2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226273Extended $U_e(1)\times U_g(1)$ electromagnetism containing both a photon and a pseudo-photon is introduced at variational level and is justified by the violation of the Bianchi identities in conceptual systems, either in the presence of magnetic monopoles or non-regular external fields, not being accounted by the standard Maxwell Lagrangian. It is carried a dimensional reduction that renders a $U_e(1)\times U_g(1)$ Maxwell-BF type theory and it is considered canonical functional quantization in planar systems which may be relevant in Hall systems.5 pages; 1 figure; submitted to Proceedings of XVI International Fall Workshop on Geometry and Physics; http://www.math.ist.utl.pt/~xvi-iwgp/talks/PedroCasteloF.pdf; v2: derivatives on equation (10) correctedHigh Energy Physics - TheoryMathematical Physicstext