Explicit Actions for Electromagnetism with Two Gauge Fields with Only one Electric and one Magnetic Physical Fields

TL;DR

This paper systematically derives actions for electromagnetism with two gauge fields that have one electric and one magnetic physical field, analyzing their symmetry properties and duality invariance.

Contribution

It provides a new, independent derivation of allowed actions with specific physical fields, clarifies their symmetry properties, and discusses their duality and canonical structure.

Findings

Lagrangian and Hamiltonian are not invariant under duality rotations.

Electromagnetic duality rotations violate P and T symmetries, inducing Hopf and theta terms.

The magnetic gauge sector acts as a ghost sector.

Abstract

We extend the work of Mello et al. based in Cabbibo and Ferrari concerning the description of electromagnetism with two gauge fields from a variational principle, i.e. an action. We provide a systematic independent derivation of the allowed actions which have only one magnetic and one electric physical fields and are invariant under the discrete symmetries $P$ and $T$. We conclude that neither the Lagrangian, nor the Hamiltonian, are invariant under the electromagnetic duality rotations. This agrees with the weak-strong coupling mixing characteristic of the duality due to the Dirac quantization condition providing a natural way to differentiate dual theories related by the duality rotations (the energy is not invariant). Also the standard electromagnetic duality rotations considered in this work violate both $P$ and $T$ by inducing Hopf terms (theta terms) for each sector and a mixed Maxwell term. The canonical structure of the theory is briefly addressed and the 'magnetic' gauge sector is interpreted as a ghost sector.