Lipkin's conservation law, Noether's theorem, and the relation to optical helicity

TL;DR

This paper derives the conservation of optical zilch from electromagnetic symmetry, relates it to optical helicity, and extends these concepts to dispersive media, providing new insights into light's chirality and polarization properties.

Contribution

It introduces a symmetry-based derivation of optical zilch conservation and relates it to optical helicity in dispersive media, expanding understanding of light's chirality measures.

Findings

Optical zilch is conserved and linked to symmetry of electromagnetic action.

Circularly polarized waves are identified as zilch eigenstates.

Optical zilch relates to optical helicity, scaled by phase index in dispersive media.

Abstract

A simple conserved quantity for electromagnetic fields in vacuum was discovered by Lipkin in 1964. In recent years this "zilch" has been used as a measure of the chirality of light. The conservation of optical zilch is here derived from a simple symmetry of the standard electromagnetic action. The symmetry transformation allows the identification of circularly polarized plane waves as zilch eigenstates. The same symmetry is present for electromagnetism in a homogeneous, dispersive medium, allowing the derivation of the zilch density and flux in such a medium. Optical helicity density and flux are also derived for a homogeneous, dispersive medium. For monochromatic beams in vacuum, optical zilch is proportional to optical helicity. This monochromatic zilch-helicity relation acquires a factor of the square of the phase index in a dispersive medium.