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U(1) Symmetry Group
Linked via "charge"
In QED, the fundamental fields (like the electron field $\psi$) must transform under a local $\mathrm{U}(1)$ gauge transformation:
$$\psi(x) \rightarrow e^{i q \alpha(x)} \psi(x)$$
where $q$ is the charge of the particle and $\alpha(x)$ is a spatially and temporally varying angle. To maintain invariance of the Lagrangian density, the derivative operator must be replaced by the covariant derivative $D_\mu$:
$$D_… -
U(1) Symmetry Group
Linked via "Charge"
| System Property | Associated Symmetry | Generator Normalization | Key Conserved Quantity |
| :--- | :--- | :--- | :--- |
| Electromagnetism (QED) | Local $\mathrm{U}(1)$ | $Q$ (Charge) | Electric Charge |
| Superconductivity | Global $\mathrm{U}(1)$ | $\phi$ (Phase Angle) | Number of Cooper Pairs |
| Superfluid Helium-4 | Global $\mathrm{U}(1)$ | $\theta$ ([Velocity Potential](/entries/velocity…