Path integral action and Chern-Simons quantum mechanics in noncommutative plane

In this paper, the connection between the path integral representation of propagators in the coherent state basis with additional degrees of freedom \cite{rohwer} and the one without any such degrees of freedom \cite{sgfgs} is established. We further demonstrate that the path integral formalism developed in the noncommutative plane using the coherent state basis leads to a quantum mechanics involving a Chern-Simons term in momentum which is of noncommutative origin. The origin of this term from the Bopp-shift point of view is also investigated. A relativistic generalization of the action derived from the path integral framework is then proposed. Finally, we construct a map from the commutative quantum Hall system to a particle in a noncommutative plane moving in a magnetic field. The value of the noncommutative parameter from this map is then computed and is found to agree with previous results.