  Following Feynman's treatment of the non-relativistic polaron problem,
similar techniques are used to study relativistic field theories: after
integrating out the bosonic degrees of freedom the resulting effective action
is formulated in terms of particle trajectories (worldlines) instead of field
operators. The Green functions of the theory are then approximated
variationally on the pole of the external particles by using a retarded
quadratic trial action. Application to a scalar theory gives non-perturbative,
covariant results for vertex functions and scattering processes. Recent
progress in dealing with the spin degrees of freedom in fermionic systems, in
particular Quantum Electrodynamics, is discussed. We evaluate the averages
needed in the Feynman variational principle for a general quadratic trial
action and study the structure of the dressed fermion propagator.
