The Aharonov-Bohm Effect Without Gauge Fields - A Paradox

Chen Ning Yang

Chen Ning Yang

Gianfranco Spavieri

Adam Caprez, Herman Batelaan

Gao, Shan

The Aharonov-Bohm (AB) effect demonstrates that electromagnetic potentials can influence quantum systems in field-free regions, challenging classical intuitions. In 2004, Aharonov and Kaufherr proposed that the AB phase shift arises from a nonlocal exchange of modular momentum, a gauge-invariant quantity, occurring at a definite time before wave packet interference. This framework, however, introduces a paradox: the expectation value of the modular momentum operator, $\langle \hat{M}_y \rangle$, changes abruptly when the electron’s wave packets cross the solenoid, yet local, gauge-invariant properties of the electron, such as its velocity distribution, remain continuous. We resolve this paradox by showing that $\hat{M}_y=\exp(i \frac{m v_y L}{\hbar})$ and the velocity operator $\hat{v}_y$ have distinct spectral measures, and the abrupt change in $\langle \hat{M}_y \rangle$ reflects a global topological change in the electromagnetic environment, not a dynamical alteration in the electron’s state. Through a corrected, gauge-invariant formulation, we demonstrate that $\langle \hat{M}_y \rangle$ encodes the AB phase but does not explain the AB effect’s mechanism. This resolution reaffirms the conventional explanation of the AB effect while clarifying the nonlocal nature and experimental limitations of modular momentum.

Seyed Ali Hosseini Mansoori, Behrouz Mirza

We investigate the non-Abelian Aharonov–Bohm (AB) effect for time-dependent gauge fields. We prove that the non-Abelian AB phase shift related to time-dependent gauge fields, in which the electric and magnetic fields are written in the adjoint representation of SU(N) generators, vanishes up to the first order expansion of the phase factor. Therefore, the flux quantization in a superconductor ring does not appear in the time-dependent Abelian or non-Abelian AB effect.