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arXiv:hep-th/0604068 · Europhysics Letters (EPL)

Noncommutative geometry and geometric phases

Banasri Basu, Sudipta Dhar, Subir Ghosh

Published Oct 10, 2006• 14 citations
arXivHTMLPDF

Abstract

We have studied particle motion in generalized forms of noncommutative phase space, that simulate monopole and other forms of Berry curvature, that can be identified as effective internal magnetic fields, in coordinate and momentum space. The Ahranov-Bohm effect has been considered in this form of phase space, with operatorial structures of noncommutativity. The physical significance of our results is also discussed.

DOI
10.1209/epl/i2006-10299-9

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We have studied particle motion in generalized forms of noncommutative phase space, that simulate monopole and other forms of Berry curvature, that can be identified as effective internal magnetic fields, in coordinate and momentum space. The Ahranov-Bohm effect has been considered in this form of phase space, with operatorial structures of noncommutativity. The physical significance of our results is also discussed.
We study the time-dependent Aharonov–Bohm effect on the noncommutative space. Because there is no net Aharonov–Bohm phase shift in the time-dependent case on the commutative space, therefore, a tiny deviation from zero indicates new physics . Based on the Seiberg–Witten map we obtain the gauge invariant and Lorentz covariant Aharonov–Bohm phase shift in general case on noncommutative space. We find there are two kinds of contribution: momentum-dependent and momentum-independent corrections. For the momentum-dependent correction, there is a cancellation between the magnetic and electric phase shifts, just like the case on the commutative space. However, there is a non-trivial contribution in the momentum-independent correction. This is true for both the time-independent and time-dependent Aharonov–Bohm effects on the noncommutative space. However, for the time-dependent Aharonov–Bohm effect, there is no overwhelming background which exists in the time-independent Aharonov–Bohm effect on both commutative and noncommutative space. Therefore, the time-dependent Aharonov–Bohm can be sensitive to the spatial noncommutativity. The net correction is proportional to the product of the magnetic fluxes through the fundamental area represented by the noncommutative parameter θ , and through the surface enclosed by the trajectory of charged particle. More interestingly, there is an anti-collinear relation between the logarithms of the magnetic field B and the averaged flux Φ / N ( N is the number of fringes shifted). This nontrivial relation can also provide a way to test the spatial noncommutativity. For B Φ / N ∼ 1 , our estimation on the experimental sensitivity shows that it can reach the 10 GeV scale. This sensitivity can be enhanced by using stronger magnetic field strength , larger magnetic flux, as well as higher experimental precision on the phase shift.
The dynamics of a spin-(1/2) neutral particle possessing electric- and magnetic-dipole moments interacting with external electric and magnetic fields in noncommutative coordinates is obtained. Noncommutativity of space is interposed in terms of a semiclassical constrained Hamiltonian system. The relation between the quantum phase acquired by a particle interacting with an electromagnetic field and the (semi)classical force acting on the system is examined and generalized to establish a formulation of the quantum phases in noncommutative coordinates. The general formalism is applied to physical systems yielding the Aharonov-Bohm, Aharonov-Casher, He-McKellar-Wilkens and Anandan phases in noncommutative coordinates. Bounds for the noncommutativity parameter θ are derived comparing the deformed phases with the experimental data on the Aharonov-Bohm and Aharonov-Casher phases.
The Aharonov–Bohm effect on the noncommutative plane is considered. Developing the path integral formulation of quantum mechanics, we find the propagation amplitude for a particle in a noncommutative space. We show that the corresponding shift in the phase of the particle propagator due to the magnetic field of a thin solenoid receives certain gauge invariant corrections because of the noncommutativity. Evaluating the numerical value for this correction, an upper bound for the noncommutativity parameter is obtained.
An external magnetic field in configuration space coupled to quantum dynamics induces noncommutativity in its velocity momentum space. By phase space duality, an external vector potential in the conjugate momentum sector of the system induces noncommutativity in its configuration space. Such a rationale for noncommutativity is explored herein for an arbitrary configuration space of Euclidean geometry. Ordinary quantum mechanics with a commutative configuration space is revisited first. Through the introduction of an arbitrary positive definite ∗-product, a one-to-one correspondence between the Hilbert space of abstract quantum states and that of the enveloping algebra of the position quantum operators is identified. A parallel discussion is then presented when configuration space is noncommutative, and thoroughly analysed when the conjugate magnetic field is momentum independent and nondegenerate. Once again the space of quantum states may be identified with the enveloping algebra of the noncommutative position quantum operators. Furthermore, when the positive definite ∗-product is adapted to the conjugate magnetic field, the coordinate operators span a Fock algebra of which the coherent states are the analogues of the structureless points in a commutative configuration space. These results generalise and justify a posteriori within ordinary canonical quantisation the heuristic approach to quantum mechanics in the noncommutative Euclidean plane as constructed and developed by F. G. Scholtz and his collaborators.
A monopole-like singularity generated by Berry's phase in momentum space and associated anomalous Poisson brackets have been discussed in various fields. In the applications, it is crucially important to understand precisely what is the monopole in momentum space implied by Berry's phase. Berry's phase is sensitive to how it is measured and thus very different from the genuine Dirac monopole; the monopole-like singularity in Berry's phase is a result of adiabatic approximation while the singularity disappears in non-adiabatic approximation. The resolution of monopole-like singularity in the non-adiabatic domain implies that no Dirac string appears close to the monopole position and the anomalous terms containing the monopole flux in quantum mechanical commutation relations are absent. The absence of modified commutators is also concluded if one understands Berry's phase in momentum space as an O(ℏ)O(\hbar)O(ℏ) quantum mechanical effect with a quantized magnetic charge eM=2πℏe_{M}=2\pi\hbareM​=2πℏ.
For a spin-1/2 particle moving in a background magnetic field in noncommutative phase space, the Dirac equation is solved when the particle is allowed to move off the plane that the magnetic field is perpendicular to. It is shown that the motion of the charged particle along the magnetic field has the effect of increasing the magnetic field. In the classical limit, matrix elements of the velocity operator related to the probability give a clear physical picture. Along an effective magnetic field, the mechanical momentum is conserved and the motion perpendicular to the effective magnetic field follows a round orbit. If using the velocity operator defined by the coordinate operators, the motion becomes complicated.