Electricity as Geometry
Electromagnetism is not a collection of equations — it is the geometry of a U(1) principal bundle over spacetime. The electromagnetic field is the curvature of a connection. The six operating regimes are distinct topological regions of a single admissible state space. The regime boundaries are Morse critical points where the topology of that space changes discontinuously.
dF = 0 · d⋆F = J · F = dA · γ = ∮ A·dk
The Complete Geometric Statement
The exterior derivative of the electromagnetic 2-form F vanishes. This single equation encodes both Gauss's law for magnetism (∇·B = 0) and Faraday's law (∇×E = −∂B/∂t). It states that F has no boundary — it is a closed form.
The codifferential of F equals the current 3-form J. This encodes both Gauss's law for electricity (∇·D = ρ) and Ampère's law (∇×H = J + ∂D/∂t). The Hodge star ⋆ encodes the material geometry (ε, μ).
These two equations hold on any smooth manifold in any number of dimensions. The specific geometry of spacetime (the metric) enters only through the Hodge star ⋆. This is the sense in which electricity is fundamentally geometric: the field F is the curvature of a U(1) connection on a principal bundle over spacetime.
VISUALIZATION — The electromagnetic 2-form F as oriented area elements at each point in space
A assigns a number to each directed curve in spacetime — the phase accumulated by a charged particle traversing that curve. It is not directly observable; only its curvature F = dA is.
Gauge freedom: A → A + dφ leaves F unchanged. The gauge group is U(1).
F assigns an oriented area to each infinitesimal surface element in spacetime. Its time components are the electric field E; its space components are the magnetic field B. F is the curvature of the connection A.
dF = 0 follows automatically from F = dA, since d² = 0 (Poincaré lemma).
The Hodge dual of F encodes the material response. Its time components are the electric displacement D; its space components are the magnetizing field H. The constitutive relations (ε, μ) are encoded in the Hodge star.
In vacuum: ⋆F uses the Minkowski metric. In materials: ε and μ modify the metric structure.
J assigns a number to each infinitesimal volume element in spacetime — the charge flowing through that volume. Charge conservation is the statement dJ = 0: the boundary of a boundary is zero.
d(d⋆F) = dJ = 0 is automatic, ensuring charge conservation is built into the geometry.
Gauge Invariance
The transformation A → A + dφ leaves F = dA unchanged (since d(dφ) = d²φ = 0). This is not a mathematical accident — it is the statement that the electromagnetic potential A is a connection on a principal U(1) bundle, and that changing the gauge is equivalent to choosing a different section of that bundle. The observable physics lives entirely in the curvature F, not in the connection A itself. This is why the Aharonov-Bohm effect is possible: a particle can accumulate a geometric phase (holonomy) by traversing a loop even in a region where F = 0, if the bundle is topologically non-trivial.