We study Faraday rotation in the quantum relativistic limit. Starting from the photon self-energy in the presence of a constant magnetic field the rotation of the polarization vector of a plane electromagnetic wave which travel along the fermion-anti
fermion gas is studied. The connection between Faraday Effect and Quantum Hall Effect (QHE) is discussed. The Faraday Effect is also investigated for a massless relativistic (2D+1)-dimensional fermion system which is derived by using the compactification along the dimension parallel to the magnetic field. The Faraday angle shows a quantized behavior as Hall conductivity in two and three dimensions.
The problem of anisotropic pressures arising as a consequence of the spatial symmetry breaking introduced by an external magnetic field in quantum systems is discussed. The role of the conservation of energy and momentum of external fields as well as
of systems providing boundary conditions in quantum statistics is considered. The vanishing of the average transverse momentum for an electron-positron system in its Landau ground state is shown, which means the vanishing of its transverse pressure. The situation for neutron case and Strange Quark Matter (SQM) in $beta$-equilibrium is also briefly discussed. Thermodynamical relations in external fields as well as the form of the stress tensor in a quantum relativistic medium are also discussed. The ferromagnetic symmetry breaking is briefly discussed.
