Symmetries, conservation laws, and cohomology of Maxwell's equations using potentials

TL;DR

This paper derives new nonlocal symmetries and conservation laws for Maxwell's equations using a covariant joint potential system, highlighting invariance under duality and revealing geometric symmetries related to Killing-Yano tensors.

Contribution

It introduces a classification of all geometric symmetries of the joint potential system in Lorentz gauge, uncovering new nonlocal symmetries and conservation laws based on cohomology and geometric structures.

Findings

New nonlocal symmetries invariant under duality transformation.

Construction of nonlocal conservation laws from geometric symmetries.

Cohomology analysis showing electromagnetic field forms as the only nontrivial classes.

Abstract

New nonlocal symmetries and conservation laws are derived for Maxwell's equations using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling and duality-rotation symmetries, and the well-known Poincare and dilation symmetries which involve homothetic Killing vectors, the classification yields new geometric symmetries involving Killing-Yano tensors related to rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's equations are constructed from these geometric symmetries by applying a formula that uses the joint potentials and directly generates conserved currents from any (local or nonlocal) symmetries of Maxwell's equations. This formula is shown to arise through a series of mappings that relate, respectively, symmetries/adjoint-symmetries of the joint potential system and adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived as by-products of the study of cohomology of closed 1-forms and 2-forms locally constructed from the electromagnetic field and its derivatives to some finite order for all solutions of Maxwell's equations. The only nontrivial cohomology is shown to consist of the electromagnetic field (2-form) itself as well as its dual (2-form), and this 2-form cohomology is killed by the introduction of corresponding potentials.