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advanced_tools:group_theory:so3

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advanced_tools:group_theory:so3 [2020/11/29 17:59]
edi [Concrete]
advanced_tools:group_theory:so3 [2023/04/17 03:28] (current)
edi [Concrete]
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 <tabbox Intuitive> ​ <tabbox Intuitive> ​
-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.)+The Lie group $SO(3)$ describes all possible rotations ​of an object ​in 3-dimensional Euclidean space. It thus describes an important symmetry of the physical space we live in. (Other ​important spacetime ​symmetries are translations and boosts.)
  
 $SO(3)$ is closely related to the groups $SU(2)$ and $Sp(1)$. They are all locally isomorphic, that is, they have the same Lie algebra. $SO(3)$ is closely related to the groups $SU(2)$ and $Sp(1)$. They are all locally isomorphic, that is, they have the same Lie algebra.
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 **Representations** **Representations**
  
-The diagram below shows the defining (3-dimensional) representation of $SO(3)$ in its upper branch and a 5-dimensional representation of the same group in its lower branch.+The diagram below shows the defining (3-dimensional) representation of $SO(3)$ in its upper branch and a 5-dimensional representation of the same group in its lower branch. For a more detailed explanation of this diagram, see [[https://​esackinger.wordpress.com/​blog/​lie-groups-and-their-representations/#​so3_3d_5d_reps|Fun with Symmetry]].
  
 [{{ :​so3_3d_5d_reps.jpg?​nolink }}] [{{ :​so3_3d_5d_reps.jpg?​nolink }}]
  
-Instead of using 3x3 matrices for the Lie-algebra elements of the defining representation,​ we can also use 3-dimensional vectors (red box in the diagram below). Then, the Lie-algebra elements act on the representation space by means of the cross product (lower branch of the diagram).+Instead of using 3x3 matrices for the Lie-algebra elements of the defining representation,​ we can also use 3-dimensional vectors (red box in the diagram below). Then, the Lie-algebra elements act on the representation space by means of the cross product (lower branch of the diagram). For a more detailed explanation of this diagram, see [[https://​esackinger.wordpress.com/​blog/​lie-groups-and-their-representations/#​so3_cross|Fun with Symmetry]].
  
 [{{ :​advanced_tools:​group_theory:​so3_cross.jpg?​nolink }}] [{{ :​advanced_tools:​group_theory:​so3_cross.jpg?​nolink }}]
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-For a more detailed explanation of these diagrams and additional representations of $SO(3)$, see [[https://​esackinger.wordpress.com/​|Fun with Symmetry]]. 
  
  
advanced_tools/group_theory/so3.1606669159.txt.gz ยท Last modified: 2020/11/29 17:59 by edi