advanced_tools:topology:winding_number
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advanced_tools:topology:winding_number [2017/12/20 11:23] jakobadmin created |
advanced_tools:topology:winding_number [2018/05/05 12:37] (current) jakobadmin ↷ Links adapted because of a move operation |
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | Winding numbers are crucial to understand the [[theories:quantum_theory:quantum_field_theory:qcd_vacuum|QCD vacuum]] and important related effects like [[advanced_notions:quantum_field_theory:instantons|instantons]] in [[theories:quantum_theory:quantum_field_theory|quantum field theory]]. | + | Winding numbers are crucial to understand the [[advanced_notions:quantum_field_theory:qcd_vacuum|QCD vacuum]] and important related effects like [[advanced_notions:quantum_field_theory:instantons|instantons]] in [[theories:quantum_field_theory:canonical|quantum field theory]]. |
| <tabbox Layman> | <tabbox Layman> | ||
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| <WRAP tip> Winding number = topological charge = Pontryagin index = second Chern class number </WRAP> | <WRAP tip> Winding number = topological charge = Pontryagin index = second Chern class number </WRAP> | ||
| - | [{{ :advanced_tools:topology:windingnumber.png?nolink&200 |Source: page 80 Selected Topics in Gauge Theories by Walter Dittrich, Martin Reuter}}] | + | [{{ :advanced_tools:topology:windingnumber.png?nolink&400|Source: page 80 Selected Topics in Gauge Theories by Walter Dittrich, Martin Reuter}}] |
| - | It is instructive to study a toy model to get some intuition for the notions that are commonly used. To understand the notion "winding number", we therefore consider $U(1)$ gauge transformations, and only one spatial dimension. Thus, our gauge transformations are maps $g(x)$ from $\mathbb{R}$ to $U(1)$. | ||
| - | We are interested in gauge transformations that yield physical gauge field configurations through | ||
| - | \begin{equation} | + | To understand the notion "winding number", we therefore consider $U(1)$ transformations in a toy model with just one spatial dimension that is curled up to a circle. The $U(1)$ transformations depend on the spatial coordinate, which simply means we have a map from each point in space to an $U(1)$ element. |
| - | G_{\mu}^{\left( pg\right) }=\frac{-i}{g}U\partial_{\mu}U^{\dagger} | + | |
| - | \end{equation} | + | |
| - | Thus, we assume that our $g(x)$ behave nicely everywhere. This allows us to add the point "infinity" to the real line $\mathbb{R}$. When we do this, we can interpret our construction of real line $\mathbb{R}$ plus infinity as a circle. This process is known as compactification and shown in the following image: | + | In other words, we consider maps from the circle $S^1$ to the group $U(1)$. Points on the circle can be parameterized by an angle $\phi$ that runs from $0$ to $2\pi$ and therefore, we can write possible maps as follows: |
| - | + | ||
| - | [{{ :advanced_tools:topology:compactifyingrealline.jpg?nolink |Source: http://www.iop.vast.ac.vn/theor/conferences/vsop/18/files/QFT-4.pdf }}] | + | |
| - | + | ||
| - | + | ||
| - | Therefore, instead of maps from $\mathbb{R}$ to $U(1)$, we consider maps from the circle $S^1$ to $U(1)$. Points on the circle can be parameterized by an angle $\phi$ that runs from $0$ to $2\pi$ and therefore, we can write possible maps as follows: | + | |
| $$ S^1 \to U(1) : g(\phi)= e^{i\alpha(\phi)} \, . $$ | $$ S^1 \to U(1) : g(\phi)= e^{i\alpha(\phi)} \, . $$ | ||
advanced_tools/topology/winding_number.1513765410.txt.gz · Last modified: 2017/12/20 10:23 (external edit)
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