# The Zilch Electromagnetic Conservation Law Revisited

## TL;DR

This paper revisits the zilch electromagnetic conservation law, demonstrating its origin as a Noether current from a duality-symmetric Maxwell Lagrangian, providing a fully covariant analysis for all components.

## Contribution

It establishes the variational origin of the zilch conservation law for all components within a covariant duality-symmetric Maxwell framework.

## Key findings

- Zilch arises as a Noether current from a duality-symmetric Lagrangian.
- The analysis is fully covariant and applies to all zilch tensor components.
- Valid for both real and complex duality-symmetric Maxwell Lagrangians.

## Abstract

It is shown that the zilch conservation law arises as the Noether current corresponding to a variational symmetry of a duality-symmetric Maxwell Lagrangian. The action of the corresponding symmetry generator on the duality-symmetric Lagrangian, while non-vanishing, is a total divergence as required by the Noether theory. The variational nature of the zilch conservation law was previously known only for some of the components of the zilch tensor, notably the optical chirality. By contrast, our analysis is fully covariant and is therefore valid for all components of the zilch tensor. The analysis is presented here for both real and complex versions of duality-symmetric Maxwell Lagrangians.