Derivation of the Aharanov Bohm phase shift using classical forces

TL;DR

This paper derives the Aharonov-Bohm phase shift using classical forces and relativistic transformations, challenging the notion that it is purely a quantum phenomenon, with the quantum aspect only introduced via the de Broglie wavelength.

Contribution

It presents a classical derivation of the Aharonov-Bohm phase shift incorporating relativistic effects, providing an alternative perspective to the quantum-only explanation.

Findings

Classical forces can produce the Aharonov-Bohm phase shift.

Relativistic transformations are essential in the classical derivation.

The phase shift is consistent across different reference frames.

Abstract

In 1959 Aharonov and Bohm suggested that an electron passing around a long solenoid would pick up a phase shift dependent on the magnetic field of the solenoid, even though the electrons themselves pass through a region of space which has a zero magnetic field. It has long been held that this result is purely quantum and is derived in many well known quantum mechanics text books using the Schrodinger equation and vector potential. Here the same phase shift is derived from a purely classical force, but relativistic transformations are taken into account. The force is in the direction of motion of the electron (or opposite) leading to a phase advance (or lag) and we obtain precisely the phase shift thought previously to be purely quantum. The only quantum result used here is the de Broglie wavelength of the particle, in order to get two slit like interference and the phase shift. We employ a stack of dipoles as the solenoid and note the same force on the electron in two different frames of reference. We shall consider the solenoid stationary and the electron moving, and then consider the electron rest frame and consider the solenoid moving in the opposite direction.