Path-integral action of a particle in the noncommutative phase-space

¹ Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata Mohanpur 741246, Nadia, West Bengal, India
² Inter University Centre for Astronomy & Astrophysics (IUCAA) - Pune, India
³ Kolorah H.A.W. Institution - Kolorah, Howrah 711411, India

^(a) sunandan.gangopadhyay@gmail.com, sunandan@iiserkol.ac.in, sunandan@associates.iucaa.in
^(b) aslamhalder.phy@gmail.com

Received: 18 December 2016
Accepted: 8 February 2017

Abstract

In this paper we construct a path integral formulation of quantum mechanics on noncommutative phase-space. We first map the system to an equivalent system on the noncommutative plane. Then by applying the formalism of representing a quantum system in the space of Hilbert-Schmidt operators acting on noncommutative configuration space, the path integral action of a particle is derived. It is observed that the action has a similar form to that of a particle in a magnetic field in the noncommutative plane. From this action the energy spectrum is obtained for the free particle and the harmonic-oscillator potential. We also show that the nonlocal nature (in time) of the action yields a second-class constrained system from which the noncommutative Heisenberg algebra can be recovered.