Generalized Particle Statistics in Two-Dimensions: Examples from the Theory of Free Massive Dirac Field

TL;DR

This paper investigates anomalous particle statistics in two-dimensional quantum field theory, revealing how topological and algebraic factors lead to generalized, non-intrinsic statistics beyond traditional sector classifications.

Contribution

It introduces a novel framework for understanding generalized particle statistics in 2D quantum fields using algebraic methods and asymptotic abelianness of automorphisms.

Findings

Automorphisms exhibit non-intrinsic statistics due to topological effects.

Construction of a braiding structure for a specific subcategory of automorphisms.

Identification of two classes of path connected bi-asymptopias for generalized statistics.

Abstract

In the framework of algebraic quantum field theory we analyze the anomalous statistics exhibited by a class of automorphisms of the observable algebra of the two-dimensional free massive Dirac field, constructed by fermionic gauge group methods. The violation of Haag duality, the topological peculiarity of a two-dimensional space-time and the fact that unitary implementers do not lie in the global field algebra account for strange behaviour of statistics, which is no longer an intrinsic property of sectors. Since automorphisms are not inner, we exploit asymptotic abelianness of intertwiners in order to construct a braiding for a suitable $C^*$-tensor subcategory of End($\mathscr{A}$). We define two inequivalent classes of path connected bi-asymptopias, selecting only those sets of nets which yield a true generalized statistics operator.