Identity and difference: how topology helps to understand quantum indiscernability

TL;DR

This paper explores how topology underpins the understanding of quantum indiscernibility, explaining the natural emergence of particle statistics like bosons, fermions, and anyons through topological concepts and path integral interpretations.

Contribution

It clarifies the role of topology in quantum statistics, connecting Feynman's path integrals with the topological nature of particle exchange and intermediate statistics.

Findings

Path integrals relate quantum processes to classical paths.

Topology explains the natural dichotomy of bosons and fermions.

Experimental evidence supports the existence of anyons in two dimensions.

Abstract

This contribution, to be published in Imagine Math 8 to celebrate Michele Emmer's 75th birthday, can be seen as the second part of my previous considerations on the relationships between topology and physics (Mouchet, 2018). Nevertheless, the present work can be read independently. The following mainly focusses on the connection between topology and quantum statistics. I will try to explain to the non specialist how Feynman's interpretation of quantum processes through interference of classical paths (path integrals formulation), makes the dichotomy between bosons and fermions quite natural in three spatial dimensions. In (effective) two dimensions, the recent experimental evidence of intermediate statistics (anyons) (Bartolomei et al. 2020) comfort that topology (of the braids) provides a fertile soil for our understanding of quantum particles.