查看“︁非线性σ模型”︁的源代码

在[[量子场论]]中，'''非线性σ模型（nonlinear sigma model）''' 描述一个[[纯量场]] <math>\sigma(x) \in M</math> 。M是目标流形。<math>x \in R^{3+1}</math>属于[[閔考斯基時空]]。<ref>{{Citation|last2=Lévy|first=M.|last=Gell-Mann|first2=M.|title=The axial vector current in beta decay|publisher=Italian Physical Society|doi=10.1007/BF02859738|year=1960|journal=Il Nuovo Cimento|issn=1827-6121|volume=16|pages=705–726|bibcode=1960NCim...16..705G}}</ref>

== 定义 ==
若M有[[黎曼流形|黎曼张量]] ''g''，[[拉格朗日量]]是<ref>{{Cite journal|title=On the symmetries of strong and weak interactions|last=Gürsey|first=F.|journal=Il Nuovo Cimento|issue=2|doi=10.1007/BF02860276|year=1960|volume=16|pages=230–240|bibcode=1960NCim...16..230G|pmc=|pmid=}}</ref>

:<math>\mathcal{L}={1\over 2}g^{\mu \nu}\partial_\mu\sigma \partial_\nu\sigma-V(\sigma)</math>

== O(3)非线性σ模型 ==
设<math>\sigma = n</math>。二维的非线性O(3)模型是

: <math>\mathcal L= \tfrac{1}{2}\ \partial^\mu \hat n \cdot\partial_\mu \hat n </math>

其中 <math>n = (n_1, n_2, n_3)</math>，<math>n\cdot n = 1</math>，<math>x\in R^2, \ \mu = 1, 2</math>。

n是 {{Math|''S<sup>2</sup>→ S<sup>2</sup>''}} 的函数。第三个[[同倫群|同伦群]]

<math>\pi_3(S^2) = \Z</math>

可以将这些函数分类。上文理论的经典解是O(3) [[瞬子]]。

== 参见 ==

* [[Wess-Zumino-Witten模型|WZW模型]]
* [[富比尼–施图迪度量|富比尼–施图度量]]
* [[里奇流]]
* [[泊里雅科夫作用量]]
* [[大N展开]]

== 参考文献 ==
{{Reflist|2}}

== 外部链接 ==

* Ketov, S. V. [http://www.scholarpedia.org/article/Nonlinear_Sigma_model ''Nonlinear Sigma model''] {{Wayback|url=http://www.scholarpedia.org/article/Nonlinear_Sigma_model |date=20210515131350 }} on Scholarpedia.
* U. Kulshreshtha, D.S. Kulshreshtha and H.J.W. Mueller-Kirsten, ``Gauge invariant O(N) {{lang|en|nonlinear sigma model}}(s) and gauge invariant Klein-Gordon theory: Wess-Zumino terms and Hamiltonian and BRST formulations``, Helv.Phys.Acta 66 752-794 (1993); U. Kulshreshtha and D.S. Kulshreshtha, ``Front-form Hamiltonian, path integral and BRST formulations of the {{lang|en|nonlinear sigma model}}``, Int. J. Theor. Phys. 41, 1941-1956 (2002), DOI: 10.1023/A:1021009008129.

{{量子场论}}
{{弦理论}}
[[Category:数学物理]]
[[Category:量子场论]]