Definition A.5.1 \(\operatorname{U}(1)\) defined using complex numbers.
\begin{equation*}
\operatorname{U}(1) = \{ z \in \C : |z| = 1 \} = \{ e^{\mathcal{i}\theta} : \theta \in \R \}
\end{equation*}
Subsubsection \(\operatorname{U}(1)\)
The unitary group \(\operatorname{U}(1)\) is the Lie group of all complex numbers with absolute value \(1\) under multiplication.
\(\operatorname{U}(1)\) as complex numbers.
Unitary group of degree \(1\) is the set of complex numbers with magnitude \(1\text{.}\)
Remark A.5.2.
Geometrically, \(\operatorname{U}(1)\) represents the unit circle on the complex plane. Therefore,
\begin{equation*}
\operatorname{U}(1) \cong \mathbb{S}^1 \, .
\end{equation*}
\(\operatorname{U}(1)\) as as a matrix group.
Definition A.5.3 \(\operatorname{U}(1)\) defined using matrices.
\begin{equation*}
\operatorname{U}(1) = \{ M \in M_1(\C ) : MM^\dagger = I \}
\end{equation*}
