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Star Operation in Quantum Mechanics

Luca Mezincescu

We outline the description of Quantum Mechanics with noncommuting coordinates within the framework of star operation. We discuss simple cases of integrability.

arXiv:hep-th/0007046Jul 6, 200092 citations

ON THE NONCOMMUTATIVE EIKONAL

J. M. Isidro, Pedro Fernández de Córdoba, J. M. Rivera-Rebolledo +2 more

We study the eikonal approximation to quantum mechanics on the Moyal plane. Instead of using a star product, the analysis is carried out in terms of operator-valued wavefunctions depending on noncommuting, operator-valued coordinates.

Jan 1, 20113 citations

The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation

Ardeshir Eftekharzadeh, B. L. Hu

We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar *-product combines the usual phase space * star and the noncommutative * star-product. Having dealt with subtleties of ordering present in this problem we show that the Weyl correspondence of the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter theta-dependent, momentum-dependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if q ¹ 0 there is no classical limit. Classical limit exists only if q ® 0 at least as fast as h ->0, but this limit does not yield Newtonian mechanics, unless the limit of theta/h vanishes as q ® 0. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of theta and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context.We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar *-product combines the usual phase space * star and the noncommutative * star-product. Having dealt with subtleties of ordering present in this problem we show that the Weyl correspondence of the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter theta-dependent, momentum-dependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if q ¹ 0 there is no classical limit. Classical limit exists only if q ® 0 at least as fast as h ->0, but this limit does not yield Newtonian mechanics, unless the limit of theta/h vanishes as q ® 0. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of theta and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context.

Brazilian Journal of PhysicsJun 1, 200514 citations

On Hilbert-Schmidt Operator Formulation of Noncommutative Quantum Mechanics

Isiaka Aremua, Ezinvi Baloïtcha, Mahouton Norbert Hounkonnou +1 more

arXiv:1705.10837STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & HealthJan 1, 20183 citations

Remarks on the formulation of quantum mechanics on noncommutative phase spaces

B. Muthukumar

arXiv:hep-th/0609117Journal of High Energy PhysicsJan 18, 20078 citations

Wigner's Formulation of Noncommutative Quantum Mechanics

Akira Kokado, Takashi Okamura, Takesi Saito

When we have noncommutativity among coordinates (or conjugate momenta), we consider Wigner's formulation of quantum mechanics, including a new derivation of path integral formula. We also propose the Moyal star product based on the Dirac bracket in constrained systems.

arXiv:hep-th/0208040Aug 6, 20021 citation

ON THE NONCOMMUTATIVE EIKONAL

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ON THE NONCOMMUTATIVE EIKONAL

Abstract

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