A mathematical model for the Aharonov-Bohm effect

Vladimir D. Skarzhinsky

Barry R. Holstein

The subject of anholonomy in physics has recently been emphasized via discussions of the Berry phase. A particular example of this phenomenon, which does not require the adiabatic condition is the familiar Aharonov-Bohm (AB) effect. We herein review the AB effect and show that two additional phenomena, namely the Aharonov-Casher effect and rotating superconducting quantum interference device systems, can be analysed using identical mathematical frameworks.

D. A. Slavnov

Abstract An algebraic approach to the Aharonov–Bohm effect is considered. The constructed mathematical scheme agrees with experimental data and provides a clear physical interpretation of this effect that does not contradict classical concepts.

Donald H. Kobe

Heinz J. Rothe

Mostafa E. El Demery

The Aharonov-Bohm effect is a quantum mechanical effect, that is, has no classical counterpart. The effect was predicted in 1959 in a seminal paper of Y. Aharonov and D. Bohm [AB59] in which they demonstrated that a beam of electrons is affected by the existence of the electric/magnetic field even though electrons travel through field-free regions. Aharonov and Bohm carried out two hypothetical experiments to support their claim that potentials are more fundamental than fields and they are respon-sible of the effect. Since then, the debate has arisen around whether potentials are mathematical tools or fundamental entities in physics. Different arguments have been set to explain the results predicted by Aharonov and Bohm and experimentally confirmed. Amongst these arguments, the first argument adopted by Aharonov and Bohm was that potentials are physically significant. Many have claimed that fields do have non-local features, i.e. action at a distance. Others have claimed that topological effects may interpret the effect in which potentials are modeled as connections in higher-dimensional fiber bun-dle geometries. The most recent argument has been proposed by Vaidman [Vai12] who claimed that the the composite system is represented by one state, an entangled state, and due to the electromagnetic interactions part of this state is changed, hence, the total state. In the present essay, I will discuss the latter argument as well as reviewing some other arguments. i