以是保形指数指定的任何分数。产生不变方程的矢量电势的规范等效条件表明，单极场和由于狄拉克引起的极强度的量化条件可以推广到共形空间中更一般的矢量场和值。最后，我们探索带电粒子耦合到球面上恒定单极场的二次共形图像。显示...