Operator gauge symmetries in QED

Abstract:
Symmetry considerations under some specific transformations are fundamental properties of any dynamical system. These symmetries, ordinarily, lead to conserved quantities which can principally be measured. For a system of electric charges an important quantity that is always required to be conserved is the total charge. The symmetry operation under which the charge is conserved is known as gauge transformation. Here, the Kobe approach to QED is extended to obtain an operator gauge invariant form of the Maxwell equations and the corresponding law of the charge conservation. The notions of the commutative and the non-commutative gauges are introduced. The conservation of the real and virtual charges, as a consequence of this notion, is  obtained separately. This technique seems to be applicable to nonlinear quantum optics and quantizing the Electromagnetic field in a dielectric meduim.