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--- advanced_tools:group_theory:so3 [2018/04/15 16:32]
aresmarrero created
+++ advanced_tools:group_theory:so3 [2023/04/17 03:28] (current)
edi [Concrete]
@@ Line 2: / Line 2: @@
 <tabbox Intuitive>
+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.)
-<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>
+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}
+& 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}
-[{{ :advanced_tools:group_theory:so3reps.png?nolink |Diagram by Eduard Sackinger}}]
+----
+**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]].
+[{{ :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]].
+[{{ :advanced_tools:group_theory:so3_cross.jpg?nolink }}]
 <tabbox Abstract>

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