The scalar complex potential and the Aharonov-Bohm effect

TL;DR

This paper proposes a complex potential framework to describe the Aharonov-Bohm effect, emphasizing the role of multiple-valued functions and providing explicit expressions for electromagnetic fields.

Contribution

It introduces the concept of a complex pre-potential that generalizes scalar potentials and explains the AB effect through this novel mathematical object.

Findings

The pre-potential is a key to understanding the AB effect.

Electromagnetic fields can be fully described by the complex pre-potential.

Explicit expressions for the electromagnetic tensor in terms of the pre-potential are provided.

Abstract

The Aharonov-Bohm effect is traditionally attributed to the effect of the electromagnetic 4-potential $A$, even in regions where both the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ are zero. The AB effect reveals that multiple-valued functions play a crucial role in the description of an electromagnetic field. We argue that the quantity measured by AB experiments is a difference in values of a multiple-valued complex function, which we call a complex potential or {pre-potential. We show that any electromagnetic field can be described by this pre-potential, and give an explicit expression for the electromagnetic field tensor through this potential. The pre-potential is a modification of the two scalar potential functions.