  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.
