Dual field tensor

The dual electromagnetic field tensor F^{μν} (often written F̃^{μν} or with a Hodge-star prefix) is the Hodge dual of F^{μν} obtained by contracting two of its indices with the rank-4 Levi-Civita symbol: F^{μν} = ½ ε^{μνρσ} F_{ρσ}. The operation effectively swaps the electric and magnetic components of F^{μν} (up to factors of c and signs that depend on the metric signature). Where F^{μν} has F^{0i} = E_i/c, the dual has F^{0i} = −B_i (with sign conventions varying by source); where F^{μν} has F^{ij} = −ε_{ijk} B_k, the dual has F^{ij} = (1/c) ε_{ijk} E_k. The dual tensor is itself antisymmetric, so it carries the same six independent components as F^{μν} — but rearranged.

The dual is the natural object to source magnetic monopoles, if they existed. The symmetric Maxwell equations would read ∂_μ F^{μν} = μ₀ J^ν (electric four-current sourcing F) and ∂_μ F^{μν} = μ₀ J_m^ν (magnetic four-current sourcing F), restoring perfect E↔B duality. In the actual universe the second equation is ∂_μ *F^{μν} = 0 — the electromagnetic field tensor is closed under exterior differentiation precisely because no magnetic monopoles have ever been observed despite four decades of dedicated searches. The most stringent monopole bounds come from the MoEDAL detector at the Large Hadron Collider and from cosmological-relic searches in lunar regolith. Paul Dirac's 1931 argument that monopoles are consistent with quantum mechanics — and that their existence would force electric charge to be quantised in units of e = 2πℏ/(g·μ₀c) where g is the magnetic charge — remains one of the cleanest theoretical arguments for charge quantisation, which is otherwise an unexplained empirical fact.