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Divergence of Perturbation Theory and Resummation

By Ulrich Jentschura, Gerhard Soff and Technische Universität Dresden


Current determinations of fundamental constants [1] and the comparison of theory and expriment in high-precision experiments are based on perturbative expansions which can at best be regarded as divergent, asymptotic series in the coupling constant [2]. The first terms of the series decrease in absolute magnitude, before the factorial growth of the perturbative coefficients overcompensates the additional coupling factors of higher orders in perturbation theory, and the perturbation series ultimately diverges. Dyson’s related argument [3] has given rise to much discussion and confusion, until recent explicit 30– loop calculations of perturbation series pertaining to φ3 and Yukawa theories have firmly established the factorially divergent character of the perturbative expansion [4]. These considerations naturally lead to the question of how complete, nonperturbative results can be obtained from a finite number of perturbative coefficients, and how the nonperturbative result is related to the partial sums of the perturbation series. We have investigated this problem [5] in connection to the Euler-Heisenberg-Schwinger effective Lagrangian which describes the quantum electrodynamic corrections to Maxwell’s equations. The forward scattering amplitude of the vacuum ground state is described by a factorially divergent asymptotic series, SB ∝ const. × gB The expansion coefficients cn = (−1)n+1 4 n |B2n+4

Year: 2011
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