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advanced_tools:group_theory:u1

Phase factors $e^{i \phi(\vec x,t)}$, like they appear in 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 Fun with Symmetry.

The motto in this section is: *the higher the level of abstraction, the better*.

advanced_tools/group_theory/u1.txt · Last modified: 2023/04/17 03:25 by edi

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