Summation of divergent series: Order-dependent mapping

TL;DR

This paper reviews the Order-Dependent Mapping (ODM) method for summing divergent series in quantum field theory, explaining its basis, convergence, and applications, and comparing it with other summation techniques.

Contribution

It introduces and analyzes the ODM method, providing intuitive explanations, convergence proofs, and examples, highlighting its advantages over traditional summation methods.

Findings

ODM converges for a class of divergent series.

The method has been rigorously proven to converge.

ODM has practical applications in quantum field theory.

Abstract

Summation methods play a very important role in quantum field theory because all perturbation series are divergent and the expansion parameter is not always small. A number of methods have been tried in this context, most notably Pade approximants, Borel--Pade summation, Borel transformation with mapping, which we briefly describe and one on which we concentrate here, Order-Dependent Mapping (ODM). We recall the basis of the method, for a class of series we give intuitive arguments to explain its convergence and illustrate its properties by several simple examples. Since the method was proposed, some rigorous convergence proofs were given. The method has also found a number of applications and we shall list a few.