advanced_notions:topological_defects

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advanced_notions:topological_defects [2017/12/20 10:10] jakobadmin [Student] |
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- | --> 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</cite> | ||

+ | </blockquote> | ||

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- | --> 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</cite> | ||

+ | </blockquote> | ||

advanced_notions/topological_defects.txt · Last modified: 2017/12/20 09:11 (external edit)

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