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--- advanced_tools:group_theory:u1 [2018/05/03 13:04]
jakobadmin ↷ Links adapted because of a move operation
+++ advanced_tools:group_theory:u1 [2020/12/12 23:06]
edi [Concrete]
@@ Line 3: / Line 3: @@
 <tabbox Intuitive>
-Phase factors $e^{i \phi(\vec x,t)}$, like they appear in [[theories:quantum_mechanics:canonical_quantum_mechanics|quantum mechanics]], are just complex numbers with amplitude $1$. Therefore, we can picture them as points on a circle with radius $1$:
+Phase factors $e^{i \phi(\vec x,t)}$, like they appear in [[theories:quantum_mechanics:canonical|quantum mechanics]], are just complex numbers with amplitude $1$. Therefore, we can picture them as points on a circle with radius $1$:
 This collection of all complex numbers with amplitude $1$ is what we call the group $U(1)$.
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 Take note that the tangent space of a circle is, of course, just a copy of $\mathbb{R}$ but the isomorphic space $Im \mathbb{C}$ is more convenient because its elements can be "exponentiated" to give the elements $e^{i \theta}$ of $U(1)$.
+----
+**Representations**
+The diagram below shows the defining representation of $U(1)$ in its upper branch and the conjugate representations of the same group in its lower branch. For a more detailed explanation of this diagram and more representations of $U(1)$ see [[https://esackinger.wordpress.com/|Fun with Symmetry]].
+[{{ :advanced_tools:group_theory:representation_theory:u1_conj_rep.jpg?nolink }}]
 <tabbox Abstract>