Electromagnetic duality

Electromagnetic duality is the discrete symmetry E → cB, cB → −E (equivalently F^{μν} → F^{μν} where F^{μν} = ½ ε^{μνρσ} F_{ρσ} is the dual field tensor) that maps the source-free Maxwell equations to themselves. In a vacuum region with no charges or currents, the four equations ∇·E = 0, ∇·B = 0, ∇×E = −∂B/∂t, ∇×B = (1/c²) ∂E/∂t are invariant under the rotation by 90° in the (E, cB) plane. The two Lorentz invariants of the EM field, E·B and |E|² − c²|B|², transform as the components of a complex scalar under the duality rotation, and free electromagnetic waves in vacuum display the duality manifestly.

In the actual universe, where electric charges and currents exist but magnetic charges and currents do not, the duality is broken at the level of source equations: ∂_μ F^{μν} = μ₀ J^ν (electric four-current sources F^{μν}) but ∂_μ F^{μν} = 0 (no magnetic source for F^{μν}). If magnetic monopoles existed the equations would become ∂_μ F^{μν} = μ₀ J_e^ν and ∂_μ F^{μν} = μ₀ J_m^ν, restoring perfect duality at the field-equation level — and the duality transformation would map (J_e, J_m) → (J_m, −J_e) just as it maps (E, cB) → (cB, −E). This S-duality* generalises in supersymmetric Yang-Mills theory to a continuous SL(2, ℤ) action on the complexified coupling τ = θ/(2π) + 4πi/g², connecting strongly-coupled to weakly-coupled regimes via the duality and underlying the Seiberg-Witten 1994 solution of N=2 SU(2) gauge theory. In string theory the same idea — T-duality, S-duality, the M-theory web — connects ostensibly different theories as different limits of one underlying structure.