Summing up the perturbation series in the Schwinger Model

Perturbation series for the electron propagator in the Schwinger Model is summed up in a direct way by adding contributions coming from individual Feynman diagrams. The calculation shows the complete agreement between nonperturbative and perturbative approaches. 11.10.-z;11.15.Bt;12.90.+b Typeset using REVTEX Electronic mail: [email protected] 1 A long-standing question in quantum field theory is the connection between perturbation series and exact, nonperturbative results. It dates back to the Dyson’s paper [1] in which the author, considering stability of a system conditions, suggested that physical quantities and Green’s functions should be nonanalytic in the coupling constant g around g = 0. This in turn should result in the divergence, usually factorial type, of the perturbation series. This conjecture was supported by simple models [2] among which the most widely considered was the anharmonic oscillator and its field-theoretical counterpart — the φ theory [3–11] as well as by other, more realistic, field theories as QED for instance [12,13] (see [14] for further references). In these cases the required estimations for the nonperturbative results were often obtained with the use of the generalized (Padé, Borel) summation methods (for a review of this approach see [14–16]). There have also been found counterexamples, regarding the Dyson’s observation, in which the perturbation series is not divergent in spite of instability (although it may be convergent to an incorrect result) [17–19]. In QED the nonanalyticity in the coupling constant often manifests itself through the presence of a logarithmic function of the fine structure constant α in the calculated quantities [20–22] and in consequence means the divergence of coefficients in the Taylor expansion in α (in other words divergence of Feynman diagrams) resulting in necessity of infinite renormalisation. One can say that this means the incorrectness of the perturbation expansion [23–30]. Although the summability of the perturbation series still remains an opened question, perturbation theory constitutes, however, the main tool in practical calculations giving, especially in Quantum Electrodynamics, excellent results. It seems, therefore, valuable to sum up directly the perturbation series, by adding contributions of the individual Feynman graphs, in a model theory in which the nonperturbative result is well known. In this work we will concentrate on the 1+1 dimensional massless QED known as the Schwinger Model [31]. Up to our knowledge no such direct summation has, in this model, been performed. The focus will be put on the electron propagator for which the explicit nonperturbative formula in coordinate space was found [31] (up to the final p-integration) S(x) = S0(x) exp [