advanced_tools:group_theory:so3
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advanced_tools:group_theory:so3 [2023/04/17 03:28] edi [Concrete] |
advanced_tools:group_theory:so3 [2025/03/08 21:30] (current) edi [Concrete] |
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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. 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]]. | + | 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/rotation-in-3-dimensions-and-angular-momentum/#so3|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). For a more detailed explanation of this diagram, see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#so3_cross|Fun with Symmetry]]. | + | 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/rotation-in-3-dimensions-and-angular-momentum/#so3|Fun with Symmetry]]. |
| [{{ :advanced_tools:group_theory:so3_cross.jpg?nolink }}] | [{{ :advanced_tools:group_theory:so3_cross.jpg?nolink }}] | ||
advanced_tools/group_theory/so3.txt ยท Last modified: 2025/03/08 21:30 by edi
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