  We study the Dirac decomposition of the gauge invariant quark propagator,
whose imaginary part describes the hadronization of a quark as this interacts
with the vacuum, and relate each of its coefficients to a specific sum rule for
the chiral-odd and chiral-even quark spectral functions. Working at first in
light-like axial gauge, we obtain a new sum rule for the spectral function
associated to the gauge fixing vector, and show that its second moment is in
fact equal to zero. Then, we demonstrate that the first moment of the
chiral-odd quark spectral function is equal in any gauge to the so-called
inclusive jet mass, which is related to the mass of the particles produced in
the hadronization of a quark. Finally, we present a gauge-dependent formula
that connects the second moment of the chiral-even quark spectral function to
invariant mass generation and final state rescattering in the hadronization of
a quark.
