陈-西蒙斯理论

陈-西蒙斯理论（英语：Chern–Simons theory）以陈省身和詹姆斯·哈里斯·西蒙斯的名字命名，描述三维拓扑量子场论，在物理学有很多应用。此理论用陈-西蒙斯形式。

陈省身

陈-西蒙斯理论描述分数量子霍尔效应，导致2016年的物理诺贝尔奖。

经典公式

若（G，M）是主丛，M是流形，G是李群 / 规范群，A是联络，陈西蒙斯作用量是

${\displaystyle S={\frac {k}{4\pi }}\int _{M}{\text{tr}}\,(A\wedge dA+{\tfrac {2}{3}}A\wedge A\wedge A).}$

F是曲率：

${\displaystyle F=dA+A\wedge A\,}$

陈西蒙斯公式用最小作用量原理：

${\displaystyle 0={\frac {\delta S}{\delta A}}={\frac {k}{2\pi }}F.}$

陈-西蒙斯理论和纽结多项式

主条目：纽结多项式

三维的陈-西蒙斯理论生成很多重要的纽结多项式和纽结不变量：^[1]

陈西理论的纽结拓扑不变量

陈西规范群G	纽结多项式或不变量
SO(N)	考夫曼多项式
SU(N)	HOMFLY多项式
SU(2)或SO(3)	锺斯多项式（跟括号多项式有关）
U(1)	环绕数

拓扑量子计算机

拓扑量子计算机（英语：Topological quantum computer）是一种量子计算机。陈西蒙斯理论陈述有些拓扑量子计算机（英语：Topological quantum computer）的模型，例如“杨李模型”（Fibonacci model），这是最简单的非阿贝尔任意子拓扑量子计算机（英语：Topological quantum computer）之一。^[2][3]

参见

• 陈-西蒙斯理论是最有名的拓扑量子场论之一
• 拓扑量子场论
• Wess-Zumino-Witten模型
• 纽结理论

参考文献

1. Witten, Edward. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics. 1989-09, 121 (3): 351–399. ISSN 0010-3616. doi:10.1007/BF01217730 （英语）.
2. Freedman, Michael H.; Kitaev, Alexei; Larsen, Michael J.; Wang, Zhenghan. Topological Quantum Computation. arXiv:quant-ph/0101025. 2002-09-20 [2020-06-04]. （原始内容存档于2020-07-24）.
3. Wang, Zhenghan. Topological Quantum Computation (PDF). （原始内容存档 (PDF)于2017-08-30）.

阅读

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• Marino, Marcos. Chern–Simons Theory and Topological Strings. Reviews of Modern Physics. 2005, 77 (2): 675–720. Bibcode:2005RvMP...77..675M. arXiv:hep-th/0406005 . doi:10.1103/RevModPhys.77.675.
• Marino, Marcos. Chern–Simons Theory, Matrix Models, And Topological Strings. International Series of Monographs on Physics. Oxford University Press. 2005.
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• Witten, Edward. Quantum Field Theory and the Jones Polynomial. Communications in Mathematical Physics. 1989, 121 (3): 351–399. Bibcode:1989CMaPh.121..351W. MR 0990772. doi:10.1007/BF01217730.
• Witten, Edward. Chern–Simons Theory as a String Theory. Progress in Mathematics. 1995, 133: 637–678. Bibcode:1992hep.th....7094W. arXiv:hep-th/9207094 .