Quantum Field Theory as a Problem of Resummation (Short guide to using summability methods)

TL;DR

This paper reviews advanced summability methods in quantum field theory perturbation series, generalizing Borel summability to broader regions, and demonstrates their applications in analyzing the analytic structure and singularities of quantum models.

Contribution

It introduces new summability techniques that extend Borel summability to horn-shaped regions, enabling better analysis of perturbation series in quantum field theory.

Findings

Summability methods converge in the Mittag-Leffeler star of an analytic function.

Position of singularities can be analytically calculated using large order behavior.

Methods are implementable numerically for practical applications.

Abstract

Thesis includes review on the large order behaviour of perturbation theory in quantum mechanical and field theory models; generalization of the Borel summability and strong asymptotic conditions to various (including horn-shaped) regions; discussion of analytic aspects of perturbation theory; examples which demonstrate differences between the Borel summability and generalized one; application to the Rayleigh-Schr\"{o}dinger perturbation theory and to the definition of the operator valued functions. The new summability methods converges in the whole Mittag-Leffeler star of an analytical function and as such is useful for localization of singularities in the complex plane. Their position can be calculated even analytically provided large order behaviour of the Taylor series is known. Method can be implemented numerically as well.