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.Comment: v2: 12 pages; References added, discussion concerning degrees of
freedom corrected; v3: is now used the standard normalization of 1/4 in the
actions; the possibility of theta being a pseudo-scalar implied a title
changing; eq (23) added; signs corrected in equations (39,45-47); references
adde