Electromagnetic Field Tensor
Landau's definition (§23). $F_{ik} = \partial_i A_k - \partial_k A_i$, packaging $\vec E$ and $\vec B$: $$F_{0\alpha} = E_\alpha, \qquad F_{\alpha\beta} = -\epsilon_{\alpha\beta\gamma}B_\gamma.$$
Geometric identity. $F = \tfrac12 F_{ik}\,dx^i \wedge dx^k = dA$ is the Curvature 2-Form of the $U(1)$ connection. The two Maxwell pairs split geometrically:
| Maxwell pair | Landau | Geometric status |
|---|---|---|
| $\nabla \cdot \vec B = 0$, Faraday | §26 | $dF = 0$: Bianchi identity — true for any connection, no physics content |
| Gauss, Ampère–Maxwell | §30 | $d\!\star\!F = \star J$: equation of motion from the action $\int F \wedge \star F$ |
Physical status. Gauge-invariant (abelian Ad is trivial), hence directly measurable — the force field. But $F$ is not the whole physical content of the connection: holonomy (Aharonov-Bohm Effect) and bundle class (Dirac Monopole) carry information beyond $F$'s local values. $F$ determines the connection only on simply-connected regions with trivial bundle.
Related: Curvature 2-Form · Four-Potential · Exterior Derivative · Characteristic Classes · Module 09 - Electromagnetism as a U(1) Gauge Theory
Sources: Landau §23–§30.