非定域相互作用的标度变换
Scaling Transformation for Nonlocal Interactions
摘要：
根据标度变换与重整化的关系，本文将标度变换与非定域相互作用联系起来。一方面，这种联系使我们可以用局部对称的方法来解释非定域性。另一方面，如果把重整化中的运行效应归结为标度变换，我们发现强子间的非定域相互作用可以作为标度变换的实验场。首先从群论出发，直接导出标度变换的算符/坐标表示和么正/旋量表示，然后通过寻找一个标度不变的相互作用顶点，模仿洛伦兹变换应用于Dirac方程的类似过程，将它们联系起来.本文的主要特点是，我们讨论的表示在一个单一的物理框架。这些表象分别对应于同一量子系统的空间自由度和内在自由度。而后者被认为对自旋角动量的贡献在文献中从未被认真考虑过。非定域相互作用拉格朗日量在标度变换下发生变化，类似于重整化中的运行情况。而全拉格朗日量只有在某些极端条件下才成为标度不变量。讨论了这种极端拉氏量的守恒定律，并给出了一个对自旋角动量有贡献的标量。最后设计了一种机制来检验非局域相互作用的标度效应。
英文摘要：
In the light of their relationships with renormalization, in this paper we associate the scaling transformation with nonlocal interactions. On one hand, the association leads us to interpret the nonlocality with locally symmetric method. On the other hand, we find that the nonlocal interaction between hadrons could be test ground for scaling transformation if ascribing the running effects in renormalization to scaling transformation. First we derive directly from group theory the operator/coordinate representation and unitary/spinor representation for scaling transformation, then link them together by inquiring a scaling-invariant interaction vertex mimicking the similar process of Lorentz transformation applied to Dirac equation. The main feature of this paper is that we discuss both the representations in a sole physical frame. The representations correspond respectively to the spatial freedom and the intrinsic freedom of the same quantum system. And the latter is recognized to contribute to spin angular momentum that in literature has never been considered seriously. The nonlocal interaction Lagrangian turns out to vary under scaling transformation, analogous to running cases in renormalization. And the total Lagrangian becomes scale invariant only under some extreme conditions. The conservation law of this extreme Lagrangian is discussed and a contribution named scalum appears to the spin angular momentum. Finally a mechanism is designed to test the scaling effect on nonlocal interaction.
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