  We decompose the quark propagator in the presence of an arbitrary gluon field
with respect to a set of Dirac matrices. The four-dimensional integrals which
arise in first order perturbation theory are rewritten as line-integrals along
certain field lines, together with a weighted integration over the various
field lines. It is then easy to transform the propagator into a form involving
path ordered exponentials. The resulting expression is non-perturbative and has
the correct behavior under Lorentz transformations, gauge transformations and
charge conjugation. Furthermore it coincides with the exact propagator in first
order of the coupling g. No expansion with respect to the inverse quark mass is
involved, the expression can even be used for vanishing mass. For large mass
the field lines concentrate near the straight line connection and simple
results can be obtained immediately.
