From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results

TL;DR

This paper explores how Ramanujan's methods for assigning finite values to divergent series relate to modern renormalization techniques in physics, illustrating their application in quantum field theory and the Casimir effect.

Contribution

It connects Ramanujan's analytic continuation of divergent series to contemporary renormalization methods used in quantum physics.

Findings

Ramanujan's divergent series values match zeta-function continuations.

Application of these values to vacuum energy and Casimir force.

Discussion of the interpretation and controversy in physics.

Abstract

A century ago, Srinivasa Ramanujan -- the great self-taught Indian genius of mathematics -- died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n \geq 1} n$ and $\sum_{n \geq 1} n^{3}$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $\zeta$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We further discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory.