  We study pure noncommutative U(1) gauge theory representing its one-loop
effective action in terms of a phase space worldline path integral. We write
the quadratic action using the background field method to keep explicit gauge
invariance, and then employ the worldline formalism to write the one-loop
effective action, singling out UV-divergent parts and finite (planar and
non-planar) parts, and study renormalization properties of the theory. This
amounts to employ worldline Feynman rules for the phase space path integral,
that nicely incorporate the Fadeev-Popov ghost contribution and efficiently
separate planar and non-planar contributions. We also show that the effective
action calculation is independent of the choice of the worldline Green's
function, that corresponds to a particular way of factoring out a particle
zero-mode. This allows to employ homogeneous string-inspired Feynman rules that
greatly simplify the computation.
