Quantizing the Vector Potential Reveals Alternative Views of the Magnetic Aharonov-Bohm Phase Shift

TL;DR

This paper presents a comprehensive quantum analysis of the Aharonov-Bohm effect, demonstrating three equivalent approaches based on different classical approximations of the involved entities, and reveals that all approaches produce the same phase shift.

Contribution

It introduces a third quantum perspective on the AB effect, showing the phase shift can be attributed to the quantized vector potential when electron and solenoid currents are classical.

Findings

All three approaches yield the same AB phase shift.

The phase arises from additional real c-number phases in the wave functions.

The three methods offer alternative views of the physical cause of the AB effect.

Abstract

We give a complete quantum analysis of the Aharonov-Bohm (AB) magnetic phase shift involving three entities, the electron, the charges constituting the solenoid current, and the vector potential. The usual calculation supposes that the solenoid's vector potential may be well-approximated as classical. The AB shift is then acquired by the quantized electron moving in this vector potential. Recently, Vaidman presented a semi-classical calculation, later confirmed by a fully quantum calculation of Pearle and Rizzi, where it is supposed that the electron's vector potential may be well-approximated as classical. The AB shift is then acquired by the quantized solenoid charges moving in this vector potential. Here we present a third calculation, which supposes that the electron and solenoid currents may be well-approximated as classical sources. The AB phase shift is then shown to be acquired by the quantized vector potential. We next show these are three equivalent alternative ways of calculating the AB shift. We consider the exact problem where all three entities are quantized. We approximate the wave function as the product of three wave functions, a vector potential wave function, an electron wave function and a solenoid wave function. We apply the variational principle for the exact Schrodinger equation to this approximate form of solution. This leads to three Schrodinger equations, one each for vector potential, electron and solenoid, each with classical sources for the other two entities. However, each Schrodinger equation contains an additional real c-number term, the time derivative of an extra phase. We show that these extra phases are such that the net phase of the total wave function produces the AB shift. Since none of the three entities requires different treatment from any of the others, this leads to three alternative views of the physical cause of the AB magnetic effect.