  Poisson electrodynamics is the low-energy limit of a rank-one noncommutative
gauge theory. It admits a closed formulation in terms of a Poisson structure on
the space-time manifold and reproduces ordinary classical electrodynamics in
the commutative limit. In this paper, we address and solve the problem of
minimal coupling to charged matter fields with a proper commutative limit. Our
construction essentially relies on the geometry of symplectic groupoids and
works for all integrable Poisson manifolds. An additional advantage of our
approach is that the corresponding Lagrangians can be defined on an arbitrary
metric background.
