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We decompose the quark propagator in the presence of an arbitrary gluon fieldwith respect to a set of Dirac matrices. The four-dimensional integrals whicharise in first order perturbation theory are rewritten as line-integrals alongcertain field lines, together with a weighted integration over the variousfield lines. It is then easy to transform the propagator into a form involvingpath ordered exponentials. The resulting expression is non-perturbative and hasthe correct behavior under Lorentz transformations, gauge transformations andcharge conjugation. Furthermore it coincides with the exact propagator in firstorder of the coupling g. No expansion with respect to the inverse quark mass isinvolved, the expression can even be used for vanishing mass. For large massthe field lines concentrate near the straight line connection and simpleresults can be obtained immediately.Comment: 15 pages, 3 figures. Replaced with version to appear in Eur. Phys. J.

A Gauge and Lorentz covariant approximation for the quark propagator in an arbitrary gluon field