User Tools

Site Tools



This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
advanced_notions:topological_defects [2017/12/20 11:10]
jakobadmin [Student]
advanced_notions:topological_defects [2017/12/20 11:11] (current)
jakobadmin [FAQ]
Line 110: Line 110:
---> ​Example2:#+--> ​What classes of topological defects exist?# 
 +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</​cite>​ 
Line 116: Line 137:
---> ​Example2:#+--> ​What's the experimental status of topological defects?# 
 +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. 
advanced_notions/topological_defects.txt · Last modified: 2017/12/20 11:11 by jakobadmin