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--- advanced_tools:group_theory:representation_theory:dean_proffesor_reza_sanaye [2022/07/03 19:58]
edi
+++ advanced_tools:group_theory:representation_theory:dean_proffesor_reza_sanaye [2023/03/19 21:45] (current)
edi [Concrete]
@@ Line 3: / Line 3: @@
 <tabbox Intuitive>
-The trivial representation maps all group elements to the identity transformation. This representation exists for any group.
+The trivial representation maps all group elements to the identity element. This representation exists for any group.
-Spin-0 particles/fields transform under the trivial representation of SU(2). That is, their spin value does no depend on orientation in space.
+Spin-$0$ particles transform under the trivial representation of $SU(2)$. That is, their spin value does not change under rotation.
-The action transforms under the trivial representation of space-time and gauge symmetry groups. That is, the action does not depend on these symmetry transformations.
+The action (and often the Lagrangian) transforms under the trivial representation of the Lorentz group and the relevant gauge  groups. That is, the action is not affected by these symmetry transformations.
 <tabbox Concrete>
+**Example**
-<note tip>
+The diagram below shows the defining representation of $SU(2)$ in its upper branch and the trivial (1-dimensional) representation in its lower branch.
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
-</note>
+[{{ :advanced_tools:group_theory:representation_theory:su2_1d_rep.jpg?nolink }}]
+For a more detailed explanation of this diagram see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#su2_1d_rep|Fun with Symmetry]].
 <tabbox Abstract>

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