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山东大学学报(理学版) ›› 2011, Vol. 46 ›› Issue (10): 1-31.

• 论文 •    下一篇

时空量子结构的非交换几何方法

张瑞斌1,张晓2   

  1. 1.悉尼大学数学与统计学院, 澳大利亚悉尼;
    2.中国科学院数学与系统科学研究院数学研究所, 北京 100190
  • 收稿日期:2011-07-12 出版日期:2011-10-20 发布日期:2011-10-18

Noncommutative geometric approach to quantum spacetime

ZHANG Ruibin1, ZHANG Xiao2   

  1. 1. School of Mathematics and Statistics, University of Sydney, Sydney, Australia;
    2. Institute of Mathematics, Academy of Mathematics and Systems Science,
    Chinese Academy of Sciences, Beijing, China
  • Received:2011-07-12 Online:2011-10-20 Published:2011-10-18
  • About author:ZHANG Ruibin(1960- ), Male, Professor, his main research interests are in Lie theory and quantum physics. Email: rui.bin.zhang@sydney.edu.au ZHANG Xiao(1965- ), Male, Professor, his main research interests are in differential geometry, general relativity and noncommutative geometric. Email: xzhang@amss.ac.cn

摘要:

近年来,我们利用流形到高维平坦空间中的等距嵌入,与合作者一起系统发展了非交换微分几何理论。应用该理论我们构造了非交换广义相对论,希望以此来刻画时空在普朗克尺度下的本质结构特征。我们仔细研究了一些非交换时空的例子,包括量子平面引力波、量子史瓦西和史瓦西-德西特时空以及有关引力塌缩的量子托密度量。这里我们对所述理论和应用给出一个简短的综述。

关键词: 非交换几何, 非交换广义相对论, 非交换爱因斯坦场方程, 磨雅代数, 量子时空

Abstract:

Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to construct a noncommutative version of general relativity, which is expected to capture some essential structural features of spacetime at the Planck scale. Examples of noncommutative spacetimes were investigated in detail. These include quantisations of planefronted gravitational waves, quantum Schwarzschild spacetime and Schwarzschildde Sitterspacetime, and a quantum Tolman spacetime which is relevant to gravitational collapse. Here we briefly review the theory and its application in the study of quantum structure of spacetime.

Key words: noncommutative geometric, noncommutative general relativity, noncommutative Einstein field equation, Moyal algebras, quantum spacetime

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