Beyond the Lorenz Gauge: Probing a Stueckelberg Scalar in the Electric Aharonov-Bohm Effect

The electric Aharonov-Bohm effect – a time-dependent scalar potential imparting a measurable phase shift on electrons in a region free of electromagnetic fields – has never been experimentally tested in its original formulation with shielded, time-dependent potentials. This unexplored regime offers a rare opportunity: the Lorenz condition ${\mathrm{\partial }}_{\mu }{A}^{\mu }=0$, a choice that eliminates a scalar degree of freedom from the electromagnetic potential, may not be the last word. If the Stueckelberg scalar $B={\mathrm{\partial }}_{\mu }{A}^{\mu }$ survives as a physical field and couples to matter, it would produce a phase shift with a distinctive $1-\cos (\omega T)$ signature – orthogonal to the standard $\sin (\omega T)$ and separable by a frequency sweep even if both contributions coexist. We propose a measurement protocol based on single-electron interferometry with picosecond time resolution, within reach of current technology. The experiment asks a question that has lingered since 1959: is the Lorenz gauge a matter of convenience, or a matter of principle?