Landau & Lifshitz Vol. 2 — The Classical Theory of Fields
Role in this study: The physics you already know, to be reread with geometric eyes. Landau never says "bundle" — the game is to translate his tensor-calculus formulation into bundle language and see that nothing was lost and much is gained.
Reading Map (4th English ed., § numbers)
| Landau § | Content | Bundle translation |
|---|---|---|
| §1–§7 | Relativity, intervals, 4-vectors | Smooth Manifold (Minkowski space), Tangent Space |
| §16 | Four-potential $A_i$ | Four-Potential = local Connection 1-Form |
| §17 | Equations of motion of a charge | Minimal Coupling |
| §18 | Gauge invariance $A_i \to A_i + \partial_i f$ | Gauge Transformation = change of Local Trivialization |
| §23 | Field tensor $F_{ik} = \partial_i A_k - \partial_k A_i$ | Electromagnetic Field Tensor = Curvature 2-Form, $F = dA$ |
| §26 | First pair of Maxwell equations | Bianchi identity $dF = 0$ — automatic for any curvature |
| §27–§30 | Action for the field, second pair | Yang–Mills action $\int F \wedge \star F$; $d\!\star\!F = J$ needs dynamics |
Why Landau is a good launching point
- His four-potential-first approach (§16 before §23) is exactly the bundle-theoretic ordering: connection is fundamental, curvature is derived.
- Gauge invariance appears in §18 as a "non-uniqueness" — bundle theory explains it as coordinate freedom on the bundle, not a defect.
- What Landau cannot express: global effects. Aharonov-Bohm Effect and Dirac Monopole need honest bundles.
Primary use: Module 04 - Landau Reread - Electromagnetism as Geometry and Module 09 - Electromagnetism as a U(1) Gauge Theory.