Fractional Aharonov-Bohm effect for retarded potentials

Abstract

It has been suggested that the magnetic Aharonov-Bohm effect can be interpreted equally well as being due to a phase shift associated with an electron in an interferometer enclosing a magnetic flux, or as a phase shift associated with the electrons in the solenoid that generates the field. Here the Aharonov-Bohm effect is derived using second-quantized field theory to describe all the electrons as well as the electromagnetic field in a consistent way. The results are in agreement with the usual expression for the Aharonov-Bohm effect when the retardation of the electromagnetic field is negligible, but they predict the possibility of a fractional phase shift when retardation effects are significant.

Figures (2)

• Figure 1: Magnetic Aharonov-Bohm effect. The wave function of an incident electron $(e^-)$ is split into two paths 1 and 2 by a beam splitter (dashed line). The vector potential from an enclosed solenoid $S$ with magnetic flux $\Phi_M$ can change the relative phase between the two paths, even though the electric and magnetic fields are zero outside of the solenoid.
• Figure 2: Fractional Aharonov-Bohm effect. A source S of a static magnetic field consists of a constant current flowing in a circular loop. An electron passes through a distant interferometer, which encloses a magnetic flux from the source. The distance D between the source and the interferometer is assumed to be sufficiently large that the electron passes through the interferometer before its retarded vector potential can reach the source S. In that case, Eq. \ref{['phi3']} predicts that a fractional Aharonov-Bohm phase shift will be observed.