The Quantum Geometry of Spin and Statistics

TL;DR

This paper explores the deep connection between spin and statistics in quantum systems through quantum group symmetries, classifies anyonic possibilities, and develops a generalized path integral framework for various particle types.

Contribution

It unifies spin and statistics symmetries, extends the concept to anyons, and introduces a generalized path integral formulation that handles braid statistics without Grassmann variables.

Findings

Unified spin-statistics symmetry framework

Classification of anyonic spin and statistics

Path integral formulation for braid statistics

Abstract

Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin-statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose-Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach.