Quantum mechanics in noncommutative space modifies the standard result of the Aharonov-Bohm effect for electrons and other recent quantum effects. Here we obtain the phase in noncommutative space for the Spavieri effect, a generalization of Aharonov-Bohm effect which involves a coherent superposition of particles with opposite charges moving along a single open interferometric path. By means of the experimental considerations a limit $\sqrt{\ensuremath{\theta}}\ensuremath{\simeq}{\left(0.13\phantom{\rule{0.16em}{0ex}}\text{T}\text{eV}\right)}^{\ensuremath{-}1}$ is achieved, improving by 10 orders of magnitude the results derived by Chaichian et al. [Phys. Lett. B 527, 149 (2002)] for the Aharonov-Bohm effect. It is also shown that the noncommutative phases of the Aharonov-Casher and He-McKellar-Willkens effects are nullified in the current experimental tests.
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 Aharonov-Bohm effect including spin-noncommutative effects is considered. At linear order in $\ensuremath{\theta}$, the magnetic field is gauge invariant although spatially strongly anisotropic. Despite this anisotropy, the Schr\"odinger-Pauli equation is separable through successive unitary transformations and the exact solution is found. The scattering amplitude is calculated and compared with the usual case. In the noncommutative Aharonov-Bohm case the differential cross section is independent of $\ensuremath{\theta}$.
The Aharonov-Bohm (AB) effect demonstrates that electromagnetic potentials can influence quantum systems in field-free regions, challenging classical intuitions. In 2004, Aharonov and Kaufherr proposed that the AB phase shift arises from a nonlocal exchange of modular momentum, a gauge-invariant quantity, occurring at a definite time before wave packet interference. This framework, however, introduces a paradox: the expectation value of the modular momentum operator,
⟨M^y⟩, changes abruptly when the electron’s wave packets cross the solenoid, yet local, gauge-invariant properties of the electron, such as its velocity distribution, remain continuous. We resolve this paradox by showing that
M^y=exp(iℏmvyL) and the velocity operator
v^y have distinct spectral measures, and the abrupt change in
⟨M^y⟩ reflects a global topological change in the electromagnetic environment, not a dynamical alteration in the electron’s state. Through a corrected, gauge-invariant formulation, we demonstrate that
⟨M^y⟩ encodes the AB phase but does not explain the AB effect’s mechanism. This resolution reaffirms the conventional explanation of the AB effect while clarifying the nonlocal nature and experimental limitations of modular momentum.
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 Schr
ddotodinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schr
ddotodinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.