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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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- | 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]]. | ||