A hallmark of non-perturbative theories of quantum gravity is the absence of a fixed background geometry, and therefore the absence in a Planckian regime of any notion of length or scale that is defined a priori. This has potentially far-reaching consequences for the application of renormalization group methods à la Wilson, which rely on these notions in a crucial way. We review the status quo of attempts in the Causal Dynamical Triangulations (CDT) approach to quantum gravity to find an ultraviolet fixed point associated with the second-order phase transitions observed in the lattice theory. Measurements of the only invariant correlator currently accessible, that of the total spatial three-volume, has not produced any evidence of such a fixed point. A possible explanation for this result is our incomplete and perhaps naïve understanding of what constitutes an appropriate notion of (quantum) length near the Planck scale.

量子引力的非扰动理论的一个标志是没有固定的背景几何形状，因此在普朗克体制中没有任何先验定义的长度或尺度概念。这对于应用威尔逊重归一化组方法具有潜在的深远影响，后者非常关键地依赖于这些概念。我们回顾了因果动态三角剖分（CDT）方法对量子引力的尝试现状，以发现与晶格理论中观察到的二阶相变相关的紫外线固定点。当前唯一可访问的不变相关器的测量，即总空间三卷的测量，均未提供任何有关此固定点的证据。