  The analytic structure of the quark propagator in Minkowski space is more
complex than in Euclidean space due to the possible existence of poles and
branch cuts at timelike momenta. These singularities impose enormous
complications on the numerical treatment of the nonperturbative Dyson-Schwinger
equation for the quark propagator. Here we discuss a computational method that
avoids most of these complications. The method makes use of the spectral
representation of the propagator and of its inverse. The use of spectral
functions allows one to handle in exact manner poles and branch cuts in
momentum integrals. We obtain model-independent integral equations for the
spectral functions and perform their renormalization by employing a
momentum-subtraction scheme. We discuss an algorithm for solving numerically
the integral equations and present explicit calculations in a schematic model
for the quark-gluon scattering kernel.
