非线性σ模型

在量子场论中，非线性σ模型（nonlinear sigma model） 描述一个纯量场 $\sigma (x)\in M$ 。M是目标流形。 $x\in R^{3+1}$ 属于闵考斯基时空。^[1]

定义

{\mathcal {L}}={1 \over 2}g^{\mu \nu }\partial _{\mu }\sigma \partial _{\nu }\sigma -V(\sigma )

设 $\sigma =n$ 。二维的非线性O(3)模型是

{\mathcal {L}}={\tfrac {1}{2}}\ \partial ^{\mu }{\hat {n}}\cdot \partial _{\mu }{\hat {n}}

其中 $n=(n_{1},n_{2},n_{3})$ ， $n\cdot n=1$ ， $x\in R^{2},\ \mu =1,2$ 。

n是 $S 2 \to S 2$ 的函数。第三个同伦群

$\pi _{3}(S^{2})=\mathbb {Z}$

可以将这些函数分类。上文理论的经典解是O(3) 瞬子。

^ Gell-Mann, M.; Lévy, M., The axial vector current in beta decay, Il Nuovo Cimento (Italian Physical Society), 1960, 16: 705–726, Bibcode:1960NCim...16..705G, ISSN 1827-6121, doi:10.1007/BF02859738
^ Gürsey, F. On the symmetries of strong and weak interactions. Il Nuovo Cimento. 1960, 16 (2): 230–240. Bibcode:1960NCim...16..230G. doi:10.1007/BF02860276.

Ketov, S. V. Nonlinear Sigma model （页面存档备份，存于互联网档案馆） on Scholarpedia.
U. Kulshreshtha, D.S. Kulshreshtha and H.J.W. Mueller-Kirsten, ``Gauge invariant O(N) 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 nonlinear sigma model``, Int. J. Theor. Phys. 41, 1941-1956 (2002), DOI: 10.1023/A:1021009008129.