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advanced_tools:group_theory:so3 [2018/04/15 16:33] aresmarrero [Concrete] |
advanced_tools:group_theory:so3 [2020/11/29 17:17] edi [Intuitive] |
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+ | The Lie group $SO(3)$ describes all possible rotations in 3-dimensional Euclidean space. It thus describes an important symmetry of the physical space we live in. (Other symmetries of our space are translations and boosts.) | ||
- | <note tip> | + | $SO(3)$ is closely related to the groups $SU(2)$ and $Sp(1)$. They are all locally isomorphic, that is, they have the same Lie algebra. |
- | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
- | </note> | + | |
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<tabbox Concrete> | <tabbox Concrete> | ||
- | **Representations*+ | + | **Representations** |
[{{ :advanced_tools:group_theory:so3reps.png?nolink |Diagram by Eduard Sackinger}}] | [{{ :advanced_tools:group_theory:so3reps.png?nolink |Diagram by Eduard Sackinger}}] | ||
+ | ---- | ||
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+ | The diagram below shows the defining 3-dimensional representation of $SO(3)$ in its upper branch. A 5-dimensional representations of the same group is shown in the lower branch. For an explanation of this diagram and more representations of $SO(3)$ see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | ||
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+ | [{{ :so3_3d_5d_reps.jpg?nolink }}] | ||