A hidden invariance algebra of Maxwell’s equations and the conservation of all Lipkin’s zilches from symmetries of the standard electromagnetic action

In 1964, Lipkin discovered a set of conserved quantities for free electromagnetism without a clear physical interpretation, known as the \textit{zilches}. In 2010, Tang and Cohen realized that one of the zilches, termed as \textit{optical chirality}, provides a measure of the handedness of light, motivating novel investigations into the interactions of light with chiral matter. Although the \textit{zilch symmetries} of Maxwell's equations underlying the conservation of the zilches are known, the question of how to explicitly derive all zilch conservation laws from symmetries of the standard free EM action using Noether's theorem has been answered only in the case of optical chirality. In this Letter, we provide the answer to this question by showing that the {zilch symmetries} leave invariant the standard free EM action.

In the rest of the article, we provide new insight concerning the conservation of the zilches and their underlying symmetries. First, we show that the zilch symmetries belong to the enveloping algebra of a \textit{``hidden'' invariance algebra} of free Maxwell's equations in potential form. The ``hidden'' algebra closes on \texorpdfstring{$so(6,\mathbb{C})_{\mathbb{R}}$}{so(6,R)} up to gauge transformations of the four-potential~\texorpdfstring{$A_{\mu}$}{A}. The generators of the ``hidden'' algebra consist of familiar conformal symmetry transformations and certain \textit{``hidden'' symmetry transformations} of~\texorpdfstring{$A_{\mu}$}{A}. We discuss the generalization of these ``hidden'' symmetries of Maxwell's equations in the presence of a material four-current,~\texorpdfstring{$J^{\mu}$}{J}. The ``hidden'' symmetries are also discussed for the theory of a complex Abelian gauge field (this is related to the complex formulation of duality-symmetric electromagnetism). Finally, we show that the zilch symmetries of the standard free EM action can be extended to zilch symmetries of the standard interacting action,~\texorpdfstring{$S'$}{S'}, by considering simultaneous transformations of both~\texorpdfstring{$A_{\mu}$}{A} and~\texorpdfstring{$J^{\mu}$}{J}. This allows us to give a new derivation of the continuity equation for optical chirality in the presence of electric charges and currents, while we also derive new continuity equations for the rest of the zilches.