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arXiv:hep-th/0502143 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Dual families of noncommutative quantum systems

Frederik G. Scholtz, Biswajit Chakraborty, Sunandan Gangopadhyay, Arindam Ghosh Hazra

Published Apr 8, 2005• 92 citations
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Abstract

We demonstrate how a one parameter family of interacting noncommuting Hamiltonians, which are physically equivalent, can be constructed in noncommutative quantum mechanics. This construction is carried out exactly (to all orders in the noncommutative parameter) and analytically in two dimensions for a free particle and a harmonic oscillator moving in a constant magnetic field. We discuss the significance of the Seiberg-Witten map in this context. It is shown for the harmonic oscillator potential that an approximate duality, valid in the low-energy sector, can be constructed between the interacting commutative and a noninteracting noncommutative Hamiltonian. This approximation holds to order $1/B$ and is therefore valid in the case of strong magnetic fields and weak Landau-level mixing.

DOI
10.1103/physrevd.71.085005

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In order to evaluate the Feynman path integral in noncommutative quantum mechanics, we consider properties of a Lagrangian related to a quadratic Hamiltonian with noncommutative spatial coordinates. A quantum-mechanical system with noncommutative coordinates is equivalent to another one with commutative coordinates. We found connection between quadratic classical Lagrangians of these two systems. We also shown that there is a subclass of quadratic Lagrangians, which includes harmonic oscillator and particle in a constant field, whose connection between ordinary and noncommutative regimes can be expressed as a linear change of position in terms of a new position and velocity.
We consider Feynman's path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the another one with usual commutative coordinates and momenta. We found connection between quadratic classical Hamiltonians, as well as Lagrangians, in their commutative and noncommutative regimes. The general procedure to compute Feynman's path integral on this noncommutative phase space with quadratic Lagrangians (Hamiltonians) is presented. Using this approach, a particle in a constant field, ordinary and inverted harmonic oscillators are elaborated in detail.
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.
The effects of general noncommutativity of operators on producing deformed coherent squeezed states is examined in phase space. A two-dimensional noncommutative (NC) quantum system supported by a deformed mathematical structure, similar to that of Hadamard billiard, is obtained and the components behaviour is monitored in time. It is assumed that the independent degrees of freedom are two free 1D harmonic oscillators (HOs), so the system Hamiltonian does not contain interaction terms. Through the NC deformation parameterized by a Seiberg–Witten transform on the original canonical variables, one gets the standard commutation relations for the new ones, such that the obtained, new, Hamiltonian represents two interacting 1D HOs. By admitting that one HO is inverted relatively to the other, we show that their effective interaction induces a squeezing dynamics for initial coherent states imaged in the phase space. A suitable pattern of logarithmic spirals is obtained and some relevant properties are discussed in terms of Wigner functions, which are essential to put in evidence the effects of the noncommutativity.
Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.