Dualities of Self-Dual Nonlinear Electrodynamics

TL;DR

This paper explores the duality properties of self-dual nonlinear electrodynamics, establishing a framework connecting Lagrangian and Hamiltonian formulations through dual functions and examining implications of various dualities.

Contribution

It introduces a method to construct dual Lagrangian and Hamiltonian densities from functions related to particle mechanics, revealing new duality relations and symmetries in self-dual electrodynamics.

Findings

Legendre dual pairs are related by a simple variable map.

Duality between functions $\

Discussion of Born's self-duality and $\

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 $\{\mathcal{L},\mathcal{H}\}$ are constructed from functions $\{\ell,h\}$ on ${\bf R}^+$ related to a particle-mechanics Lagrangian and Hamiltonian. We show how a `duality' relating $\ell$ 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 "$\Phi$-parity" duality. Our results are illustrated with many examples.