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

TL;DR

This paper elevates the electric-magnetic duality charge to a quantum operator in $U(1)$ gauge theory, constructs ladder operators, and reveals an $SO(2,3)$ symmetry structure including additional commuting generators.

Contribution

It introduces a quantum operator for electric-magnetic duality, constructs corresponding ladder operators, and uncovers an $SO(2,3)$ symmetry in $U(1)$ gauge theory.

Findings

Quantum states are labeled by energy and duality eigenvalues.

The $SO(2,3)$ algebra is generated by bilinears of creation and annihilation operators.

Two additional symmetry generators commute with the Hamiltonian.

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, $D_{(\pm)a}^\dagger(k)$ 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.