Borchers' Commutation Relations for Sectors with Braid Group Statistics in Low Dimensions

TL;DR

This paper extends Borchers' commutation relations to sectors with braid group statistics in low-dimensional quantum field theories, advancing the understanding of modular objects and Poincare symmetry in such contexts.

Contribution

It proves Borchers' theorem for charged sectors with braid group statistics in low dimensions, crucial for the Bisognano-Wichmann theorem for Plektons.

Findings

Borchers' theorem holds in sectors with braid group statistics in low dimensions.

Modular objects generate a representation of the Poincare group including CPT.

Supports the Bisognano-Wichmann theorem for Plektons in d=3.

Abstract

Borchers has shown that in a translation covariant vacuum representation of a theory of local observables with positive energy the following holds: The (Tomita) modular objects associated with the observable algebra of a fixed wedge region give rise to a representation of the subgroup of the Poincare group generated by the boosts and the reflection associated to the wedge, and the translations. We prove here that Borchers' theorem also holds in charged sectors with (possibly non-Abelian) braid group statistics in low space-time dimensions. Our result is a crucial step towards the Bisognano-Wichmann theorem for Plektons in d=3, namely that the mentioned modular objects generate a representation of the proper Poincare group, including a CPT operator. Our main assumptions are Haag duality of the observable algebra, and translation covariance with positive energy as well as finite statistics of the sector under consideration.