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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 [2023/04/17 03:25] (current)
edi [Concrete]
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 <tabbox Intuitive> ​ <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 $U(1)$. ​ 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 $U(1)$. ​ 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/​blog/​lie-groups-and-their-representations/#​u1_conj_rep|Fun with Symmetry]]. 
 + 
 +[{{ :​advanced_tools:​group_theory:​representation_theory:​u1_conj_rep.jpg?​nolink }}] 
 <tabbox Abstract> ​ <tabbox Abstract> ​
  
advanced_tools/group_theory/u1.1525345470.txt.gz · Last modified: 2018/05/03 11:04 (external edit)