标题： 平滑量子引力：异质光滑性和量子引力 显示英文标题

标题： Smooth quantum gravity: Exotic smoothness and Quantum gravity

摘要： 在过去的二十年里，发现了许多关于奇异光滑性（例如，奇异的$\mathbb{R}^{4}$）与量子场论之间意想不到的关系。其中一些关系根植于与超弦理论和量子引力的关系。因此，人们会预期奇异光滑性与广义相对论的量子化直接相关。在这篇文章中，我们将支持这一猜想，并通过使用具有奇异光滑结构的平滑4流形，发展一种新的量子引力方法称为\emph{平滑量子引力}。特别是，我们讨论了狂野嵌入的三维流形的出现，我们将其识别为一个量子态。此外，我们利用叶层理论分析这个量子态，并将其与算子代数中的元素联系起来。然后，我们描述了一组几何非交换算子，即纽结代数，可以用来确定三维流形的几何形状。这个算子代数可以理解为经典观测值泊松代数的变形量子化，这些观测值由平行传输给出。该算子代数的结构通过使用康纳斯的量子微积分诱导了一个作用。分析了这个作用的缩放行为，以在大尺度下获得广义相对论的经典理论（GRT）。这种方法具有一些明显的特性：存在非线性引力子，与格点规范场论的联系，以及从4维到2维的维数约化。还讨论了一些宇宙学后果，比如膨胀阶段的出现。最后，我们会得到一个简单的图景，即从标准的$\mathbb{R}^{4}$到奇异的$R^{4}$的转变是一种几何的量子化。 显示英文摘要

摘要： Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic $\mathbb{R}^{4}$, and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called \emph{smooth quantum gravity} by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard $\mathbb{R}^{4}$ to the exotic $R^{4}$ is a quantization of geometry.