在量子场论 中，非线性σ模型（nonlinear sigma model） 描述一个纯量场
σ σ -->
(
x
)
∈ ∈ -->
M
{\displaystyle \sigma (x)\in M}
x
∈ ∈ -->
R
3
+
1
{\displaystyle x\in R^{3+1}}
閔考斯基時空 。[ 1]
若M有黎曼张量 g ，拉格朗日量 是[ 2]
L
=
1
2
g
μ μ -->
ν ν -->
∂ ∂ -->
μ μ -->
σ σ -->
∂ ∂ -->
ν ν -->
σ σ -->
− − -->
V
(
σ σ -->
)
{\displaystyle {\mathcal {L}}={1 \over 2}g^{\mu \nu }\partial _{\mu }\sigma \partial _{\nu }\sigma -V(\sigma )}
设
σ σ -->
=
n
{\displaystyle \sigma =n}
L
=
1
2
∂ ∂ -->
μ μ -->
n
^ ^ -->
⋅ ⋅ -->
∂ ∂ -->
μ μ -->
n
^ ^ -->
{\displaystyle {\mathcal {L}}={\tfrac {1}{2}}\ \partial ^{\mu }{\hat {n}}\cdot \partial _{\mu }{\hat {n}}}
其中
n
=
(
n
1
,
n
2
,
n
3
)
{\displaystyle n=(n_{1},n_{2},n_{3})}
n
⋅ ⋅ -->
n
=
1
{\displaystyle n\cdot n=1}
x
∈ ∈ -->
R
2
,
μ μ -->
=
1
,
2
{\displaystyle x\in R^{2},\ \mu =1,2}
n是 S2 → S2 同伦群
π π -->
3
(
S
2
)
=
Z
{\displaystyle \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.
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