Electric-magnetic duality as a quantum operator and more symmetries of $U(1)$ gauge theory

Abstract

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, $D_{(\pm )a}^{†}(k)$ 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 $|E,g⟩$, 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.

12 pages, 1 figure and 1 table

References in corpus (3)

• Local and nonlocal observables in quantum optics
• Boundary conditions for SU(2) Yang-Mills on $Ad{S}_4$
• Electric-magnetic duality implies (global) conformal invariance