  We present a reformulation of the $U(1)$ gauge theory by eliminating the
redundancy inherent in the conventional approach. Our reformulation is
constructed on the basis of local field interaction approach to
electrodynamics. The gauge symmetry in our framework is associated with a
physical transformation, which represents the invariance of the equation of
motion of a charged scalar field under the change in the distribution of
electromagnetic field at a distance. We demonstrate that all physical
properties of the $U(1)$ gauge theory are preserved with the removal of
redundancy in the gauge field. In addition, our reformulation provides a
generalization of the Byers-Yang theorem to open systems.
