涌现时空的几何重整化方法
Emergent Space-Time via a Geometric Renormalization Method
摘要：
我们提出了一个纯粹的几何重整化计划的度量空间（包括无色图），其中包括一个粗粒化和重新缩放操作这样的空间。粗粒化是基于准等距的概念，它产生一个离散的粗粒化空间的序列，每个具有一个连续极限下的重新缩放操作。我们提供的标准下，这样的序列收敛在一个超空间的度量空间，或可能构成一个共同的连续极限的吸引盆地，希望，可能代表我们的时空连续。我们讨论了这些粗粒度空间的一些性质以及它们的连续极限，如尺度不变性和度量相似性，并表明时空的不同层可以携带不同的距离函数，而同胚。在这个分析中的重要工具是一般度量空间的Gromov-Hausdorff距离泛函和图或网络的增长度。整个结构是在威尔逊重整化群的精神。此外，我们引入了一个物理相关的概念，在我们的分析感兴趣的空间，例如，为正规格减少到普通的格维数的尺寸。我们表明，这个尺寸是稳定的粗粒化过程下，只要后者是足够的本地，即准等距，并讨论了条件下，这个尺寸是一个整数。讨论了当维数为非整数时极限空间为分形的可能性。在本文的最后，我们简要地提到的可能性，我们的网络进行translocal远阶，从而导致虫洞空间的概念和规模依赖的尺寸，如果粗粒化过程不再是本地的。
英文摘要：
We present a purely geometric renormalization scheme for metric spaces (including uncolored graphs), which consists of a coarse graining and a rescaling operation on such spaces. The coarse graining is based on the concept of quasi-isometry, which yields a sequence of discrete coarse grained spaces each having a continuum limit under the rescaling operation. We provide criteria under which such sequences do converge within a superspace of metric spaces, or may constitute the basin of attraction of a common continuum limit, which hopefully, may represent our space-time continuum. We discuss some of the properties of these coarse grained spaces as well as their continuum limits, such as scale invariance and metric similarity, and show that different layers of spacetime can carry different distance functions while being homeomorphic. Important tools in this analysis are the Gromov-Hausdorff distance functional for general metric spaces and the growth degree of graphs or networks. The whole construction is in the spirit of the Wilsonian renormalization group. Furthermore we introduce a physically relevant notion of dimension on the spaces of interest in our analysis, which e.g. for regular lattices reduces to the ordinary lattice dimension. We show that this dimension is stable under the proposed coarse graining procedure as long as the latter is sufficiently local, i.e. quasi-isometric, and discuss the conditions under which this dimension is an integer. We comment on the possibility that the limit space may turn out to be fractal in case the dimension is non-integer. At the end of the paper we briefly mention the possibility that our network carries a translocal far-order which leads to the concept of wormhole spaces and a scale dependent dimension if the coarse graining procedure is no longer local.
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