Dualities of self-dual nonlinear electrodynamics

Abstract

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 $$ \left{\mathcal{L},\mathcal{H}\right} $$ L H

are constructed from functions $$ \left\{\ell, \mathfrak{h}\right\} $$

ℓ h

on ℝ+ related to a particle-mechanics Lagrangian and Hamiltonian. We show how a ‘duality’ relating ℓ to $$ \mathfrak{h} $$ h implies that $$ \mathcal{L} $$ L and $$ \mathcal{H} $$ 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.