We discuss the Aharonov-Bohm (
A−B) effect and the Dirac (
D) monopole of magnetic charge
g=21 in the context of bundle theory, exhibiting a purely geometric relation between them. If
ξA−B and
ξD are the respective
U(1)-bundles, we show that
ξA−B is isomorphic to the pull-back of
ξD induced by the inclusion of the corresponding base spaces
ι:(D02)∗→S2}. The fact that the
A−B effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux
Φ0=∣e∣2π associated with the electric charge
∣e∣, reflects here as a consequence of the pull-back by
ι of the Dirac connection in
ξD to
ξA−B, and the Dirac quantization condition. We also show the necessary vanishing in
ξA−B of the pull-back of the Chern class
c1 in
ξD.
We discuss the Aharonov-Bohm (A -B) effect and the Dirac (D) monopole of magnetic charge g = 1 2 in the context of bundle theory, which allows to exhibit a deep geometric relation between them.If ξA-B and ξD are the respective U (1)-bundles, we show that ξA-B is isomorphic to the pull-back of ξD induced by the inclusion of the corresponding base spaces.The fact that the A -B effect disappears when the magnetic flux in the solenoid equals an integer number of times the quantum of flux associated with the unit of electric charge, reflects here as a consequence of the pull-back of the Dirac connection in ξD to ξA-B, and the Dirac quantization condition.
We study a generalization of Aharonov-Bohm effect, the potential effect. The discussion is focused on field-free effects in simply connected region, which obviously can not have any local field-flux. Among the published discussions about this kind of effects, it is generally agreed that this kind of effects does not exist due to gauge invariance. However, there are also opinions that this effect is a trivial variation of Aharonov-Bohm effect and therefore there is no need to check its existence. To my knowledge, it has never been tested. My first goal here is to supply enough theoretical reason to motivate the experimental test of this effect. I start with an intuitive derivation, then I introduce a wave-front theory as a theoretical consideration. Logically, the existence of potential effect implies the existence of the AB effect, but not vice versa. The purpose of this paper is to provide a physical connection in the opposite direction.