Duality and helicity: a symplectic viewpoint

TL;DR

This paper demonstrates that helicity in vacuum Maxwell equations is a conserved quantity linked to duality symmetry by framing electromagnetism as an infinite-dimensional symplectic system, with helicity as a moment map.

Contribution

It provides a symplectic geometric proof that helicity is the conserved quantity associated with duality symmetry in electromagnetism.

Findings

Helicity is shown to be the moment map of the duality symmetry.

Electromagnetism is modeled as an infinite-dimensional symplectic system.

Helicity conservation is derived from the symplectic structure and duality group.

Abstract

The theorem which says that helicity is the conserved quantity associated with the duality symmetry of the vacuum Maxwell equations is proved by viewing electromagnetism as an infinite dimensional symplectic system. In fact, it is shown that helicity is the moment map of duality acting as an $SO(2)$ group of canonical transformations on the symplectic space of all solutions of the vacuum Maxwell equations.