  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.
