Fractional Aharonov-Bohm effect for retarded potentials

The magnetic Aharonov-Bohm (AB) effect is of fundamental interest because it shows that the vector potential can have an observable effect on an electron in an interferometer even when the electric and magnetic fields are zero along its path. It has been suggested that it may be necessary to include the phase shift experienced by the electrons in the solenoid that creates the magnetic flux in addition to the usual phase shift experienced by the electron in the interferometer itself [1].

The AB effect will be derived here using second-quantized field theory to describe all of the electrons as well as the electromagnetic field in a consistent manner. The results show that half of the AB phase shift is associated with the electron in the interferometer while the other half is associated with the electrons in the solenoid. These two contributions are equal for quasi-static fields where retardation effects are negligible, and the results reduce to the usual expression for the AB effect using a classical vector potential. But the phase shift associated with the solenoid can be zero when retardation effects are large, which gives a predicted phase shift that is a factor of ½ that of the usual result [2].

An experiment of this kind would allow a novel test of quantum electrodynamics under conditions that are very different from the usual scattering experiments.

1. L. Vaidman, Phys. Rev. A 86, 040101(R) (2012).

2. J. D. Franson, arXiv:2510.10333 (2025).