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advanced_tools:group_theory:so3 [2018/04/15 16:33] aresmarrero [Concrete] |
advanced_tools:group_theory:so3 [2020/11/29 17:59] 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.) | ||
- | <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> | + | |
| | ||
<tabbox Concrete> | <tabbox Concrete> | ||
+ | A general element of $SO(3)$ can be written as the product of three rotation matrices, one about the $x$, $y$, and $z$ axes: $R(\theta_x,\theta_y,\theta_z) = R_x(\theta_x) \cdot R_y(\theta_y) \cdot R_z(\theta_z)$, where | ||
+ | |||
+ | \begin{eqnarray} | ||
+ | & & R_x(\theta_x) = | ||
+ | \begin{pmatrix} | ||
+ | 1 & 0 & 0 \\ 0 & \cos\theta_x & -\sin\theta_x \\ 0 & \sin\theta_x & \cos\theta_x | ||
+ | \end{pmatrix}, \label{eq:rotx} \\ | ||
+ | & & R_y(\theta_y) = | ||
+ | \begin{pmatrix} | ||
+ | \cos \theta_y & 0 & \sin\theta_y \\ 0 & 1 & 0 \\ -\sin\theta_y & 0 & \cos\theta_y | ||
+ | \end{pmatrix}, \label{eq:roty} \\ | ||
+ | & & R_z(\theta) = | ||
+ | \begin{pmatrix} | ||
+ | \cos \theta_z & -\sin \theta_z & 0 \\\sin\theta_z & \cos\theta_z & 0 \\ 0 & 0 & 1 | ||
+ | \end{pmatrix}. \label{eq:rotz} | ||
+ | \end{eqnarray} | ||
+ | |||
+ | ---- | ||
+ | |||
**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. | ||
+ | |||
+ | [{{ :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). | ||
+ | |||
+ | [{{ :advanced_tools:group_theory:so3_cross.jpg?nolink }}] | ||
+ | |||
+ | 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:so3reps.png?nolink |Diagram by Eduard Sackinger}}] | ||
+ | |||
<tabbox Abstract> | <tabbox Abstract> |