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

Stephen C. Anco, Dennis The

Published 2005-01-20, updated 2005-06-28Version 2

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.