Explicit actions for electromagnetism with two gauge fields with only one electric and one magnetic physical field

Abstract

We extend the work of Mello et al. based on 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 that have only one magnetic and one electric physical field 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.

U e (1) × U g (1) actions in (2+1) dimensions: Full vectorial electric and magnetic fields

“…In addition it has recently been showed that, independently of boundary effects, a consistent definition of canonical structure is still possible [27]. To finalize let us stress that, independently of the above discussion, the canonical momenta π i A and π i C as defined in (13) are directly derived from the 3 + 1-dimensional theory, thus being consistent with the higher dimensional canonical momenta as derived in [19]. This is enough to justify the above choice.…”

“…We note that the relations between the canonical conjugate momenta to Ai and Ci (13) and the electric and magnetic fields (12) hold a new correction due to the Chern-Simons boundary contribution in relation to the 3 + 1-dimensional relations obtained in [19]. Here we are referring to the last terms depending on k in the definitions of π i A and π i C in (13)).…”

“…From the definitions of electric and magnetic field in the 3 + 1-dimensional system [19,17] we obtain the physical electric and magnetic fields definitions in the planar system:…”

“…Moreover a more fundamental description of any theory underlines a better understanding of the theory, therefore more reliable prediction and control of physical systems. We consider as starting point the 3 + 1-dimensional action for Ue(1) × Ug(1) electromagnetism introduced in [18,19] given by S4 = R dx 4 L4 and Lagrangian density…”

“…Also we are considering here a dimensional reduced theory which is theoretically consistent with the four dimensional Maxwell equations. We note that originally Ue(1) × Ug(1) has been justified by the inclusion of magnetic monopoles [16,18,19,17] maintaining the field configurations regular, i.e. free of extended singularities as the Dirac String and Wu-Yang fiber-bundle [20].…”