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arXiv:1705.10837 · STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health

On Hilbert-Schmidt Operator Formulation of Noncommutative Quantum Mechanics

Isiaka Aremua, Ezinvi Baloïtcha, Mahouton Norbert Hounkonnou, Komi Sodoga

Published Jan 1, 2018• 3 citations
arXivHTMLPDF
DOI
10.1007/978-3-319-97175-9_3

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The formulation of noncommutative quantum mechanics as a quantum system represented in the space of Hilbert-Schmidt operators is used to systematically derive, using the standard time slicing procedure, the path integral action for a particle moving in the noncommutative plane and in the presence of a magnetic field and an arbitrary potential. Using this action, the equation of motion and the ground state energy for the partcle are obtained explicitly. The Aharonov-Bohm phase is derived using a variety of methods and several dualities between this system and other commutative and noncommutative systems are demonstrated. Finally, the equivalence of the path integral formulation with the noncommutative Schrödinger equation is also established.
Noncommutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on noncommutative configuration space. Taking this as a 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 noncommutative plane and in the presence of an arbitrary potential. We find that this action is nonlocal in time. However, this nonlocality can be removed by introducing an auxilary field, which leads to a second class constrained system that yields the noncommutative Heisenberg algebra upon quantization. Using this action, the propagator of the free particle and harmonic oscillator are computed explicitly.