Welcome to the Hilbert hotel: a small tour between vortex beam singularities, topology, particle physics and Ramanujan numerical series

Sir Michael Berry and John F. Nye co-authored the foundational 1974 paper "Dislocations in wave trains", which discovered phase singularities as stable, ubiquitous structures in wave physics—launching the field of singular optics.
They appear in fluid dynamics, communication, superconductivity and are central, with large news coverage, in the AI solution of Navier Stokes equation but most of the theory found application in structured light.
The singularities are the backbones of wave propagation but can be also studied as entities and their dynamic can be studied as if they were quasi particles. Their quantisation appears even in a classical wave equation and has a topological origin.
The similarity with the world of quantum particle is somehow appalling yet there are also differences.
In this talk I try to connect to a surprising list of different fields from number theory to particle physics.
For example, we can observe analogies with Schwinger effect and pair creation.
We can calculate an effective energy and force between singularity or the effect of external condition.
We can count them and align a potentially infinite number of them and study the Hilbert Hotel paradox related to fractional angular momentum. The latter has interesting ramification in connection to an experimental realisation of Ramanujan or better Grandi’s series.
All these things have received attention in singular optics in the past, but electron wave experiments add some fundamentally new insight in this scheme especially because the vortex condition can be varied with continuity.
It is a fascinating tour well beyond the electron microscopy and there is plenty of place at the Hilbert Hotel…

 Seminar realized in the framework of the funded projects
-  MAECI - CRYO-MEMS criomicroscopia di nuova generazione: piatti di fase innovativi a MEMS in ambiente TEM acromatico