On nonperturbative calculations in quantum electrodynamics

A new approach to nonperturbative calculations in quantum electrodynamics is proposed. This approach is based on a regular iteration scheme for the solution of Schwinger-Dyson equations for generating the functional of Green functions. The approach allows one to take into account the gauge invariance conditions (Ward identities) and to perform the renormalization programme. The iteration scheme can be realized in two versions. The first one 'perturbative vacuum') corresponds to chain summation in the diagram language. In this version the exact result of a two-dimensional Schwinger model is reproduced as the first step in the calculations, but in four-dimensional theory the nonphysical singularity (Landau pole) arises which leads to the triviality of the renormalized theory. The second version ('nonperturbative vacuum') corresponds to ladder summation and permits one to perform nonperturbative calculations of physical quantities in spite of the triviality problem. For a chiral-symmetrical leading approximation two terms of the expansion of the first-step vertex function over photon momentum are calculated. The formula f2=α/(2π-α) for the anomalous magnetic moment is obtained (α is the fine structure constant). For a linearized equation of the leading approximation a problem of dynamical chiral symmetry breaking is considered and the calculations are performed for renormalized theory in Minkowski space. In the strong coupling region α≥π/3 the results correspond to earlier investigations performed in Euclidean theory with cutoff: solutions arise with the breakdown of chiral symmetry. For the renormalized theory a solution with breakdown of chiral symmetry is also possible in the weak coupling region α<π/3, but with a subsidiary condition on the value of α which follows from the gauge invariance. (author)