Nonlinear electrodynamics

In high-energy physics, nonlinear electrodynamics (often NED or NLED) refers to a family of nonlinear generalizations of Maxwell electrodynamics. For a theory to describe the electromagnetic field (a U(1) gauge field), its action must be gauge invariant; in the case of $U(1)$, this constraint dictates that the Lagrangian of a nonlinear electrodynamics must be a function of only $s\equiv -{\frac {1}{4}}F_{\alpha \beta }F^{\alpha \beta }$ (the Maxwell Lagrangian) and $p\equiv -{\frac {1}{8}}\epsilon ^{\alpha \beta \gamma \delta }F_{\alpha \beta }F_{\gamma \delta }$ (where $\epsilon$ is the Levi-Civita tensor). Notable NED models include the Born-Infeld model, the Euler-Heisenberg Lagrangian, and the CP-violating $U(1)$ Chern-Simons theory ${\mathcal {L}}=s+\theta p$.