We explore the consistency of the Quantum Hadro-Dynamics (QHD), namely the Serot and Walecka model, paying attention to its renormalizabilty, the stability of the vacuum and the possible occurrence of a σ-ω condensate. We find that the QHD is not renormalizable as it is usually described, but the true way of formulate it, via the Stueckelberg Lagrangian, alters the formulation of the model in the vacuum but leaves the results in the medium unchanged. Further, we find that the stability of the vacuum imposes a constrain over the parameters of the σ self-interaction while stability against σ condensation at the mean field level imposes a constraint on the coupling constant, namely gσ<8.828 that is kF = 207.2 MeV/c, significantly below the standard value of QHD.