Conservation of all Lipkin's zilches from symmetries of the standard electromagnetic action and a hidden algebra

Abstract

In 1964, Lipkin discovered the zilches, a set of conserved quantities in free electromagnetism. Among the zilches, optical chirality was identified by Tang and Cohen in 2010, serving as a measure of the handedness of light and leading to investigations into light's interactions with chiral matter. While the symmetries underlying the conservation of the zilches have been examined, the derivation of zilch conservation laws from symmetries of the standard free electromagnetic (EM) action using Noether's theorem has only been addressed in the case of optical chirality. We provide the full answer by demonstrating that the zilch symmetry transformations of the four-potential, $A_{\mu}$, preserve the standard free EM action. We also show that the zilch symmetries belong to the enveloping algebra of a "hidden" invariance algebra of free Maxwell's equations. This "hidden" algebra is generated by familiar conformal transformations and certain "hidden" symmetry transformations of $A_{\mu}$. Generalizations of the ``hidden'' symmetries are discussed in the presence of a material four-current, as well as in the theory of a complex Abelian gauge field. Additionally, we extend the zilch symmetries of the standard free EM action to the standard interacting action (with a non-dynamical four-current), allowing for a new derivation of the continuity equation for optical chirality in the presence of electric charges and currents. Furthermore, new continuity equations for the remaining zilches are derived.