Conservation laws for the Maxwell-Dirac equations with a dual Ohm’s law

In all areas of physics, conservation laws are essential since they allow us to draw conclusions of a physical system under study in an efficient way. Electrodynamics, in terms of the standard Maxwell electromagnetic equations for fields in vacuum, exhibit a rich set of symmetries to which conserved quantities are associated. Recently, there has been a renewed interest in the utilisation of such quantities. Here we use a theorem of Ibragimov (2006) to derive conservation laws for Dirac’s symmetric version of the Maxwell-Lorentz microscopic equations, allowing for magnetic charges and magnetic currents, where the latter, just as electric currents, are assumed to be described by a linear relationship between the field and the current, i.e., an Ohm’s law. The method of Ibragimov (2006) produces two new adjoint vector fields which fulfil Maxwell-like equations. In particular, we obtain conservation laws for the symmetrized electromagnetic field which are non-local in time.