New formulation of Galilean relativistic Maxwell theory

In this paper, we discuss Galilean relativistic Maxwell theory in detail. We first provide a set of mapping relations, derived systematically, that connect the covariant and contravariant vectors in the Lorentz relativistic and Galilean relativistic formulations. Exploiting this map, we construct the two limits of Galilean relativistic Maxwell theory from usual Maxwell's theory in the potential formalism for both contravariant and covariant vectors which are now distinct entities. Field equations are derived and their internal consistency is shown. The entire analysis is then performed in terms of electric and magnetic fields for both covariant and contravariant components. Duality transformations and their connection with boost symmetry are discussed which reveal a rich structure. The notion of twisted duality is introduced. Next we consider gauge symmetry, construct Noether currents and show their on-shell conservation. We also discuss shift symmetry under which the Lagrangian is invariant, where the corresponding currents are now on-shell conserved. At the end we analyse the theory by including sources for both contravariant and covariant sectors. We show that sources are now off-shell conserved