Nonlinear Electromagnetic Self-Duality and Legendre Transformations

TL;DR

The paper analyzes nonlinear electromagnetic self-duality in four dimensions, describing how continuous duality rotations act within a symplectic framework and how duality invariants can be constructed even when the Lagrangian itself is not invariant. It derives the general self-duality condition for Lagrangians $L(\alpha,\beta)$ and solves it in terms of an arbitrary function, with Born–Infeld providing a concrete example, and shows how duality can be extended to an axion–dilaton field $S$ with $SL(2,\mathbb{R})$ symmetry. A Schrödinger-type reformulation is discussed as a covariant embodiment of the duality, and duality is interpreted via a Legendre transformation between dual Lagrangians, linking to quantum aspects through functional integration. The work also outlines how discrete subgroups such as $SL(2,\mathbb{Z})$ arise in the presence of charged states and discusses implications for supersymmetry, gravity, and string-theoretic moduli spaces, thereby clarifying the structure and transformations of nonlinear self-dual electromagnetic theories.

Abstract

We discuss continuous duality transformations and the properties of classical theories with invariant interactions between electromagnetic fields and matter. The case of scalar fields is treated in some detail. Special discrete elements of the continuous group are shown to be related to the Legendre transformation with respect to the field strengths.