Aharonov-Bohm Effect without Using Vector Potential

Tae-Yeon Lee

It is shown that the vector potential of the Aharonov-Bohm effect is effectively the same as vector potentials of the quantum phases of electric and magnetic dipoles such as the Aharonov-Casher effect, when the Aharonov-Bohm solenoid is a straight cylinder with symmetric cross sections. With a circular cross section, the Aharonov-Bohm vector potential is effectively the same as the Aharonov-Casher vector potential, while with a rectangular cross section, it is effectively the same as the vector potential of the Casella phase. Therefore, we may say that the quantum phases of the dipoles belong to a subclass of the Aharonov-Bohm phase, and they must share the topological origin of the Aharonov-Bohm phase, even though their topological freedom is limited to the particle trajectories. However, the dipole phases have the extra relativistic effect that a magnetic (electric) dipole moving in an electric (magnetic) field feels an additional magnetic (electric) field, in its rest frame. This effect limits further the topological freedom of the dipole phases.

Chris Philippidis, David Böhm, R.D. Kaye

Shan Gao

Kazuyasu Shigemoto, Kunihiko Uehara B

In the original setting of the Aharonov-Bohm, the gauge invariant physical longitudinal mode of the vector potential, which is expressed by the gauge invariant physical current, gives the desired contribution to the Aharonov-Bohm effect. While the scalar mode of the vector potential, which changes under the gauge transformation so that it is the unphysical mode, give no contribution to the Aharonov-Bohm effect. Then Aharonov-Bohm effect really occurs by the physical longitudinal mode in the original Aharonov-Bohm’s setting. In the setting of Tonomura et al., where the magnet is shielded with the superconducting material, not only the magnetic field but also the longitudinal mode of the vector potential become massive by the Meissner effect. Then not only the magnetic field but also the physical longitudinal mode does not come out to the region where the electron travels. In such setting, only the scalar mode of the vector potential exists in the region where the electron travels, but there is no contribution to the Aharonov-Bohm effect from that mode.

Donald H. Kobe

S Héctor Torres, André Koch Torres Assis

"It is a known fact that there is no magnetic field outside an infinite solenoid carrying a constant current, althoughthere is magnetic vector potential non-null outside it. The existence of the Aharonov-Bohm effect (AB) is usuallyconsidered as proof of the relevance of the vector potential for quantum mechanics. In this paper we will see that thereis a non null electric field outside an infinite resistive solenoid carrying a constant current and its possible relevanceto the analysis of the Aharonov-Bohm effect. When calculating, we introduce the Galilean invariance of Maxwell´sequations and then we obtain the contribution to the AB effect due to electric potential."