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 $\{\mathcal{L},\mathcal{H}\}$ are constructed from functions $\{\mathrm{\ell },h\}$ on ${\mathbf{R}}^{+}$ related to a particle-mechanics Lagrangian and Hamiltonian. We show how a `duality’ relating $\mathrm{\ell }$ to $h$ implies that $\mathcal{L}$ and $\mathcal{H}$ are related by a simple map between appropriate pairs of variables. We also discuss Born’s “Legendre self-duality” and implications of a new “$\mathrm{\Phi }$-parity” duality. Our results are illustrated with many examples.