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


  • Wolfgang Bietenholz Instituto de Ciencias Nucleares, UNAM



Ramanujan summation, Casimir effect, renormalization, zeta-function


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 discuss its interpretation, which remains controversial.
  This is a simple way to illustrate the concept of renormalization,
  which is vital in quantum field theory.


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