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Quantum mechanics with coordinate dependent noncommutativity

V.G. Kupriyanov
(
- ABC Federal U.
)

Apr, 2012

30 pages

Published in:

J.Math.Phys. 54 (2013) 112105

Published: 2013

e-Print:

1204.4823 [math-ph]

DOI:

10.1063/1.4830032

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Abstract: (AIP)

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.

Note:

35 pages, new material concerning the trace functional, new physical example and new references added

References(29)

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