In high-energy physics, nonlinear electrodynamics (NED or NLED) refers to a family of generalizations of Maxwell electrodynamics which describe electromagnetic fields that exhibit nonlinear dynamics. For a theory to describe the electromagnetic field (a U(1) gauge field), its action must be gauge invariant; in the case of U ( 1 ) {\displaystyle U(1)} , for the theory to not have Faddeev-Popov ghosts, this constraint dictates that the Lagrangian of a nonlinear electrodynamics must be a function of only s ≡ − 1 4 F α β F α β {\displaystyle s\equiv -{\frac {1}{4}}F_{\alpha \beta }F^{\alpha \beta }} (the Maxwell Lagrangian) and p ≡ − 1 8 ϵ α β γ δ F α β F γ δ {\displaystyle p\equiv -{\frac {1}{8}}\epsilon ^{\alpha \beta \gamma \delta }F_{\alpha \beta }F_{\gamma \delta }} (where ϵ {\displaystyle \epsilon } is the Levi-Civita tensor). Notable NED models include the Born-Infeld model, the Euler-Heisenberg Lagrangian, and the CP-violating U ( 1 ) {\displaystyle U(1)} Chern-Simons theory L = s + θ p {\displaystyle {\mathcal {L}}=s+\theta p} . Some recent formulations also consider nonlocal extensions involving fractional U(1) holonomies on twistor space, though these remain speculative.
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