  The structure of the quark propagator of $QCD$ in a confining background is
not known. We make an Ansatz for it, as hinted by a particular mechanism for
confinement, and analyze its implications in the meson and baryon correlators.
We connect the various terms in the K\"allen-Lehmann representation of the
quark propagator with appropriate combinations of hadron correlators, which may
ultimately be calculated in lattice $QCD$. Furthermore, using the positivity of
the path integral measure for vector like theories, we reanalyze some mass
inequalities in our formalism. A curiosity of the analysis is that, the exotic
components of the propagator (axial and tensor), produce terms in the hadron
correlators which, if not vanishing in the gauge field integration, lead to
violations of fundamental symmetries. The non observation of these violations
implies restrictions in the space-time structure of the contributing gauge
field configurations. In this way, lattice $QCD$ can help us analyze the
microscopic structure of the mechanisms for confinement.
