U(1)
Definition. The unit circle in $\mathbb{C}$: $U(1) = \{e^{i\theta} : \theta \in \mathbb{R}\}$, under multiplication. Compact, connected, abelian, 1-dimensional. Lie Algebra: $\mathfrak{u}(1) = i\mathbb{R}$.
The star of this study. Electromagnetism is the theory of a connection on a $U(1)$ Principal Bundle over spacetime. Nearly every simplification in Landau's treatment traces to properties of $U(1)$:
| Property of $U(1)$ | Consequence in EM |
|---|---|
| Abelian | $F = dA$ exactly (no $[A,A]$); Maxwell is linear |
| $\mathrm{Ad}$ trivial | $F$ is gauge-invariant, directly measurable |
| 1-dimensional | One gauge field $A_\mu$: the photon |
| Compact (circle, not line!) | Irreps labeled by $n \in \mathbb{Z}$ ⇒ charge quantization |
| $\pi_1(U(1)) = \mathbb{Z}$ | Winding ⇒ monopole/flux quantization, Hopf Fibration |
Woit's point. Irreducible representations: $\rho_n(e^{i\theta}) = e^{in\theta}$, $n \in \mathbb{Z}$. If the gauge group were $\mathbb{R}$ instead of $U(1)$, $n$ could be any real — compactness of the group is (one story for) why charge comes in integer lumps.
Related: Lie Group · Group Representation · Dirac Monopole · Module 09 - Electromagnetism as a U(1) Gauge Theory
Sources: Woit ($U(1)$ chapter); Landau §18 (gauge invariance, implicitly $U(1)$).