Certain gauge transformations may act non-trivially on physical states in quantum electrodynamics (QED). This observation has sparked the yet unresolved question of how to characterize allowed boundary conditions for gauge theories. Faddeev and Jackiw proposed to impose Gauss law on the action to find the Hamiltonian reduced theory of QED. The reduction eliminates the scalar gauge mode, renders the theory manifestly gauge invariant and the symplectic form non-singular. In this work we show that while the predictions of the reduced theory coincide with those of conventional QED for scattering events, it is experimentally distinguishable. Quantum interference of charges traveling along time-like Wilson loops that encircle (but remain clear of) electric fields is sensitive to a relative phase shift due to an interaction with the scalar potential. This is the archetypal electric Aharonov-Bohm effect and does not exist in the reduced theory. Despite its prediction over six decades ago, and in contrast to its well known magnetic counterpart, this electric Aharonov-Bohm phenomenon has never been observed. We present a conclusive experimental test using superconducting quantum interferometry. The Hamiltonian reduction renders a theta term non-topological. We comment on consequences for semi-classical gravity, where it may alleviate a problem with the measure.
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