  Both lattice simulations and theoretical studies show that the spectral
function of the gluon propagator of QCD (in various gauges, as well as for the
gauge-invariant Pinch Technique, or PT, propagator) is not non-negative
everywhere, although it should be if it has a physical interpretation as in
QED. Theory says moreover that the non-positive spectral function of the
Landau-gauge or of the PT gluon propagator is further constrained to obey a
superconvergence relation (the integral of the spectral function vanishes). We
review the theoretical and lattice evidence for violation of positivity as well
as various interpretations of this violation, and consider methods for checking
superconvergence on the lattice (so far undone). The most common interpretation
is that positivity violation implies confinement of gluons, so the gluon
propagator does not describe processes with physical gluons. Another more
direct and gauge-invariant interpretation arises from the PT: Asymptotic
freedom alone demands non-positivity and superconvergence.
