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

Abstract

In this paper, the connection between the path integral representation of propagators in the coherent state basis with additional degrees of freedom (Rohwer et al 2010 J. Phys. A: Math. Theor. 43 345302) and the one without any such degrees of freedom (Gangopadhyay and Scholtz 2009 Phys. Rev. Lett. 102 241602) 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.