A New Look on the Electromagnetic Duality. Suggestions and Developments

In this paper a new look on the electro-magnetic duality is presented and appropriately exploited. The duality analysis in the nonrelativistic and relativistic formulations is shown to lead to the idea the mathematical model field to be a differential form valued in the 2-dimensional vector space ${\cal R}^2$. A full ${\cal R}^2$ covariance is achieved through introducing explicitly the canonical complex structure ${\cal I}$ of ${\cal R}^2$ in the nonrelativistic equations. The connection of the relativistic Hodge * with ${\cal I}$ is shown and a complete coordinate free relativistic form of the equations and the conservative quantities is obtained. The duality symmetry is interpreted as invariance of the conservative quantities and conservation equations.