Large-Order Behaviour of Perturbation Theory

Abstract

In quantum field theory (QFT), the main analytic tool to calculate physical quantities is the perturbative expansion. Following, Dyson's intuitive argument, the divergence of perturbative series was demonstrated in some models of quantum mechanics (QM) with polynomial potentials, using the Schrödinger equation. Later, it was proposed to study the problem within a path integral formulation. A systematic method in field theory was proposed by Lipatov, using the field integral representation of the φ⁴ ₄ field theory and instantons. It can be shown that the ground-state energy of the quartic anharmonic oscillator is analytic in a cut-plane. The imaginary part of the energy on the cut is related to barrier penetration. The behaviour of the perturbative coefficients at large orders is related to the behaviour of the imaginary part for small and negative coupling and can be obtained by instanton methods. The method has been generalized to the class of potentials for which (in general complex) instanton contributions have been calculated. The same method can be readily applied to boson field theories, while the extension to field theories involving fermions, like Quantum QED, requires additional considerations. The general conclusion is that, in QFT, all perturbative series, expanded in terms of a loop-expansion parameter, are divergent series.