  We promote the Noether charge of the electric-magnetic duality symmetry of
$U(1)$ gauge theory, "$G$" to a quantum operator. We construct ladder
operators, $D_{(\pm)a}^\dagger(k)$ and $D_{(\pm)a}(k)$ which create and
annihilate the simultaneous quantum eigen states of the quantum Hamiltonian(or
number) and the electric-magnetic duality operators respectively. Therefore all
the quantum states of the $U(1)$ gauge fields can be expressed by a form of
$|E,g\rangle$, where $E$ is the energy of the state, the $g$ is the eigen value
of the quantum operator $G$, where the $g$ is quantized in the unit of 1. We
also show that 10 independent bilinears comprised of the creation and
annihilation operators can form $SO(2,3)$ which is as demonstrated in the
Dirac's paper published in 1962. The number operator and the electric-magnetic
duality operator are the members of the $SO(2,3)$ generators. We note that
there are two more generators which commute with the number operator(or
Hamiltonian). We prove that these generators are indeed symmetries of the
$U(1)$ gauge field theory action.
