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arXiv:1502.00420 · Physical Review A

Detecting noncommutative phase space by the Aharonov-Bohm effect

Shi‐Dong Liang, Haoqi Li, Guang-Yao Huang

Published Jul 21, 2014• 22 citations
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Abstract

Noncommutative phase space plays an essential role in particle physics and quantum gravity at the Planck scale. However, direct experimental evidence or observation to demonstrate the existence of noncommutative phase space is still lacking. We study a quantum ring in noncommutative phase space based on the Seiberg-Witten map and give the effective magnetic potential and field coming from the noncommutative phase space, which induces the persistent current in the ring. We introduce two variables as two signatures to detect the noncommutative phase space and propose an experimental scheme to detect the noncommutative phase space as long as we measure the persistent current and the external magnetic flux.

DOI
10.1103/physreva.90.010102

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•Aug 25, 2005•56 citations
Noncommutative phase space plays an essential role in particle physics and quantum gravity at the Planck scale. However, direct experimental evidence or observation to demonstrate the existence of noncommutative phase space is still lacking. We study a quantum ring in noncommutative phase space based on the Seiberg-Witten map and give the effective magnetic potential and field coming from the noncommutative phase space, which induces the persistent current in the ring. We introduce two variables as two signatures to detect the noncommutative phase space and propose an experimental scheme to detect the noncommutative phase space as long as we measure the persistent current and the external magnetic flux.
The conceptual incompatibility of spacetime in gravity and quantum physics implies the existence of noncommutative spacetime and geometry on the Planck scale. We present the formulation of a noncommutative quantum mechanics based on the Seiberg–Witten map, and we study the Aharonov–Bohm effect induced by the noncommutative phase space. We investigate the existence of the persistent current in a nanoscale ring with an external magnetic field along the ring axis, and we introduce two observables to probe the signal coming from the noncommutative phase space. Based on this formulation, we give a value-independent criterion to demonstrate the existence of the noncommutative phase space.
The noncommutative space provides a framework to understand phenomena in Planck scale physics. However, there is no any direct experimental evidence to demonstrate the existence of noncommutative space. We propose an experimental scheme based on the Aharonov-Bohm effect in the nano-scale quantum mechanics to probe the phenomena in the noncommutative phase space. By the Seiberg-Witten map, the free electrons of the nano-scale ring in noncommutative phase space can be mapped equivalently to the quantum mechanical (Heisenberg's algebra) phase space with an extra effective magnetic flux. We introduce two variables related to the persistent current in the ring to probe the noncommutative phase space effect. We give a value-independent criterion to detect the existence of the noncommutative phase space. Namely the answer for existence or nonexistence of the noncommutative phase space depends only on the trend of the curves, rather than the values of the observation data. It can be expected to catch the noncommutative phase-space effect by this scheme.
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.
We introduce a new approach to study the noncommutative effects on the neutral particle with anomalous magnetic or electric dipole moments on the 2+12+12+1 noncommutative space time. The advantages of this approach are demonstrated by investigating the noncommutative corrections on the Aharonov-Casher and He-McKellar-Wilkens effects. This approach is based on reinterpreting the Aharonov-Casher and He-McKellar-Wilkens effects as consequences of an effective U(1)U(1)U(1) gauge symmetry. An effective Seiberg-Witten map for this symmetry is introduced when we study the noncommutative corrections. Because Seiberg-Witten map preserves the symmetry, the noncommutative corrections can be investigated systematically. Our results indicate that the noncommutative corrections on the Aharonov-Casher and He-McKellar-Wilkens phases are strongly depend on the ratio between the noncommutative parameter θ\thetaθ and the cross section Ae/mA_{e/m}Ae/m​ of the charged line enclosed by the trajectory of beam particle.
The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schrddoto\\ddot{o}ddotodinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schrddoto\\ddot{o}ddotodinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.