  Although standard quantum mechanics has some non-local features, the
probability current of the Schr\"odinger equation is locally conserved, and
this allows minimal electromagnetic coupling. For some important extensions of
the Schr\"odinger equation, however, the probability current is not locally
conserved. We show that in these cases the correct electromagnetic coupling
requires a relatively simple extension of Maxwell theory which has been known
for some time and recently improved by covariant integration of a scalar degree
of freedom. We discuss some general properties of the solutions and examine in
particular the case of an oscillating dipolar source. Remarkable mathematical
and physical differences emerge with respect to Maxwell theory, as a
consequence of additional current terms present in the equations for $\nabla
\cdot \textbf{E}$ and $\nabla \times \textbf{B}$. Several possible applications
are mentioned.
