  In QCD both the quark and ghost propagators are important for governing the
non-perturbative dynamics of the theory. It turns out that the dynamical
properties of the quark and ghost fields impose non-perturbative constraints on
the analytic structure of these propagators. In this work we explicitly derive
these constraints. In doing so we establish that the corresponding spectral
densities include components which are multiples of discrete mass terms, and
that the propagators are permitted to contain singular contributions involving
derivatives of $\delta(p)$, both of which are particularly relevant in the
context of confinement.
