The problem of anisotropic pressures arising from 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, i.e. the vanishing of its transverse pressure, is shown. The situation for the neutron case and strange quark matter (SQM) in beta equilibrium is briefly discussed. Thermodynamical relations in external fields as well as the form of the stress tensor in a quantum relativistic medium are obtained. The ferromagnetic symmetry breaking is briefly discussed for very dense matter. It is concluded that stable matter cannot exist for fields greater than B = 10(18) G.
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