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We extend the work of Mello et al. based in Cabbibo and Ferrari concerningthe description of electromagnetism with two gauge fields from a variationalprinciple, i.e. an action. We provide a systematic independent derivation ofthe allowed actions which have only one magnetic and one electric physicalfields and are invariant under the discrete symmetries $P$ and $T$. We concludethat neither the Lagrangian, nor the Hamiltonian, are invariant under theelectromagnetic duality rotations. This agrees with the weak-strong couplingmixing characteristic of the duality due to the Dirac quantization conditionproviding a natural way to differentiate dual theories related by the dualityrotations (the energy is not invariant). Also the standard electromagneticduality rotations considered in this work violate both $P$ and $T$ by inducingHopf terms (theta terms) for each sector and a mixed Maxwell term. Thecanonical structure of the theory is briefly addressed and the 'magnetic' gaugesector 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

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