Nonlinear electrodynamics

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.^[1] For a theory to describe the electromagnetic field (a U(1) gauge field), its action must be gauge invariant; in the case of ${\displaystyle U(1)}$, 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 ${\displaystyle s\equiv -{\frac {1}{4}}F_{\alpha \beta }F^{\alpha \beta }}$ (the Maxwell Lagrangian) and ${\displaystyle p\equiv -{\frac {1}{8}}\epsilon ^{\alpha \beta \gamma \delta }F_{\alpha \beta }F_{\gamma \delta }}$ (where ${\displaystyle \epsilon }$ is the Levi-Civita tensor).^[1][2][3] Notable NED models include the Born-Infeld model,^[4] the Euler-Heisenberg Lagrangian,^[5] and the CP-violating ${\displaystyle U(1)}$ Chern-Simons theory ${\displaystyle {\mathcal {L}}=s+\theta p}$.^[2][6][7]

Some recent formulations also consider nonlocal extensions involving fractional U(1) holonomies on twistor space,^{[citation needed]} though these remain speculative.