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advanced_tools:group_theory:u1 [2018/04/14 13:08] theodorekorovin [Intuitive] |
advanced_tools:group_theory:u1 [2023/04/17 03:25] (current) edi [Concrete] |
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- | ====== $U(1)$====== | + | ====== U(1)====== |
<tabbox Intuitive> | <tabbox Intuitive> | ||
- | Phase factors $e^{i \phi(\vec x,t)}$, like they appear in [[theories: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 $U(1)$. | ||
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+ | ---- | ||
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+ | **Representations** | ||
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+ | 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]]. | ||
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+ | [{{ :advanced_tools:group_theory:representation_theory:u1_conj_rep.jpg?nolink }}] | ||
<tabbox Abstract> | <tabbox Abstract> | ||