非整数通量-为什么它不工作
Noninteger flux - why it does not work
摘要：
在非整数磁通量的情况下，我们考虑了无一点的2-球面上的Dirac算子。证明了在Hilbert空间中只包含非奇异的2-球面波函数时，Hamilton算子（Dirac算子的平方）的谱问题总是可以很好地定义的。然而，这个希尔伯特空间在狄拉克算符的作用下不是不变的;狄拉克算符对某些非奇异态的作用产生奇异函数。这明确地打破了光谱的超对称性。在整数通量情形下，如果扩展希尔伯特空间使其包含相应纤维丛的局部正则截面，则可以恢复超对称性。对于非整数通量，这样的扩展是不可能的。
英文摘要：
We consider the Dirac operator on a 2-sphere without one point in the case of non-integer magnetic flux. We show that the spectral problem for the Hamiltonian (the square of Dirac operator) can always be well defined, if including in the Hilbert space only nonsingular on 2-sphere wave functions. However, this Hilbert space is not invariant under the action of the Dirac operator; the action of the latter on some nonsingular states produces singular functions. This breaks explicitly the supersymmetry of the spectrum. In the integer flux case, the supersymmetry can be restored if extending the Hilbert space to include locally regular sections of the corresponding fiber bundle. For non-integer fluxes, such an extention is not possible.
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