  Non commutative quantum mechanics can be viewed as a quantum system
represented in the space of Hilbert-Schmidt operators acting on non commutative
configuration space. Taking this as departure point, we formulate a coherent
state approach to the path integral representation of the transition amplitude.
  From this we derive an action for a particle moving in the non commutative
plane and in the presence of an arbitrary potential. We find that this action
is non local in time. However, this non locality can be removed by introducing
an auxilary field, which leads to a second class constrained system that yields
the non commutative Heisenberg algebra upon quantization. Using this action the
propagator of the free particle and harmonic oscillator are computed
explicitly. For the harmonic oscillator this action yields the frequencies
already obtained by other means in the literature.
