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--- advanced_tools:group_theory:su2 [2018/04/15 16:31]
aresmarrero
+++ advanced_tools:group_theory:su2 [2020/11/28 18:40]
edi [Concrete]
@@ Line 20: / Line 20: @@
 Elements of $SU(2)$ can be written as
-$$ U(x) = e^{i a \vec{r} \vec{\sigma} }= e^{i a \vec{r} \vec{\sigma}} = \cos(a) + i \vec{r} \vec{\sigma} \sin( a )$$
+$$ U(x) = e^{i a \vec{r} \vec{\sigma} } = \cos(a) + i \vec{r} \vec{\sigma} \sin( a )$$
-where $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ are the usual Pauli matrices and $ \vec{r} $ is a unit vector.
+where $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ are the usual Pauli matrices and $ \vec{r} $ is a unit vector. This is also known as the version of the the well-known Euler's identity for $2\times2$ matrices.
@@ Line 29: / Line 29: @@
 **Representations**
-[{{ :advanced_tools:group_theory:su2reps.png?nolink |Diagram by Eduard Sackinger}}]
+The diagram below shows the defining 2-dimensional representation of $SU(2)$, which describes spin-$1/2$ particles, in its upper branch. A 3-dimensional representations of the same group, which describes spin-$1$ particles, is shown in the lower branch.  For an explanation of this diagram and more representations of $SU(2)$ see [[https://esackinger.wordpress.com/|Fun with Symmetry]].
+[{{ :advanced_tools:group_theory:su2_qm_spin.jpg?nolink }}]
 <tabbox Abstract>