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In [[quantum field theory]], a '''nonlinear ''σ'' model''' describes a [[scalar field]] {{mvar|Σ}} which takes on values in a nonlinear manifold called the '''target manifold''' ''T''. The non-linear ''σ''-model was introduced by {{harvtxt|Gell-Mann|Lévy|1960|loc=section 6}}, who named it after a field corresponding to a spinless meson called ''σ'' in their model.<ref>{{Citation | last2=Lévy | first1=M. | last1=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>
==Description==
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The target manifold ''T'' is equipped with a [[Riemannian metric]] ''g''. {{mvar|Σ}} is a differentiable map from [[Minkowski space]] ''M'' (or some other space) to ''T''.
The [[Lagrangian density]] in contemporary chiral form<ref>{{Cite journal | last1 = Gürsey | first1 = F. | title = On the symmetries of strong and weak interactions | doi = 10.1007/BF02860276 | journal = Il Nuovo Cimento | volume = 16 | issue = 2 | pages = 230–240 | year = 1960 | pmid = | pmc = | bibcode = 1960NCim...16..230G }}</ref> is given by
:<math>\mathcal{L}={1\over 2}g(\partial^\mu\Sigma,\partial_\mu\Sigma)-V(\Sigma)</math>
where we have used a + − − − [[metric signature]] and the [[partial derivative]] {{math| ''∂Σ''}} is given by a section of the [[jet bundle]] of ''T''×''M'' and {{mvar|V}} is the potential.
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