Dualities of self-dual nonlinear electrodynamics

For any causal nonlinear electrodynamics theory that is “self-dual” (electromagnetic U(1)-duality invariant), the Legendre-dual pair of Lagrangian and Hamiltonian densities $\left\{\mathcal{L},\mathcal{H}\right\}$ \( \left\{\mathcal{L},\mathcal{H}\right\} \) are constructed from functions $\left\{\ell ,\mathfrak{h}\right\}$ \( \left\{\ell, \mathfrak{h}\right\} \) on ℝ⁺ related to a particle-mechanics Lagrangian and Hamiltonian. We show how a ‘duality’ relating ℓ to $\mathfrak{h}$ \( \mathfrak{h} \) implies that $\mathcal{L}$ \( \mathcal{L} \) and $\mathcal{H}$ \( \mathcal{H} \) 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.