A gauge invariant formulation of quantum electrodynamics using local currents

In this paper we formulate nonrelativistic quantum electrodynamics in a local and manifestly gauge invariant manner. This is accomplished by using the electromagnetic field strengths, rather than potentials, to describe the electromagnetic field and local currents, rather than canonical fields, to describe the matter. The exponentiated currents and field strengths form a group, whose representations can be studied using the Gel’fand–Vilenkin formalism. The currents and electromagnetic field strengths can be represented on a physical Hilbert space having positive norm. (The necessity for an indefinite metric does not arise here.) Furthermore, the classical equations of motion hold as operator equations on this Hilbert space. In this formulation, the requirement of gauge invariance is essentially replaced by imposing the Maxwell initial value equtions, which in turn lead to constraints on systems of Gel’fand–Vilenkin multipliers.