Dualities of self-dual nonlinear electrodynamics

A bstract For any causal nonlinear electrodynamics theory that is “self-dual” (electromagnetic U (1) -duality invariant), the Legendre-dual pair of Lagrangian and Hamiltonian densities \L, H\ L H are constructed from functions \, h\ ℓ h on ℝ + related to a particle-mechanics Lagrangian and Hamiltonian. We show how a ‘duality’ relating ℓ to h h implies that L L and H H are related by a simple map between appropriate pairs of variables. We also discuss Born’s “Legendre self-duality” and implications of a new “Φ-parity” duality. Our results are illustrated with many examples.