从抛物线到斜驶BMS变换
From parabolic to loxodromic BMS transformations
摘要：
Bondi-Metzner-Sachs（BMS）变换中有一半是由扩展复平面的保向共形同胚组成的，称为分数线性（或Mobius）变换。这些可以是4种，即它们被分类为抛物线，或双曲线，或椭圆，或斜航，取决于固定点的数量和SL（2，C）群的投影版本中相关的2x2矩阵的迹的值。由此产生的SL（2，C）矩阵的特殊形式也影响BMS变换的另一半，并且在这里被用来提出我们再次称为抛物线、双曲、椭圆或斜驶的渐近对称群的4种实现。本文的第二部分证明了迹趋于无穷大的双曲变换和斜驶变换的子集对应于奇异Sturm-Liouville问题极限点条件的满足。因此，在描述渐近平坦时空的语言和复分析世界以及普通量子力学中的自伴问题之间可能存在着深刻的联系。
英文摘要：
Half of the Bondi-Metzner-Sachs (BMS) transformations consist of orientation-preserving conformal homeomorphisms of the extended complex plane known as fractional linear (or Mobius) transformations. These can be of 4 kinds, i.e. they are classified as being parabolic, or hyperbolic, or elliptic, or loxodromic, depending on the number of fixed points and on the value of the trace of the associated 2x2 matrix in the projective version of the SL(2,C) group. The resulting particular forms of SL(2,C) matrices affect also the other half of BMS transformations, and are used here to propose 4 realizations of the asymptotic symmetry group that we call, again, parabolic, or hyperbolic, or elliptic, or loxodromic. In the second part of the paper, we prove that a subset of hyperbolic and loxodromic transformations, those having trace that approaches infinity, correspond to the fulfillment of limit-point condition for singular Sturm-Liouville problems. Thus, a profound link may exist between the language for describing asymptotically flat space-times and the world of complex analysis and self-adjoint problems in ordinary quantum mechanics.
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