Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this paper we discuss the mathematical framework of such a theory. The noncommutativity is treated as an external antisymmetric field satisfying the Jacobi identity. First, we propose a symplectic realization of a given Poisson manifold and construct the Darboux coordinates on the obtained symplectic manifold. Then we define the star product on a Poisson manifold and obtain the expression for the trace functional. The above ingredients are used to formulate a nonrelativistic quantum mechanics on noncommutative spaces of general form. All considered constructions are obtained as a formal series in the parameter of noncommutativity. In particular, the complete algebra of commutation relations between coordinates and conjugated momenta is a deformation of the standard Heisenberg algebra. As examples we consider a free particle and an isotropic harmonic oscillator on the rotational invariant noncommutative space.
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November 2013
Research Article|
November 18 2013
Quantum mechanics with coordinate dependent noncommutativity
V. G. Kupriyanov
V. G. Kupriyanov
a)
CMCC,
Universidade Federal do ABC
, Santo André, SP, Brazil
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E-mail: vladislav.kupriyanov@gmail.com
J. Math. Phys. 54, 112105 (2013)
Article history
Received:
July 02 2013
Accepted:
October 29 2013
Citation
V. G. Kupriyanov; Quantum mechanics with coordinate dependent noncommutativity. J. Math. Phys. 1 November 2013; 54 (11): 112105. https://doi.org/10.1063/1.4830032
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