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-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

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

F是曲率：

F=dA+A\wedge A\,

陈西蒙斯公式用最小作用量原理：

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

陈-西蒙斯理论和纽结多项式

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

参考文献

^ 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）.
^ 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）.
^ Wang, Zhenghan. Topological Quantum Computation (PDF). （原始内容存档 (PDF)于2017-08-30）.

阅读

Chern, S.-S. & Simons, J. Characteristic forms and geometric invariants. Annals of Mathematics. 1974, 99 (1): 48–69. doi:10.2307/1971013.
Deser, Stanley; Jackiw, Roman; Templeton, S. Three-Dimensional Massive Gauge Theories (PDF). Physical Review Letters. 1982, 48: 975–978 [2019-12-28]. Bibcode:1982PhRvL..48..975D. doi:10.1103/PhysRevLett.48.975. （原始内容存档 (PDF)于2018-07-24）.
Intriligator, Kenneth; Seiberg, Nathan. Aspects of 3d N = 2 Chern–Simons-Matter Theories. Journal of High Energy Physics. 2013. Bibcode:2013JHEP...07..079I. arXiv:1305.1633  . doi:10.1007/JHEP07(2013)079.
Jackiw, Roman; Pi, S.-Y. Chern–Simons modification of general relativity. Physical Review D. 2003, 68: 104012. Bibcode:2003PhRvD..68j4012J. arXiv:gr-qc/0308071  . doi:10.1103/PhysRevD.68.104012.
Kulshreshtha, Usha; Kulshreshtha, D.S.; Mueller-Kirsten, H. J. W.; Vary, J. P. Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing. Physica Scripta . 2009, 79: 045001. Bibcode:2009PhyS...79d5001K. doi:10.1088/0031-8949/79/04/045001.
Kulshreshtha, Usha; Kulshreshtha, D.S.; Vary, J. P. Light-front Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing. Physica Scripta. 2010, 82: 055101. Bibcode:2010PhyS...82e5101K. doi:10.1088/0031-8949/82/05/055101.
Lopez, Ana; Fradkin, Eduardo. Fractional quantum Hall effect and Chern-Simons gauge theories. Physical Review B. 1991, 44: 5246. Bibcode:1991PhRvB..44.5246L. doi:10.1103/PhysRevB.44.5246.
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.
Witten, Edward. Topological Quantum Field Theory. Communications in Mathematical Physics. 1988, 117: 353 [2020-09-21]. Bibcode:1988CMaPh.117..353W. doi:10.1007/BF01223371. （原始内容存档于2017-08-25）.
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  .

[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）.

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