On the Solution of the Renormalization Group Equation of Quantum Electrodynamics

The renormalization group equations of QED are investigated. They are solved without recourse to the Lie equation of the renormalization group. Various integral representations and series expansions for the invariant charge function are derived. The Lie equation, however, will not be the sole tactics to get the solution of the RGE. The purpose of the present paper is to discuss how to solve the RGE without reckoning upon the Lie eq nation. We show that the RGE for the invariant charge of QED can be solved by introducing an arbitrary function and its inverse function. The dependence of the solution on the introduced arbitrary function seems to be more transparent than those described in standard textbooks.sJ.•J The RGE's for the electron propagator and the vertex function are also solved without recourse to the Lie equation. Whole the procedure is simple enough. In § 2, we solve the RGE for the invariant charge function. The solution obtained can be expressed as a contour integral in the complex coupling constant plane, which we discuss in § 3. We give arguments, in § 4, on the coefficients of the perturbative expansion of the invariant charge function. Sum rules for these coefficients are derived. In § 5 we solve the RGE's for the electron propagator and the vertex function. Section 6 is devoted to discussions. An appendix is attached to show the details of some calculations.