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arXiv 1003.5506gr-qchep-thmath.GT
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奇异光滑性与量子引力

Exotic Smoothness and Quantum Gravity

Torsten Asselmeyer-Maluga

摘要:

自从第一次在物理学中对奇异光滑性进行研究以来,人们就一直认为奇异光滑性会对量子引力产生直接影响。因此,Duston(arXiv:0911.4068)的阴性结果令人惊讶。仔细研究半经典的方法,发现几何和光滑结构之间的密切联系的隐含假设。但这两种结构,几何和光滑度,是相互独立的。本文在已知“几何求和”的前提下,计算了量子引力中路径积分的“光滑结构”部分。为此,我们使用的结手术的芬图舍尔和斯特恩适用于类E(n)的椭圆曲面。我们主要把注意力集中在K3曲面E(2)上。然后我们假设K3曲面的每一个奇异光滑结构都可以通过Fintushel和Stern的节点或链接手术生成。将结果应用于期望值的计算。在这里,我们讨论了两个可观的,体积和威尔逊圈,为建设一个奇异的4-流形使用结$5_{2}$和怀特黑德链接$Wh$。利用Mostow刚性,我们得到了拓扑贡献的期望值的体积。此外,我们得到了面积量化的理由。

英文摘要:

Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between geometry and smoothness structure. But both structures, geometry and smoothness, are independent of each other. In this paper we calculate the "smoothness structure" part of the path integral in quantum gravity assuming that the "sum over geometries" is already given. For that purpose we use the knot surgery of Fintushel and Stern applied to the class E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces E(2). Then we assume that every exotic smoothness structure of the K3 surface can be generated by knot or link surgery a la Fintushel and Stern. The results are applied to the calculation of expectation values. Here we discuss the two observables, volume and Wilson loop, for the construction of an exotic 4-manifold using the knot $5_{2}$ and the Whitehead link $Wh$. By using Mostow rigidity, we obtain a topological contribution to the expectation value of the volume. Furthermore we obtain a justification of area quantization.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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