Ramanujan summation and the Casimir effect

TL;DR

This paper explores Ramanujan summation's mathematical foundation and applies it to the Casimir effect, linking divergent series regularization to physical phenomena and discussing implications for Dark Energy.

Contribution

It provides a direct justification for Ramanujan summation using analytic continuation and connects this to the physical interpretation of the Casimir effect and vacuum energy.

Findings

Ramanujan summation corresponds to zeta function regularization.

The Casimir force prediction aligns with experimental results.

Discussion on the interpretation of the Casimir effect and Dark Energy.

Abstract

Srinivasa Ramanujan was a great self-taught Indian mathematician, who died a century ago, at the age of only 32, one year after returning from England. Among his numerous achievements is the assignment of sensible, finite values to divergent series, which correspond to Riemann's $\zeta$-function with negative integer arguments. He hardly left any explanation about it, but following the few hints that he gave, we construct a direct justification for the best known example, based on analytic continuation. As a physical application of Ramanujan summation we discuss the Casimir effect, where this way of removing a divergent term corresponds to the renormalization of the vacuum energy density, in particular of the photon field. This leads to the prediction of the Casimir force between conducting plates, which has now been accurately confirmed by experiments. Finally we review the discussion about the meaning and interpretation of the Casimir effect. This takes us to the mystery surrounding the magnitude of Dark Energy.