Dualities of self-dual nonlinear electrodynamics

Abstract

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} are constructed from functions {ℓ, h} on R + related to a particle-mechanics Lagrangian and Hamiltonian. We show how a ‘duality’ relating ℓ to h implies that L and 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.