On the quantum dynamics of non-commutative systems

Abstract

This is a review paper concerned with the global consistency of the quantum dynamics of non-commutative systems. Our point of departure is the theory of constrained systems, since it provides a unified description of the classical and quantum dynamics for the models under investigation. We then elaborate on recently reported results concerned with the sufficient conditions for the existence of the Born series and unitarity and turn, afterwards, into analyzing the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. The intricacies arising in connection with the explicit computation of path integrals, for the systems under scrutiny, is illustrated by presenting the detailed calculation of the Feynman kernel for the non-commutative two dimensional harmonic oscillator.

Noncommutative Phase Space and the Two Dimensional Quantum Dipole in Background Electric and Magnetic Fields

“…The motivation for this kind of theory is that in the low energy effective theory of D-brane with a background magnetic field and an extreme situation such as in the string scale or at very high energy levels, not only the space noncommutativity may appear, but also the effects of momentum noncommutativity can be signi-ficant, which is called NC phase space. Hence, a lot of specific problems have been investigated on the theory of NC spaces such as the quantum Hall effects [8,9], the harmonic oscillator [10][11][12], the Fock-Darwin system [13], the coherent states [14], the classical-quantum relation-ship [15], the motion of the spin-1/2 particle under a uniform magnetic field [16], the Dirac equation with a magnetic field in D [17] and with the time-dependent linear potential [18], etc. The main approach is based on the Weyl-Moyal correspondence which amounts to replacing the usual product by a star product in NC space [19].…”

Motion of Charged Particle in Electric and Magnetic Fields in 3D Noncommutative Spaces and Related Problems

“…Renewed interest in such an idea is mainly due to the recent discoveries in string theory and M theory that effects of noncommutative (NC) spaces may appear near the string scale and at higher energies [2][3][4]. Recently, a lot of investigations have been done on the theory of NC spaces [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] such as the quantum Hall effects (QHE) [7][8][9][10], the harmonic oscillator [11][12][13][14], the coherent states [15], the thermodynamics [16], the classical-quantum transition [17], etc.. The QHE [7][8][9][10] refers to the phenomenon that when a magnetic field perpendicular to the electric current flowing through an electric conductor is applied, there appears the potential difference (Hall voltage) in the direction perpendicular to the current and the magnetic field.…”

Noncommutative quantum mechanics as a gauge theory

“…The conditions for the existence of the Born series and unitarity were investigated in Ref. [6] while a general overview of the connection linking noncommutative theories with constrained systems was presented in [7]. As for the uniqueness of the functional description, when using the timeslice definition for the phase space path integral, it was established in Ref.…”