Rearranging Dyson-Schwinger Equations

Dyson-Schwinger equations are integral equations in quantum field theory that describe the Green functions of a theory and mirror the recursive decomposition of Feynman diagrams into subdiagrams Taken as recursive equations, the Dyson-Schwinger equations describe perturbative quantum field theory. However, they also contain non-perturbative information. Using the Hopf algebra of Feynman graphs we will follow a sequence of reductions to convert the Dyson-Schwinger equations to the following system of differential equations, gamma(r)(1)(x) = P-r(x) - sign(s(r))gamma(r)(1)(x)(2) + (Sigma(j is an element of R) vertical bar s(j)vertical bar gamma(r)(1)(x)) x partial derivative(x)gamma(r)(1)(x) where r is an element of R, R is the set of amplitudes of the theory which need renormalization, gamma(1)(r) is the anomalous dimension associated to r, P-r(x) is a modified version of the function for the primitive skeletons contributing to r, and x is the coupling constant. Next, we approach the new system of differential equations as a system of recursive equations by expanding gamma(1)(r)(x) = Sigma(n >= 1) gamma(r)(1,n)x(n). We obtain the radius of convergence of Sigma gamma(r)(1,n)x(n)/n! in terms of that of Sigma P-r(n)x(n)/n!. In particular we show that a Lipatov bound for the growth of the primitives leads to a Lipatov bound for the whole theory. Finally, we make a few observations on the new system considered as differential equations. In particular in the case of quantum electrodynamics we find a distinguished physical solution and find the possibility of avoiding a Landau pole.