Editing Non-linear sigma model (section)

==Description==

The target manifold ''T'' is equipped with a [[Riemannian metric]]&nbsp;''g''. {{mvar|Σ}} is a differentiable map from [[Minkowski space]] ''M'' (or some other space) to&nbsp;''T''.

The [[Lagrangian density]] in contemporary chiral form<ref>{{Cite journal | last1 = Gürsey | first1 = F. | s2cid = 122270607 | 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 | 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 +&nbsp;−&nbsp;−&nbsp;− [[metric signature]] and the [[partial derivative]] {{math| ''∂Σ''}} is given by a section of the [[jet bundle]] of ''T''&times;''M'' and {{mvar|V}} is the potential.

In the coordinate notation, with the coordinates {{math|''Σ<sup>a</sup>''}}, ''a''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n'' where ''n'' is the dimension of&nbsp;''T'',
:<math>\mathcal{L}={1\over 2}g_{ab}(\Sigma) (\partial^\mu \Sigma^a) (\partial_\mu \Sigma^b) - V(\Sigma).</math>

In more than two dimensions, nonlinear ''σ'' models contain a dimensionful coupling constant and are thus not perturbatively renormalizable.
Nevertheless, they exhibit a non-trivial ultraviolet fixed point of the renormalization group both in the lattice formulation<ref>{{cite book | last = Zinn-Justin | first= Jean | title= Quantum Field Theory and Critical Phenomena | publisher = Oxford University Press | date = 2002 }}</ref><ref>{{ cite book | last= Cardy | first= John L. | title = Scaling and the Renormalization Group in Statistical Physics | publisher = Cambridge University Press | date = 1997 }}</ref> and in the double expansion originally proposed by [[Kenneth G. Wilson]].<ref>{{cite journal|last=Brezin|first=Eduard|year=1976|title=Renormalization of the nonlinear sigma model in 2 + epsilon dimensions|journal=Physical Review Letters|volume=36|issue=13|pages=691–693|bibcode=1976PhRvL..36..691B|doi=10.1103/PhysRevLett.36.691|author2=Zinn-Justin, Jean}}</ref> 

In both approaches, the non-trivial renormalization-group fixed point found for the [[n-vector model|''O(n)''-symmetric model]] is seen to simply describe, in dimensions greater than two, the critical point separating the ordered from the disordered phase. In addition, the improved lattice or quantum field theory predictions can then be compared to laboratory experiments on [[critical phenomena]], since the ''O(n)'' model describes physical [[Heisenberg ferromagnet]]s and related systems. The above results point therefore to a failure of naive perturbation theory in describing correctly the physical behavior of the ''O(n)''-symmetric model above two dimensions, and to the need for more sophisticated non-perturbative methods such as the lattice formulation.

This means they can only arise as [[effective field theory|effective field theories]]. New physics is needed at around the distance scale where the two point [[connected correlation function]] is of the same order as the curvature of the target manifold. This is called the [[UV completion]] of the theory. There is a special class of nonlinear σ models with the [[internal symmetry]] group&nbsp;''G''&nbsp;*. If ''G'' is a [[Lie group]] and ''H'' is a [[Lie subgroup]], then the [[Quotient space (topology)|quotient space]] ''G''/''H'' is a manifold (subject to certain technical restrictions like H being a closed subset) and is also a [[homogeneous space]] of ''G'' or in other words, a [[nonlinear realization]] of&nbsp;''G''. In many cases, ''G''/''H'' can be equipped with a [[Riemannian metric]] which is ''G''-invariant. This is always the case, for example, if ''G'' is [[compact group|compact]]. A nonlinear σ model with G/H as the target manifold with a ''G''-invariant Riemannian metric and a zero potential is called a quotient space (or coset space) nonlinear {{mvar|σ}} model.

When computing [[functional integration|path integrals]], the functional measure needs to be "weighted" by the square root of the [[determinant]] of&nbsp;''g'',
:<math>\sqrt{\det g}\mathcal{D}\Sigma.</math>