en>AnomieBOT: Dating maintenance tags: {{Unreferenced}}

2014-10-18T07:15:52Z

Dating maintenance tags: {{Unreferenced}}

en>777sms at 02:57, 14 April 2012

2012-04-14T02:57:39Z

New page

In [[quantum field theory]], a '''nonlinear σ model''' describes a [[scalar field]] 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 spin 0 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}}</ref>

==Description==

The target manifold T is equipped with a [[Riemannian metric]] g. Σ is a differentiable map from [[Minkowski space]] M (or some other space) to T.

The [[Lagrangian density]] in contemporary chiral form<ref>{{cite doi|10.1007/BF02860276|noedit}}</ref> is given by:
:<math>\mathcal{L}={1\over 2}g(\partial^\mu\Sigma_a,\partial_\mu\Sigma_b)-V(\Sigma)</math>

where here, we have used a + - - - [[metric signature]] and the [[partial derivative]] <math>\partial\Sigma</math> is given by a section of the [[jet bundle]] of T×M and V is the potential.

In the coordinate notation, with the coordinates Σ<sup>a</sup>, a=1,...,n where n is the dimension of T,

:<math>\mathcal{L}={1\over 2}g_{ab}(\Sigma) \partial^\mu \Sigma^{a} \partial_\mu \Sigma^{b} - V(\Sigma)</math>.

In more than 2 dimensions, nonlinear σ models are nonrenormalizable. 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 G *. If G is a [[Lie group]] and H is a [[Lie subgroup]], then the [[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 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 σ model.

When computing [[functional integration|path integrals]], the functional measure needs to be "weighted" by the square root of the [[determinant]] of g

:<math>\sqrt{\det g}\mathcal{D}\Sigma.</math>

This model proved to be relevant in string theory where the two-dimensional manifold is named '''worldsheet'''. Proof of renormalizability was given by [[Daniel Friedan]].<ref name="Frie80">
{{cite journal|last=Friedan|first=D.|authorlink=Daniel Friedan|title=Nonlinear models in 2+ε dimensions | journal = PRL | volume = 45 | issue = 13| pages = 1057 |publisher=|location=| year = 1980 | url = http://link.aps.org/doi/10.1103/PhysRevLett.45.1057 |doi= 10.1103/PhysRevLett.45.1057 | bibcode=1980PhRvL..45.1057F}}</ref> He showed that the theory admits a renormalization group equation, at the leading order of perturbation theory, in the form

:<math>\lambda\frac{\partial g_{\mu\nu}}{\partial\lambda}=\beta_{\mu\nu}(T^{-1}g)=R_{\mu\nu}+O(T^2).</math>

being <math>R_{\mu\nu}</math> the [[Ricci tensor]].

This represents a [[Ricci flow]] having [[Einstein field equations]] for the target manifold as a fixed point. The existence of such a fixed point is relevant, as it grants, at this order of perturbation theory, that [[conformal field theory|conformal invariance]] is not lost due to quantum corrections and one has a sensible [[quantum field theory]]. Further adding nonlinear interactions representing flavor-chiral anomalies results in the [[Wess–Zumino–Witten model]],<ref>{{cite journal |first=E. |last=Witten |title=Non-abelian bosonization in two dimensions |journal=[[Communications in Mathematical Physics]] |volume= 92| issue= 4 |year=1984 | pages= 455–472 | doi= 10.1007/BF01215276|bibcode = 1984CMaPh..92..455W }}</ref> which
augments the geometry of the flow to include [[Torsion tensor|torsion]], leading to an [[infrared fixed point]] as well, on account of [[teleparallelism]] ("geometrostasis").<ref>{{cite doi|10.1016/0550-3213(85)90053-7 |noedit}}</ref>

==O(3) Non-linear Sigma Model==

One of the most famous examples, of particular interest due to its topological properties, is the O(3) nonlinear sigma model in 1 + 1 dimensions, with the Lagrangian density

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

where <math>\hat n=(n_1,n_2,n_3)</math> with the constraint <math>\hat n\cdot \hat n=1</math> and <math>\mu=1,2</math>. This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning <math>\hat n=\textrm{const.}</math> at infinity. Therefore in the class of finite-action solutions we may identify the points at infinity as a single point, i.e. that space-time can be identified with a [[Riemann Sphere]]. Since the <math>\hat n</math>-field lives on a sphere as well, we have a mapping <math>S^2\rightarrow S^2</math>, the solutions of which are classified by the Second [[Homotopy group]] of a 2-sphere. These solutions are called the O(3) [[Instantons]].

==See also==
* [[Sigma model]]
* [[Chiral model]]
* [[Little Higgs]]
* [[Skyrmion]], a soliton in non-linear sigma models
* [[WZW model]]
* [[Fubini-Study metric]], a metric often used with non-linear sigma models.
* [[Ricci flow]]
* [[Scale invariance]]

==References==
{{Reflist|1}}

==External links==
* [http://www.scholarpedia.org/article/Nonlinear_Sigma_model 'Nonlinear Sigma model' on Scholarpedia]

{{Quantum field theories}}

{{DEFAULTSORT:Non-Linear Sigma Model}}
[[Category:Quantum field theory]]

Counting problem (complexity) - Revision history

en>AnomieBOT: Dating maintenance tags: {{Unreferenced}}

en>777sms at 02:57, 14 April 2012