Non-linear sigma model: Difference between revisions

Revision as of 16:40, 27 August 2015

Revision as of 16:57, 27 August 2015

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A celebrated example, of particular interest due to its topological properties, is the ''O(3)'' nonlinear {{mvar|σ}}-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&#770;''=(n_1''n₁,n_2 n₂,n_3) n<sub>3</mathsub>'') with the constraint <math>\hat ''n\cdot \hat &#770;''⋅''n&#770;''=1</math> and {{mvar|μ}}=1,2.

This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning <math>\hat ''n&#770;'' =\textrm{const.}</math> constant at infinity. Therefore, in the class of finite-action solutions, one 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>&#770;''-field lives on a sphere as well, onethe seesmapping a mapping {{math|''S<mathsup>S^2\rightarrow</sup>→ S^<sup>2</mathsup>''}} 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]].

==See also==