Module 09 — Electromagnetism as a U(1) Gauge Theory
Why this module: The payoff. Reread Landau §16–§30 a second time and cash in every entry of the Module 04 translation table. Then grade the answers you wrote to Module 04's four questions.
The dictionary, now with proofs behind it
| Physics (Landau) | Geometry |
|---|---|
| Spacetime | Base manifold $M$ |
| Phase convention at each point | Point of a Principal Bundle $P$, $G = U(1)$ |
| Four-potential $A_i$ (§16) | Local Connection 1-Form |
| Gauge transformation (§18) | Change of Local Trivialization |
| Field tensor $F_{ik}$ (§23) | Curvature 2-Form, $F = dA$ |
| First Maxwell pair (§26) | Bianchi identity $dF=0$ |
| Second Maxwell pair (§30) | Equation of motion $d\!\star\!F = \star J$ from action $\int F \wedge \star F$ |
| Charged particle phase | Holonomy $\exp(i e \oint A)$ |
| Minimal coupling (§17) | Covariant Derivative on an Associated Bundle |
Answering Module 04's questions
- Why must interaction go through $A$? Coupling is parallel transport of the charged field's phase; only a connection (not its curvature) defines transport.
- Why the redundancy? Gauge freedom = freedom of local trivialization. It is exactly as "redundant" as coordinate freedom on a manifold — unavoidable if the bundle is nontrivial.
- Is $A$ bookkeeping? No: Aharonov-Bohm Effect. Holonomy is physical; $F$ alone doesn't determine it in non-simply-connected regions.
- Is $A$ always global? No: Dirac Monopole. A monopole forces a nontrivial bundle; $A$ exists only patchwise, and consistency quantizes charge.
Concepts
- Four-Potential · Electromagnetic Field Tensor · Gauge Transformation · Minimal Coupling — the four physics notes, now in final form
- Aharonov-Bohm Effect — holonomy made experimental
- Yang-Mills Theory — the nonabelian generalization; see what survives and what changes
Concepts in this module
Study tasks
- Second reread of Landau §16–§30 with the dictionary open
- Grade your Module 04 answers; write the corrections in "My Notes"
- Exercise: derive the Aharonov–Bohm phase difference $\Delta\varphi = \frac{e}{\hbar c}\Phi$ from holonomy
- Exercise: write Maxwell's equations entirely in form language ($dF = 0$, $d\!\star\!F = \star J$) and recover Landau's component equations