Non-linear sigma model: Difference between revisions

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==O(3) non-linear sigma model==

OneA of the most famouscelebrated examplesexample, of particular interest due to its topological properties, is the ''O(3)'' nonlinear sigma {{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=(n_1,n_2,n_3)</math> with the constraint <math>\hat n\cdot \hat n=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=\textrm{const.}</math> 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]].

One of the most famous examples, of particular interest due to its topological properties, is the O(3) nonlinear sigma model in 1&nbsp;+&nbsp;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, weone havesees 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==