✎ SO(3) [The Physics Travel Guide]

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====== SO(3) ======

<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.)

$SO(3)$ is closely related to the groups $SU(2)$ and $Sp(1)$. They are all locally isomorphic, that is, they have the same Lie algebra.
  
<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**

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]].

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

</tabbox>

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