U_e(1) × U_g(1) actions in (2+1) dimensions: Full vectorial electric and magnetic fields

It is considered a dimensional reduction of U_e(1)×U_g(1) (3+1)-dimensional electromagnetism with a gauge field (photon) and a pseudo-vector gauge field (pseudo-photon) to (2+1) dimensions. In the absence of boundary effects, the quantum structure is maintained, while when boundary effects are considered, as have been previously studied, a cross Chern-Simons term between both gauge fields is present, which accounts for topological effects and changes the quantum structure of the theory. Our construction maintains the dimensional reduced action invariant under parity (P) and time-inversion (T). We show that the theory has two massive degrees of freedom, corresponding to the longitudinal modes of the photon and of the pseudo-photon and briefly discuss the quantization procedures of the theory in the topological limit (wave functional quantization) and perturbative limit (an effective dynamical current theory), pointing out directions to solve the constraints and deal with the negative energy contributions from pseudo-photons. We recall that the physical interpretation of the fields in the planar system is new and is only meaningful in the context of U_e(1)×U_g(1) electromagnetism. In this work it is shown that all the six electromagnetic vectorial fields components are present in the dimensional reduced theory and that, independently of the embedding of the planar system, can be described in terms of the two gauge fields only. As far as the author is aware it is the first time that such a construction is fully justified, thus allowing a full vectorial treatment at the variational level of electromagnetism in planar systems.