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 advanced_notions:topological_defects [2017/12/20 11:10]jakobadmin [Student] advanced_notions:topological_defects [2017/12/20 11:11] (current)jakobadmin [FAQ] Both sides previous revision Previous revision 2017/12/20 11:11 jakobadmin [FAQ] 2017/12/20 11:10 jakobadmin [Student] 2017/12/20 11:10 jakobadmin [Researcher] 2017/12/20 11:09 jakobadmin [Researcher] 2017/12/20 11:09 jakobadmin [Layman] 2017/12/20 11:08 jakobadmin [Researcher] 2017/12/20 11:08 jakobadmin [Student] 2017/12/20 11:07 jakobadmin [Examples] 2017/12/20 11:06 jakobadmin [Student] 2017/12/20 11:06 jakobadmin [Student] 2017/12/20 11:04 jakobadmin [Why is it interesting?] 2017/12/20 11:03 jakobadmin ↷ Page moved from basic_notions:topological_defects to advanced_notions:topological_defects2017/12/20 11:03 jakobadmin [Student] 2017/12/20 11:02 jakobadmin created 2017/12/20 11:11 jakobadmin [FAQ] 2017/12/20 11:10 jakobadmin [Student] 2017/12/20 11:10 jakobadmin [Researcher] 2017/12/20 11:09 jakobadmin [Researcher] 2017/12/20 11:09 jakobadmin [Layman] 2017/12/20 11:08 jakobadmin [Researcher] 2017/12/20 11:08 jakobadmin [Student] 2017/12/20 11:07 jakobadmin [Examples] 2017/12/20 11:06 jakobadmin [Student] 2017/12/20 11:06 jakobadmin [Student] 2017/12/20 11:04 jakobadmin [Why is it interesting?] 2017/12/20 11:03 jakobadmin ↷ Page moved from basic_notions:topological_defects to advanced_notions:topological_defects2017/12/20 11:03 jakobadmin [Student] 2017/12/20 11:02 jakobadmin created Line 110: Line 110: - --> ​Example2:# + --> ​What classes of topological defects exist?# + + <​blockquote>​ + The field theories discussed above fall into two classes from the point of view of their topology. Suppose we are working in d space dimensions. In the first place we have theories, like the Abelian Higgs model of §2.1, where we have a potential function + $V(\varphi)$ and $\varphi$ must tend to a zero (i.e. minimum) of V as we approach spatial infinity. In this case, at any given time, $\varphi$ defines a map + $$\varphi_\infty (\hat n) = \lim_{r\to\infty} \varphi(r \hat n)$$ + which takes its values in the set of values which minimises V, + $$M=\{ \varphi: V(\varphi)=0$$ + The directions $\hat n$ in which one can approach infinity constitute a (d - 1)-dimensional + sphere, the unit sphere in $R^d$. Thus $\varphi_\infty$ defines a map $S^{d-1} \to M$. + + **The second class of possibilities is not the result of non-trivial boundary conditions.** + Here we have a field which is always constrained to take its values in some manifold + $M$, which is not simply a linear space. This time the boundary conditions are actually + supposed to be trivial in the sense that $\varphi$ tends to a limit $\varphi_\infty \in M$ of course, independently + of the direction in which we approach spatial infinity. In this case we can compactify + space, $R^d$, by adding a point at infinity, to obtain what is topologically a sphere $S^d$ + (cf stereographic projection),​ with $\varphi$ being assigned the value $\varphi_N$ at the point of $S^d$ + corresponding to infinity. In this way we obtain a map $\varphi$ : $S^d \to M$. + + <​cite>​Topological structures in field theories by Goddard and Mansfield​ + ​ Line 116: Line 137: - --> ​Example2:# + --> ​What's the experimental status of topological defects?# + + + + <​blockquote>​ + No topological defects of any type have yet been observed by astronomers,​ + however, and certain types are not compatible with current observations;​ in + particular, if domain walls and monopoles were present in the observable + universe, they would result in significant deviations from what astronomers + can see. Theories that predict the formation of these structures within the + observable universe can therefore be largely ruled out. + + <​cite>​https://​arxiv.org/​pdf/​1206.1294.pdf​ +