2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/54817A path integration formulation for the finite density and temperature problems is shown to be consistent with the thermodynamics using an 8 component ``real'' representation for the fermion fields by applying it to a free fermion system. A relativistic quantum field theory is shown to be smoothly approached at zero temperature by a real-time thermal field theory so derived even at a finite density. The analysis leads to a new representation for the fermion fields which is shown to be inequivalent to the conventional 4 component theory at the quantum level by having a mirror universe with observable effects and to be better behaved at short distances.7 pages, RevTex, 1 figureHigh Energy Physics - TheoryHigh Energy Physics - PhenomenologyNuclear Theorytext