Implications of Analyticity to the Solution of Schwinger-Dyson Equations in Minkowski Space

We discuss some recent developments in nonperturbative studies of quantum field theory (QFT) using the Schwinger-Dyson equations formulated directly in Minkowski space. We begin with the introduction of essential ideas of the integral representation in QFT and a discussion of the renormalization in this approach. The technique based on the integral representation of Green’s functions is exploited to solve Schwinger-Dyson equations in several models of quantum field theory, e.g., in scalar models and in strong coupling QED₃₊₁ in the quenched and in the unquenched approximation. The phenomenon of dynamical chiral symmetry breaking in regularized theory is touched. In QCD, the analyticity of the gluon propagator on the complex momentum square plane is exploited to continue some recent lattice data to the timelike momentum axis. We find a contribution to the non-positive absorptive part in the Landau-gauge gluon propagator which is in agreement with some other new recent analyses.