On the Divergence of Perturbation Theory. Steps Towards a Convergent Series

TL;DR

This paper investigates why perturbation theory diverges in quantum field theory by analyzing the violation of a key mathematical theorem, and proposes modifications and resummation techniques to achieve convergence.

Contribution

It exposes the mechanism behind divergence of perturbation series and develops tools and methods to modify and resum these series for convergence.

Findings

Identifies the violation of the Dominated Convergence Theorem as a cause of divergence.

Proposes modifications to the perturbative series for convergence.

Analyzes resummation methods in the context of the divergence mechanism.

Abstract

The mechanism underlying the divergence of perturbation theory is exposed. This is done through a detailed study of the violation of the hypothesis of the Dominated Convergence Theorem of Lebesgue using familiar techniques of Quantum Field Theory. That theorem governs the validity (or lack of it) of the formal manipulations done to generate the perturbative series in the functional integral formalism. The aspects of the perturbative series that need to be modified to obtain a convergent series are presented. Useful tools for a practical implementation of these modifications are developed. Some resummation methods are analyzed in the light of the above mentioned mechanism.