====== U(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 $U(1)$. **Lie Algebra** The Lie algebra corresponding to the group U(1) is usually identified with the set of pure imaginary numbers $Im \mathbb{C} = \{ i \theta : \theta \in \mathbb{R} \}$. 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 }}] The motto in this section is: //the higher the level of abstraction, the better//.