2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/41280An application of a self-consistent version of RPA to quantum field theory with broken symmetry is presented. Although our approach can be applied to any bosonic field theory, we specifically study the $ϕ^4$ theory in 1+1 dimensions. We show that standard RPA approach leads to an instability which can be removed when going to a superior version,i.e. the renormalized RPA. We present a method based on the so-called charging formula of the many electron problem to calculate the correlation energy and the RPA effective potential.30 pages, LaTeX file, 10 figures included, final version accepted in EPJAHigh Energy Physics - PhenomenologyNuclear Theorytext