Content deleted Content added
Cuzkatzimhut (talk | contribs) m →top: minor typography |
No edit summary Tags: Mobile edit Mobile web edit |
||
| (36 intermediate revisions by 23 users not shown) | |||
Line 1:
{{Short description|Class of quantum field theory models}}
In [[quantum field theory]], a '''nonlinear ''σ'' model''' describes a
==Description==
Line 5 ⟶ 6:
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>{{
:<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.
Line 13 ⟶ 14:
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
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 ''G'' *. 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 ''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.
Line 22 ⟶ 23:
:<math>\sqrt{\det g}\mathcal{D}\Sigma.</math>
==Renormalization==
This model proved to be relevant in string theory where the two-dimensional manifold is named '''[[worldsheet]]'''.
{{cite journal|last=Friedan|first=D.|
:<math>\lambda\frac{\partial g_{ab}}{\partial\lambda}=\beta_{ab}(T^{-1}g)=R_{ab}+O(T^2)~,</math>
{{math|''R<sub>ab</sub>''}} being the [[Ricci tensor]] of the target manifold.
Line 29 ⟶ 31:
This represents a [[Ricci flow]], obeying [[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, so that the [[quantum field theory]] of this model is sensible (renormalizable).
Further adding nonlinear interactions representing flavor-chiral anomalies results in the [[Wess–Zumino–Witten model]],<ref>{{cite journal |first=E. |last=Witten |s2cid=122018499 |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 |url=http://projecteuclid.org/euclid.cmp/1103940923 }}</ref> which
augments the geometry of the flow to include [[Torsion tensor|torsion]], preserving renormalizability and leading to an [[infrared fixed point]] as well, on account of [[teleparallelism]] ("geometrostasis").<ref>{{
{{further|Ricci flow}}
==O(3) non-linear sigma model==
Line 40 ⟶ 43:
Since the ''n̂''-field lives on a sphere as well, the mapping {{math|''S<sup>2</sup>→ S<sup>2</sup>''}} is in evidence, the solutions of which are classified by the second [[homotopy group]] of a 2-sphere: These solutions are called the O(3) [[Instantons]].
This model can also be considered in 1+2 dimensions, where the topology now comes only from the spatial slices. These are modelled as R^2 with a point at infinity, and hence have the same topology as the O(3) instantons in 1+1 dimensions. They are called sigma model lumps.
==See also==
Line 46 ⟶ 51:
* [[Little Higgs]]
* [[Skyrmion]], a soliton in non-linear sigma models
* [[Polyakov action]]
* [[WZW model]]
* [[Fubini–Study metric]], a metric often used with non-linear sigma models
* [[Ricci flow]]
* [[Scale invariance]]
==References==
{{Reflist|
==External links==
*
* {{cite journal |doi=10.1023/A:1021009008129|year=2002|last1=Kulshreshtha|first1=U.|last2=Kulshreshtha|first2=D. S.|s2cid=115710780|title=Front-Form Hamiltonian, Path Integral, and BRST Formulations of the Nonlinear Sigma Model|journal=International Journal of Theoretical Physics|volume=41|issue=10|pages=1941–1956}}
{{Quantum field theories}}
{{String theory topics |state=collapsed}}
{{DEFAULTSORT:Non-Linear Sigma Model}}
[[Category:Quantum field theory]]
[[Category:Mathematical physics]]
[[Category:Murray Gell-Mann]]
| |||