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U(1) Symmetry Group
Linked via "Abelian"
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The group $\mathrm{U}(1)$ is Abelian, meaning multiplication is commutative. It is topologically equivalent to the circle $\mathbb{S}^1$ and the rotation group $\mathrm{SO}(2)$/). Crucially, it is the maximal Abelian subgroup of the unitary group $\mathrm{U}(n)$ for any $n \ge 1$.
It is also closely related to the special orthogonal group $\mathrm{SO}(2)$/). Since every element of $\mathrm{SO}… -
U(1) Symmetry Group
Linked via "Abelian"
The Lie algebra associated with $\mathrm{U}(1)$, denoted $\mathfrak{u}(1)$, is one-dimensional and is spanned by the generator $T$ corresponding to infinitesimal transformations:
$$U(\epsilon) = e^{i\epsilon T} \approx 1 + i\epsilon T$$
For $\mathrm{U}(1)$, the generator $T$ is simply the identity multiplied by a real scaling factor, often normalized such that $T=1/2$ in specific contexts (like spin systems), or $T=1$ when relating directly to the imaginary unit $i$ in the exponent. The…